Revisions to Finite groups with integral character table

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occurred Aug 17, 2022 at 14:24

improved formatting

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edit approved Aug 17, 2022 at 14:04

673
1
4
13

gap> for o in [1..120] do n:=NrSmallGroups(o);; for i in [1..n] do G:=SmallGroup(o,i);; irr:=Irr(G);; s:=Size(irr);; c:=0;; for j in [1..s] do if Conductor(irr[j])<>1 then c:=1;; break; fi; od; if c=0 then Print([StructureDescription(G),[o,i]]); fi; od; od;
[ "1", [ 1, 1 ] ][ "C2", [ 2, 1 ] ][ "C2 x C2", [ 4, 2 ] ][ "S3", [ 6, 1 ] ][ "D8", [ 8, 3 ] ][ "Q8", [ 8, 4 ] ][ "C2 x C2 x C2", [ 8, 5 ] ][ "D12", [ 12, 4 ] ][ "C2 x D8", [ 16, 11 ] ]
[ "C2 x Q8", [ 16, 12 ] ][ "C2 x C2 x C2 x C2", [ 16, 14 ] ][ "(C3 x C3) : C2", [ 18, 4 ] ][ "S4", [ 24, 12 ] ][ "C2 x C2 x S3", [ 24, 14 ] ][ "(C2 x C2 x C2 x C2) : C2", [ 32, 27 ] ]
[ "(C4 x C4) : C2", [ 32, 34 ] ][ "C4 : Q8", [ 32, 35 ] ][ "C8 : (C2 x C2)", [ 32, 43 ] ][ "(C2 x Q8) : C2", [ 32, 44 ] ][ "C2 x C2 x D8", [ 32, 46 ] ][ "C2 x C2 x Q8", [ 32, 47 ] ]
[ "(C2 x C2 x C2) : (C2 x C2)", [ 32, 49 ] ][ "(C2 x Q8) : C2", [ 32, 50 ] ][ "C2 x C2 x C2 x C2 x C2", [ 32, 51 ] ][ "S3 x S3", [ 36, 10 ] ][ "C2 x ((C3 x C3) : C2)", [ 36, 13 ] ]
[ "D8 x S3", [ 48, 38 ] ][ "Q8 x S3", [ 48, 40 ] ][ "C2 x S4", [ 48, 48 ] ][ "C2 x C2 x C2 x S3", [ 48, 51 ] ][ "(C3 x C3 x C3) : C2", [ 54, 14 ] ][ "((C4 x C4) : C2) : C2", [ 64, 134 ] ]
[ "(C4 : Q8) : C2", [ 64, 137 ] ][ "((C2 x C2 x C2 x C2) : C2) : C2", [ 64, 138 ] ][ "((C4 x C4) : C2) : C2", [ 64, 177 ] ][ "(C4 : Q8) : C2", [ 64, 178 ] ][ "C8 : Q8", [ 64, 182 ] ]
[ "C2 x ((C2 x C2 x C2 x C2) : C2)", [ 64, 202 ] ][ "C2 x ((C4 x C4) : C2)", [ 64, 211 ] ][ "C2 x (C4 : Q8)", [ 64, 212 ] ][ "(C2 x C2 x D8) : C2", [ 64, 215 ] ][ "(C2 x ((C4 x C2) : C2)) : C2",
