Revisions to Where can I find a table of the exponents of the sporadic groups?

"Order" should be "Exponent"

Source Link

edited Dec 10, 2023 at 11:26

35.2k
2
110
151

I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:

$$ \begin{align*} \mathbf{Group}&&\mathbf{Order}&&\mathbf{Factorization}\\ M_{11}&&1320&&2^3\cdot3\cdot5\cdot11\\ M_{12}&&1320&&2^3\cdot3\cdot5\cdot11\\ J_1&&43890&&2\cdot3\cdot5\cdot7\cdot11\cdot19\\ M_{22}&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_2&&840&&2^3\cdot3\cdot5\cdot7\\ M_{23}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ HS&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_3&&116280&&2^3\cdot3^2\cdot5\cdot17\cdot19\\ M_{24}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ McL&&27720&&2^3\cdot3^2\cdot5\cdot7\cdot11\\ He&&14280&&2^3\cdot3\cdot5\cdot7\cdot17\\ Ru&&633360&&2^4\cdot3\cdot5\cdot7\cdot13\cdot29\\ Suz&&360360&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ O'N&&10884720&&2^4\cdot3\cdot5\cdot7\cdot11\cdot19\cdot31\\ Co_3&&637560&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Co_2&&1275120&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Fi_{22}&&720720&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ HN&&2633400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot19\\ Ly&&10651271400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot31\cdot37\cdot67\\ Th&&57886920&&2^3\cdot3^3\cdot5\cdot7\cdot13\cdot19\cdot31\\ Fi_{23}&&845404560&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\\ Co_1&&16576560&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\cdot23\\ J_4&&607938537360&&2^4\cdot3\cdot5\cdot7\cdot11\cdot23\cdot29\cdot31\cdot37\cdot43\\ F_{3+}&&24516732240&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\cdot29\\ B&&234033344344800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot31\cdot47\\ M&&1165654792878376600800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\\ &&&&\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71\\ \end{align*} $$$$ \begin{align*} \mathbf{Group}&&\mathbf{Exponent}&&\mathbf{Factorization}\\ M_{11}&&1320&&2^3\cdot3\cdot5\cdot11\\ M_{12}&&1320&&2^3\cdot3\cdot5\cdot11\\ J_1&&43890&&2\cdot3\cdot5\cdot7\cdot11\cdot19\\ M_{22}&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_2&&840&&2^3\cdot3\cdot5\cdot7\\ M_{23}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ HS&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_3&&116280&&2^3\cdot3^2\cdot5\cdot17\cdot19\\ M_{24}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ McL&&27720&&2^3\cdot3^2\cdot5\cdot7\cdot11\\ He&&14280&&2^3\cdot3\cdot5\cdot7\cdot17\\ Ru&&633360&&2^4\cdot3\cdot5\cdot7\cdot13\cdot29\\ Suz&&360360&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ O'N&&10884720&&2^4\cdot3\cdot5\cdot7\cdot11\cdot19\cdot31\\ Co_3&&637560&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Co_2&&1275120&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Fi_{22}&&720720&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ HN&&2633400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot19\\ Ly&&10651271400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot31\cdot37\cdot67\\ Th&&57886920&&2^3\cdot3^3\cdot5\cdot7\cdot13\cdot19\cdot31\\ Fi_{23}&&845404560&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\\ Co_1&&16576560&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\cdot23\\ J_4&&607938537360&&2^4\cdot3\cdot5\cdot7\cdot11\cdot23\cdot29\cdot31\cdot37\cdot43\\ F_{3+}&&24516732240&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\cdot29\\ B&&234033344344800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot31\cdot47\\ M&&1165654792878376600800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\\ &&&&\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71\\ \end{align*} $$

I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:

$$ \begin{align*} \mathbf{Group}&&\mathbf{Order}&&\mathbf{Factorization}\\ M_{11}&&1320&&2^3\cdot3\cdot5\cdot11\\ M_{12}&&1320&&2^3\cdot3\cdot5\cdot11\\ J_1&&43890&&2\cdot3\cdot5\cdot7\cdot11\cdot19\\ M_{22}&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_2&&840&&2^3\cdot3\cdot5\cdot7\\ M_{23}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ HS&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_3&&116280&&2^3\cdot3^2\cdot5\cdot17\cdot19\\ M_{24}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ McL&&27720&&2^3\cdot3^2\cdot5\cdot7\cdot11\\ He&&14280&&2^3\cdot3\cdot5\cdot7\cdot17\\ Ru&&633360&&2^4\cdot3\cdot5\cdot7\cdot13\cdot29\\ Suz&&360360&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ O'N&&10884720&&2^4\cdot3\cdot5\cdot7\cdot11\cdot19\cdot31\\ Co_3&&637560&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Co_2&&1275120&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Fi_{22}&&720720&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ HN&&2633400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot19\\ Ly&&10651271400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot31\cdot37\cdot67\\ Th&&57886920&&2^3\cdot3^3\cdot5\cdot7\cdot13\cdot19\cdot31\\ Fi_{23}&&845404560&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\\ Co_1&&16576560&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\cdot23\\ J_4&&607938537360&&2^4\cdot3\cdot5\cdot7\cdot11\cdot23\cdot29\cdot31\cdot37\cdot43\\ F_{3+}&&24516732240&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\cdot29\\ B&&234033344344800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot31\cdot47\\ M&&1165654792878376600800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\\ &&&&\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71\\ \end{align*} $$

I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:

$$ \begin{align*} \mathbf{Group}&&\mathbf{Exponent}&&\mathbf{Factorization}\\ M_{11}&&1320&&2^3\cdot3\cdot5\cdot11\\ M_{12}&&1320&&2^3\cdot3\cdot5\cdot11\\ J_1&&43890&&2\cdot3\cdot5\cdot7\cdot11\cdot19\\ M_{22}&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_2&&840&&2^3\cdot3\cdot5\cdot7\\ M_{23}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ HS&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