  [ 64, 216 ] ][ "((C4 x C4) : C2) : C2", [ 64, 217 ] ][ "(C2 x ((C4 x C2) : C2)) : C2", [ 64, 218 ] ][ "((C2 x Q8) : C2) : C2", [ 64, 224 ] ][ "(C4 : Q8) : C2", [ 64, 225 ] ]
[ "D8 x D8", [ 64, 226 ] ][ "Q8 x D8", [ 64, 230 ] ][ "Q8 x Q8", [ 64, 239 ] ][ "((C4 x C2 x C2) : C2) : C2", [ 64, 241 ] ][ "((C4 x C4) : C2) : C2", [ 64, 242 ] ][ "((C4 x C2 x C2) : C2) : C2",
  [ 64, 243 ] ][ "(C4 : Q8) : C2", [ 64, 244 ] ][ "(C2 x C2) . (C2 x C2 x C2 x C2)", [ 64, 245 ] ][ "C2 x (C8 : (C2 x C2))", [ 64, 254 ] ][ "C2 x ((C2 x Q8) : C2)", [ 64, 255 ] ]
[ "C2 x C2 x C2 x D8", [ 64, 261 ] ][ "C2 x C2 x C2 x Q8", [ 64, 262 ] ][ "C2 x ((C2 x C2 x C2) : (C2 x C2))", [ 64, 264 ] ][ "C2 x ((C2 x Q8) : C2)", [ 64, 265 ] ][ "C2 x C2 x C2 x C2 x C2 x C2",
  [ 64, 267 ] ][ "(S3 x S3) : C2", [ 72, 40 ] ][ "(C3 x C3) : Q8", [ 72, 41 ] ][ "(C3 x A4) : C2", [ 72, 43 ] ][ "C2 x S3 x S3", [ 72, 46 ] ][ "C2 x C2 x ((C3 x C3) : C2)", [ 72, 49 ] ]
[ "C2 x D8 x S3", [ 96, 209 ] ][ "C2 x Q8 x S3", [ 96, 212 ] ][ "C2 x C2 x S4", [ 96, 226 ] ][ "((C2 x C2 x C2 x C2) : C3) : C2", [ 96, 227 ] ][ "C2 x C2 x C2 x C2 x S3", [ 96, 230 ] ]
[ "((C3 x C3) : C3) : (C2 x C2)", [ 108, 17 ] ][ "((C3 x C3) : C2) x S3", [ 108, 39 ] ][ "C2 x ((C3 x C3 x C3) : C2)", [ 108, 44 ] ][ "S5", [ 120, 34 ] ]
gap> for o in [1..120] do
>       if o=1 then Print("\n","|G| ","Nr ","G ","\n","\n");fi;
>       n:=NrSmallGroups(o);;
>       for i in [1..n] do
>               G:=SmallGroup(o,i);;
>               irr:=Irr(G);;
>               s:=Size(irr);;
>               c:=0;;
>               for j in [1..s] do
>                       if Conductor(irr[j])<>1 then
>                               c:=1;;
>                               break;
>                       fi;
>               od;
>               if c=0 then
>                       Print(o,"   ",i,"   ",StructureDescription(G),"\n");
>               fi;
>       od;
> od;

|G| Nr G

1   1   1
2   1   C2
4   2   C2 x C2
6   1   S3
8   3   D8
8   4   Q8
8   5   C2 x C2 x C2
12   4   D12
16   11   C2 x D8
16   12   C2 x Q8
16   14   C2 x C2 x C2 x C2
18   4   (C3 x C3) : C2
24   12   S4
24   14   C2 x C2 x S3
32   27   (C2 x C2 x C2 x C2) : C2
32   34   (C4 x C4) : C2
32   35   C4 : Q8
32   43   C8 : (C2 x C2)
32   44   (C2 x Q8) : C2
32   46   C2 x C2 x D8
32   47   C2 x C2 x Q8
32   49   (C2 x C2 x C2) : (C2 x C2)
32   50   (C2 x Q8) : C2
32   51   C2 x C2 x C2 x C2 x C2
36   10   S3 x S3
36   13   C2 x ((C3 x C3) : C2)
48   38   D8 x S3
48   40   Q8 x S3
48   48   C2 x S4
48   51   C2 x C2 x C2 x S3
54   14   (C3 x C3 x C3) : C2
64   134   ((C4 x C4) : C2) : C2
64   137   (C4 : Q8) : C2
64   138   ((C2 x C2 x C2 x C2) : C2) : C2
64   177   ((C4 x C4) : C2) : C2
64   178   (C4 : Q8) : C2
64   182   C8 : Q8
64   202   C2 x ((C2 x C2 x C2 x C2) : C2)
64   211   C2 x ((C4 x C4) : C2)
64   212   C2 x (C4 : Q8)
64   215   (C2 x C2 x D8) : C2
64   216   (C2 x ((C4 x C2) : C2)) : C2
64   217   ((C4 x C4) : C2) : C2
64   218   (C2 x ((C4 x C2) : C2)) : C2
64   224   ((C2 x Q8) : C2) : C2
64   225   (C4 : Q8) : C2
64   226   D8 x D8
64   230   Q8 x D8
64   239   Q8 x Q8
64   241   ((C4 x C2 x C2) : C2) : C2
64   242   ((C4 x C4) : C2) : C2
64   243   ((C4 x C2 x C2) : C2) : C2
64   244   (C4 : Q8) : C2
64   245   (C2 x C2) . (C2 x C2 x C2 x C2)
64   254   C2 x (C8 : (C2 x C2))
64   255   C2 x ((C2 x Q8) : C2)
64   261   C2 x C2 x C2 x D8
64   262   C2 x C2 x C2 x Q8
64   264   C2 x ((C2 x C2 x C2) : (C2 x C2))
64   265   C2 x ((C2 x Q8) : C2)
64   267   C2 x C2 x C2 x C2 x C2 x C2
72   40   (S3 x S3) : C2
72   41   (C3 x C3) : Q8
72   43   (C3 x A4) : C2
72   46   C2 x S3 x S3
72   49   C2 x C2 x ((C3 x C3) : C2)
96   209   C2 x D8 x S3
96   212   C2 x Q8 x S3
96   226   C2 x C2 x S4
96   227   ((C2 x C2 x C2 x C2) : C3) : C2
96   230   C2 x C2 x C2 x C2 x S3
108   17   ((C3 x C3) : C3) : (C2 x C2)
108   39   ((C3 x C3) : C2) x S3
108   44   C2 x ((C3 x C3 x C3) : C2)
120   34   S5

gap> for o in [1..120] do n:=NrSmallGroups(o);; for i in [1..n] do G:=SmallGroup(o,i);; irr:=Irr(G);; s:=Size(irr);; c:=0;; for j in [1..s] do if Conductor(irr[j])<>1 then c:=1;; break; fi; od; if c=0 then Print([StructureDescription(G),[o,i]]); fi; od; od;
[ "1", [ 1, 1 ] ][ "C2", [ 2, 1 ] ][ "C2 x C2", [ 4, 2 ] ][ "S3", [ 6, 1 ] ][ "D8", [ 8, 3 ] ][ "Q8", [ 8, 4 ] ][ "C2 x C2 x C2", [ 8, 5 ] ][ "D12", [ 12, 4 ] ][ "C2 x D8", [ 16, 11 ] ]
[ "C2 x Q8", [ 16, 12 ] ][ "C2 x C2 x C2 x C2", [ 16, 14 ] ][ "(C3 x C3) : C2", [ 18, 4 ] ][ "S4", [ 24, 12 ] ][ "C2 x C2 x S3", [ 24, 14 ] ][ "(C2 x C2 x C2 x C2) : C2", [ 32, 27 ] ]
[ "(C4 x C4) : C2", [ 32, 34 ] ][ "C4 : Q8", [ 32, 35 ] ][ "C8 : (C2 x C2)", [ 32, 43 ] ][ "(C2 x Q8) : C2", [ 32, 44 ] ][ "C2 x C2 x D8", [ 32, 46 ] ][ "C2 x C2 x Q8", [ 32, 47 ] ]
[ "(C2 x C2 x C2) : (C2 x C2)", [ 32, 49 ] ][ "(C2 x Q8) : C2", [ 32, 50 ] ][ "C2 x C2 x C2 x C2 x C2", [ 32, 51 ] ][ "S3 x S3", [ 36, 10 ] ][ "C2 x ((C3 x C3) : C2)", [ 36, 13 ] ]
[ "D8 x S3", [ 48, 38 ] ][ "Q8 x S3", [ 48, 40 ] ][ "C2 x S4", [ 48, 48 ] ][ "C2 x C2 x C2 x S3", [ 48, 51 ] ][ "(C3 x C3 x C3) : C2", [ 54, 14 ] ][ "((C4 x C4) : C2) : C2", [ 64, 134 ] ]
[ "(C4 : Q8) : C2", [ 64, 137 ] ][ "((C2 x C2 x C2 x C2) : C2) : C2", [ 64, 138 ] ][ "((C4 x C4) : C2) : C2", [ 64, 177 ] ][ "(C4 : Q8) : C2", [ 64, 178 ] ][ "C8 : Q8", [ 64, 182 ] ]
[ "C2 x ((C2 x C2 x C2 x C2) : C2)", [ 64, 202 ] ][ "C2 x ((C4 x C4) : C2)", [ 64, 211 ] ][ "C2 x (C4 : Q8)", [ 64, 212 ] ][ "(C2 x C2 x D8) : C2", [ 64, 215 ] ][ "(C2 x ((C4 x C2) : C2)) : C2",
  [ 64, 216 ] ][ "((C4 x C4) : C2) : C2", [ 64, 217 ] ][ "(C2 x ((C4 x C2) : C2)) : C2", [ 64, 218 ] ][ "((C2 x Q8) : C2) : C2", [ 64, 224 ] ][ "(C4 : Q8) : C2", [ 64, 225 ] ]
[ "D8 x D8", [ 64, 226 ] ][ "Q8 x D8", [ 64, 230 ] ][ "Q8 x Q8", [ 64, 239 ] ][ "((C4 x C2 x C2) : C2) : C2", [ 64, 241 ] ][ "((C4 x C4) : C2) : C2", [ 64, 242 ] ][ "((C4 x C2 x C2) : C2) : C2",
  [ 64, 243 ] ][ "(C4 : Q8) : C2", [ 64, 244 ] ][ "(C2 x C2) . (C2 x C2 x C2 x C2)", [ 64, 245 ] ][ "C2 x (C8 : (C2 x C2))", [ 64, 254 ] ][ "C2 x ((C2 x Q8) : C2)", [ 64, 255 ] ]
[ "C2 x C2 x C2 x D8", [ 64, 261 ] ][ "C2 x C2 x C2 x Q8", [ 64, 262 ] ][ "C2 x ((C2 x C2 x C2) : (C2 x C2))", [ 64, 264 ] ][ "C2 x ((C2 x Q8) : C2)", [ 64, 265 ] ][ "C2 x C2 x C2 x C2 x C2 x C2",
  [ 64, 267 ] ][ "(S3 x S3) : C2", [ 72, 40 ] ][ "(C3 x C3) : Q8", [ 72, 41 ] ][ "(C3 x A4) : C2", [ 72, 43 ] ][ "C2 x S3 x S3", [ 72, 46 ] ][ "C2 x C2 x ((C3 x C3) : C2)", [ 72, 49 ] ]
[ "C2 x D8 x S3", [ 96, 209 ] ][ "C2 x Q8 x S3", [ 96, 212 ] ][ "C2 x C2 x S4", [ 96, 226 ] ][ "((C2 x C2 x C2 x C2) : C3) : C2", [ 96, 227 ] ][ "C2 x C2 x C2 x C2 x S3", [ 96, 230 ] ]
[ "((C3 x C3) : C3) : (C2 x C2)", [ 108, 17 ] ][ "((C3 x C3) : C2) x S3", [ 108, 39 ] ][ "C2 x ((C3 x C3 x C3) : C2)", [ 108, 44 ] ][ "S5", [ 120, 34 ] ]

gap> for o in [1..120] do
>       if o=1 then Print("\n","|G| ","Nr ","G ","\n","\n");fi;
>       n:=NrSmallGroups(o);;
>       for i in [1..n] do
>               G:=SmallGroup(o,i);;
>               irr:=Irr(G);;
>               s:=Size(irr);;
>               c:=0;;
>               for j in [1..s] do
>                       if Conductor(irr[j])<>1 then
>                               c:=1;;
>                               break;
>                       fi;
>               od;
>               if c=0 then
>                       Print(o,"   ",i,"   ",StructureDescription(G),"\n");
>               fi;
>       od;
> od;

|G| Nr G

1   1   1
2   1   C2
4   2   C2 x C2
6   1   S3
8   3   D8
8   4   Q8
8   5   C2 x C2 x C2
12   4   D12
16   11   C2 x D8
16   12   C2 x Q8
16   14   C2 x C2 x C2 x C2
18   4   (C3 x C3) : C2
24   12   S4
24   14   C2 x C2 x S3
32   27   (C2 x C2 x C2 x C2) : C2
32   34   (C4 x C4) : C2
32   35   C4 : Q8
32   43   C8 : (C2 x C2)
32   44   (C2 x Q8) : C2
32   46   C2 x C2 x D8
32   47   C2 x C2 x Q8
32   49   (C2 x C2 x C2) : (C2 x C2)
32   50   (C2 x Q8) : C2
32   51   C2 x C2 x C2 x C2 x C2
36   10   S3 x S3
36   13   C2 x ((C3 x C3) : C2)
48   38   D8 x S3
48   40   Q8 x S3
48   48   C2 x S4
48   51   C2 x C2 x C2 x S3
54   14   (C3 x C3 x C3) : C2
64   134   ((C4 x C4) : C2) : C2
64   137   (C4 : Q8) : C2
64   138   ((C2 x C2 x C2 x C2) : C2) : C2
64   177   ((C4 x C4) : C2) : C2
64   178   (C4 : Q8) : C2
64   182   C8 : Q8
64   202   C2 x ((C2 x C2 x C2 x C2) : C2)
64   211   C2 x ((C4 x C4) : C2)
64   212   C2 x (C4 : Q8)
64   215   (C2 x C2 x D8) : C2
64   216   (C2 x ((C4 x C2) : C2)) : C2
64   217   ((C4 x C4) : C2) : C2
64   218   (C2 x ((C4 x C2) : C2)) : C2
64   224   ((C2 x Q8) : C2) : C2
64   225   (C4 : Q8) : C2
64   226   D8 x D8
64   230   Q8 x D8
64   239   Q8 x Q8
64   241   ((C4 x C2 x C2) : C2) : C2
64   242   ((C4 x C4) : C2) : C2
64   243   ((C4 x C2 x C2) : C2) : C2
64   244   (C4 : Q8) : C2
64   245   (C2 x C2) . (C2 x C2 x C2 x C2)
64   254   C2 x (C8 : (C2 x C2))
64   255   C2 x ((C2 x Q8) : C2)
64   261   C2 x C2 x C2 x D8
64   262   C2 x C2 x C2 x Q8
64   264   C2 x ((C2 x C2 x C2) : (C2 x C2))
64   265   C2 x ((C2 x Q8) : C2)
64   267   C2 x C2 x C2 x C2 x C2 x C2
72   40   (S3 x S3) : C2
72   41   (C3 x C3) : Q8
72   43   (C3 x A4) : C2
72   46   C2 x S3 x S3
72   49   C2 x C2 x ((C3 x C3) : C2)
96   209   C2 x D8 x S3
96   212   C2 x Q8 x S3
96   226   C2 x C2 x S4
96   227   ((C2 x C2 x C2 x C2) : C3) : C2
96   230   C2 x C2 x C2 x C2 x S3
108   17   ((C3 x C3) : C3) : (C2 x C2)
108   39   ((C3 x C3) : C2) x S3
108   44   C2 x ((C3 x C3 x C3) : C2)
120   34   S5

remove about perfect group, covered by simple example PSp(6,2) metioned in comment

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edited Aug 17, 2022 at 9:44

Sebastien Palcoux

27k
5
74
186

Is the converse of above lemma true?
I guess that $C_2$ is the only such group which is simple. Is it true?
I guess that $C_1$ is the only such group which is perfect or of odd order. Is it true?
Is there such a group with a non-abelian and non-alternating simple normal subgroup?
Is every finite group a normal subgroup of such a group?

edit suggested by Derek Holt in comment

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edited Aug 17, 2022 at 8:33

Sebastien Palcoux

27k
5
74
186

gap> for o in [1..120] do n:=NrSmallGroups(o);; for i in [1..n] do G:=SmallGroup(o,i);; irr:=Irr(G);; s:=Size(irr);; c:=0;; for j in [1..s] do if Conductor(irr[j])<>1 then c:=1;; break; fi; od; if c=0 then Print([StructureDescription(G)],[o,i]]); fi; od; od;
[ "1", [ 1, 1 ] ][ "C2", [ 2, 1 ] ][ "C2 x C2", [ 4, 2 ] ][ "S3", [ 6, 1 ] ][ "D8", [ 8, 3 ] ][ "Q8", [ 8, 4 ] ][ "C2 x C2 x C2", [ 8, 5 ] ][ "D12", [ 12, 4 ] ][ "C2 x D8", ][[ 16, 11 ] ]
[ "C2 x Q8", [ 16, 12 ] ][ "C2 x C2 x C2 x C2", [ 16, 14 ] ][ "(C3 x C3) : C2", [ 18, 4 ] ][ "S4", [ 24, 12 ] ][ "C2 x C2 x S3", ]
[ 24, 14 ] ][ "(C2 x C2 x C2 x C2) : C2", ][[ 32, 27 ] ]
[ "(C4 x C4) : C2", [ 32, 34 ] ][ "C4 : Q8", [ 32, 35 ] ][ "C8 : (C2 x C2)", [ 32, 43 ] ][ "(C2 x Q8) : C2", [ 32, 44 ] ][ "C2 x C2 x D8", [ 32, 46 ] ][ "C2 x C2 x Q8", ][[ 32, 47 ] ]
[ "(C2 x C2 x C2) : (C2 x C2)", [ 32, 49 ] ][ "(C2 x Q8) : C2", ]
[ 32, 50 ] ][ "C2 x C2 x C2 x C2 x C2", [ 32, 51 ] ][ "S3 x S3", [ 36, 10 ] ][ "C2 x ((C3 x C3) : C2)", ][[ 36, 13 ] ]
[ "D8 x S3", [ 48, 38 ] ][ "Q8 x S3", [ 48, 40 ] ][ "C2 x S4", [ 48, 48 ] ][ "C2 x C2 x C2 x S3", [ 48, 51 ] ][ "(C3 x C3 x C3) : C2", [ 54, 14 ] ][ "((C4 x C4) : C2) : C2", [ 64, 134 ] ]
[ "(C4 : Q8) : C2", [ 64, 137 ] ][ "((C2 x C2 x C2 x C2) : C2) : C2", [ 64, 138 ] ][ "((C4 x C4) : C2) : C2", [ 64, 177 ] ][ "(C4 : Q8) : C2", [ 64, 178 ] ][ "C8 : Q8", ][[ 64, 182 ] ]
[ "C2 x ((C2 x C2 x C2 x C2) : C2)", [ 64, 202 ] ][ "C2 x ((C4 x C4) : C2)", ]
[ 64, 211 ] ][ "C2 x (C4 : Q8)", [ 64, 212 ] ][ "(C2 x C2 x D8) : C2", [ 64, 215 ] ][ "(C2 x ((C4 x C2) : C2)) : C2",
  [ 64, 216 ] ][ "((C4 x C4) : C2) : C2", [ 64, 217 ] ][ "(C2 x ((C4 x C2) : C2)) : C2", [ 64, 218 ] ][ "((C2 x Q8) : C2) : C2", [ 64, 224 ] ][ "(C4 : Q8) : C2", [ 64, 225 ] ]
[ "D8 x D8", [ 64, 226 ] ][ "Q8 x D8", [ 64, 230 ] ][ "Q8 x Q8", [ 64, 239 ] ][ "((C4 x C2 x C2) : C2) : C2", [ 64, 241 ] ][ "((C4 x C4) : C2) : C2", [ 64, 242 ] ][ "((C4 x C2 x C2) : C2) : C2",
  [ 64, 243 ] ][ "(C4 : Q8) : C2", [ 64, 244 ] ][ "(C2 x C2) . (C2 x C2 x C2 x C2)", ]
[ 64, 245 ] ][ "C2 x (C8 : (C2 x C2))", [ 64, 254 ] ][ "C2 x ((C2 x Q8) : C2)", ][[ 64, 255 ] ]
[ "C2 x C2 x C2 x D8", [ 64, 261 ] ][ "C2 x C2 x C2 x Q8", [ 64, 262 ] ][ "C2 x ((C2 x C2 x C2) : (C2 x C2))", [ 64, 264 ] ][ "C2 x ((C2 x Q8) : C2)", [ 64, 265 ] ][ "C2 x C2 x C2 x C2 x C2 x C2" ],
  [ 64, 267 ] ][ "(S3 x S3) : C2", [ 72, 40 ] ][ "(C3 x C3) : Q8", [ 72, 41 ] ][ "(C3 x A4) : C2", [ 72, 43 ] ][ "C2 x S3 x S3", [ 72, 46 ] ][ "C2 x C2 x ((C3 x C3) : C2)", ][[ 72, 49 ] ]
[ "C2 x D8 x S3", [ 96, 209 ] ][ "C2 x Q8 x S3", [ 96, 212 ] ][ "C2 x C2 x S4", ]
[ 96, 226 ] ][ "((C2 x C2 x C2 x C2) : C3) : C2", [ 96, 227 ] ][ "C2 x C2 x C2 x C2 x S3", ][[ 96, 230 ] ]
[ "((C3 x C3) : C3) : (C2 x C2)", [ 108, 17 ] ][ "((C3 x C3) : C2) x S3", [ 108, 39 ] ][ "C2 x ((C3 x C3 x C3) : C2)", [ 108, 44 ] ][ "S5", [ 120, 34 ] ]

gap> for o in [1..120] do n:=NrSmallGroups(o);; for i in [1..n] do G:=SmallGroup(o,i);; irr:=Irr(G);; s:=Size(irr);; c:=0;; for j in [1..s] do if Conductor(irr[j])<>1 then c:=1;; break; fi; od; if c=0 then Print([StructureDescription(G)]); fi; od; od;
[ "1" ][ "C2" ][ "C2 x C2" ][ "S3" ][ "D8" ][ "Q8" ][ "C2 x C2 x C2" ][ "D12" ][ "C2 x D8" ][ "C2 x Q8" ][ "C2 x C2 x C2 x C2" ][ "(C3 x C3) : C2" ][ "S4" ][ "C2 x C2 x S3" ]
[ "(C2 x C2 x C2 x C2) : C2" ][ "(C4 x C4) : C2" ][ "C4 : Q8" ][ "C8 : (C2 x C2)" ][ "(C2 x Q8) : C2" ][ "C2 x C2 x D8" ][ "C2 x C2 x Q8" ][ "(C2 x C2 x C2) : (C2 x C2)" ][ "(C2 x Q8) : C2" ]
[ "C2 x C2 x C2 x C2 x C2" ][ "S3 x S3" ][ "C2 x ((C3 x C3) : C2)" ][ "D8 x S3" ][ "Q8 x S3" ][ "C2 x S4" ][ "C2 x C2 x C2 x S3" ][ "(C3 x C3 x C3) : C2" ][ "((C4 x C4) : C2) : C2" ]
[ "(C4 : Q8) : C2" ][ "((C2 x C2 x C2 x C2) : C2) : C2" ][ "((C4 x C4) : C2) : C2" ][ "(C4 : Q8) : C2" ][ "C8 : Q8" ][ "C2 x ((C2 x C2 x C2 x C2) : C2)" ][ "C2 x ((C4 x C4) : C2)" ]
[ "C2 x (C4 : Q8)" ][ "(C2 x C2 x D8) : C2" ][ "(C2 x ((C4 x C2) : C2)) : C2" ][ "((C4 x C4) : C2) : C2" ][ "(C2 x ((C4 x C2) : C2)) : C2" ][ "((C2 x Q8) : C2) : C2" ][ "(C4 : Q8) : C2" ]
[ "D8 x D8" ][ "Q8 x D8" ][ "Q8 x Q8" ][ "((C4 x C2 x C2) : C2) : C2" ][ "((C4 x C4) : C2) : C2" ][ "((C4 x C2 x C2) : C2) : C2" ][ "(C4 : Q8) : C2" ][ "(C2 x C2) . (C2 x C2 x C2 x C2)" ]
[ "C2 x (C8 : (C2 x C2))" ][ "C2 x ((C2 x Q8) : C2)" ][ "C2 x C2 x C2 x D8" ][ "C2 x C2 x C2 x Q8" ][ "C2 x ((C2 x C2 x C2) : (C2 x C2))" ][ "C2 x ((C2 x Q8) : C2)" ][ "C2 x C2 x C2 x C2 x C2 x C2" ]
[ "(S3 x S3) : C2" ][ "(C3 x C3) : Q8" ][ "(C3 x A4) : C2" ][ "C2 x S3 x S3" ][ "C2 x C2 x ((C3 x C3) : C2)" ][ "C2 x D8 x S3" ][ "C2 x Q8 x S3" ][ "C2 x C2 x S4" ]
[ "((C2 x C2 x C2 x C2) : C3) : C2" ][ "C2 x C2 x C2 x C2 x S3" ][ "((C3 x C3) : C3) : (C2 x C2)" ][ "((C3 x C3) : C2) x S3" ][ "C2 x ((C3 x C3 x C3) : C2)" ][ "S5" ]

gap> for o in [1..120] do n:=NrSmallGroups(o);; for i in [1..n] do G:=SmallGroup(o,i);; irr:=Irr(G);; s:=Size(irr);; c:=0;; for j in [1..s] do if Conductor(irr[j])<>1 then c:=1;; break; fi; od; if c=0 then Print([StructureDescription(G),[o,i]]); fi; od; od;
[ "1", [ 1, 1 ] ][ "C2", [ 2, 1 ] ][ "C2 x C2", [ 4, 2 ] ][ "S3", [ 6, 1 ] ][ "D8", [ 8, 3 ] ][ "Q8", [ 8, 4 ] ][ "C2 x C2 x C2", [ 8, 5 ] ][ "D12", [ 12, 4 ] ][ "C2 x D8", [ 16, 11 ] ]
[ "C2 x Q8", [ 16, 12 ] ][ "C2 x C2 x C2 x C2", [ 16, 14 ] ][ "(C3 x C3) : C2", [ 18, 4 ] ][ "S4", [ 24, 12 ] ][ "C2 x C2 x S3", [ 24, 14 ] ][ "(C2 x C2 x C2 x C2) : C2", [ 32, 27 ] ]
[ "(C4 x C4) : C2", [ 32, 34 ] ][ "C4 : Q8", [ 32, 35 ] ][ "C8 : (C2 x C2)", [ 32, 43 ] ][ "(C2 x Q8) : C2", [ 32, 44 ] ][ "C2 x C2 x D8", [ 32, 46 ] ][ "C2 x C2 x Q8", [ 32, 47 ] ]
[ "(C2 x C2 x C2) : (C2 x C2)", [ 32, 49 ] ][ "(C2 x Q8) : C2", [ 32, 50 ] ][ "C2 x C2 x C2 x C2 x C2", [ 32, 51 ] ][ "S3 x S3", [ 36, 10 ] ][ "C2 x ((C3 x C3) : C2)", [ 36, 13 ] ]
[ "D8 x S3", [ 48, 38 ] ][ "Q8 x S3", [ 48, 40 ] ][ "C2 x S4", [ 48, 48 ] ][ "C2 x C2 x C2 x S3", [ 48, 51 ] ][ "(C3 x C3 x C3) : C2", [ 54, 14 ] ][ "((C4 x C4) : C2) : C2", [ 64, 134 ] ]
[ "(C4 : Q8) : C2", [ 64, 137 ] ][ "((C2 x C2 x C2 x C2) : C2) : C2", [ 64, 138 ] ][ "((C4 x C4) : C2) : C2", [ 64, 177 ] ][ "(C4 : Q8) : C2", [ 64, 178 ] ][ "C8 : Q8", [ 64, 182 ] ]
[ "C2 x ((C2 x C2 x C2 x C2) : C2)", [ 64, 202 ] ][ "C2 x ((C4 x C4) : C2)", [ 64, 211 ] ][ "C2 x (C4 : Q8)", [ 64, 212 ] ][ "(C2 x C2 x D8) : C2", [ 64, 215 ] ][ "(C2 x ((C4 x C2) : C2)) : C2",
  [ 64, 216 ] ][ "((C4 x C4) : C2) : C2", [ 64, 217 ] ][ "(C2 x ((C4 x C2) : C2)) : C2", [ 64, 218 ] ][ "((C2 x Q8) : C2) : C2", [ 64, 224 ] ][ "(C4 : Q8) : C2", [ 64, 225 ] ]
[ "D8 x D8", [ 64, 226 ] ][ "Q8 x D8", [ 64, 230 ] ][ "Q8 x Q8", [ 64, 239 ] ][ "((C4 x C2 x C2) : C2) : C2", [ 64, 241 ] ][ "((C4 x C4) : C2) : C2", [ 64, 242 ] ][ "((C4 x C2 x C2) : C2) : C2",
  [ 64, 243 ] ][ "(C4 : Q8) : C2", [ 64, 244 ] ][ "(C2 x C2) . (C2 x C2 x C2 x C2)", [ 64, 245 ] ][ "C2 x (C8 : (C2 x C2))", [ 64, 254 ] ][ "C2 x ((C2 x Q8) : C2)", [ 64, 255 ] ]
[ "C2 x C2 x C2 x D8", [ 64, 261 ] ][ "C2 x C2 x C2 x Q8", [ 64, 262 ] ][ "C2 x ((C2 x C2 x C2) : (C2 x C2))", [ 64, 264 ] ][ "C2 x ((C2 x Q8) : C2)", [ 64, 265 ] ][ "C2 x C2 x C2 x C2 x C2 x C2",
  [ 64, 267 ] ][ "(S3 x S3) : C2", [ 72, 40 ] ][ "(C3 x C3) : Q8", [ 72, 41 ] ][ "(C3 x A4) : C2", [ 72, 43 ] ][ "C2 x S3 x S3", [ 72, 46 ] ][ "C2 x C2 x ((C3 x C3) : C2)", [ 72, 49 ] ]
[ "C2 x D8 x S3", [ 96, 209 ] ][ "C2 x Q8 x S3", [ 96, 212 ] ][ "C2 x C2 x S4", [ 96, 226 ] ][ "((C2 x C2 x C2 x C2) : C3) : C2", [ 96, 227 ] ][ "C2 x C2 x C2 x C2 x S3", [ 96, 230 ] ]
[ "((C3 x C3) : C3) : (C2 x C2)", [ 108, 17 ] ][ "((C3 x C3) : C2) x S3", [ 108, 39 ] ][ "C2 x ((C3 x C3 x C3) : C2)", [ 108, 44 ] ][ "S5", [ 120, 34 ] ]

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asked Aug 17, 2022 at 6:24

Sebastien Palcoux

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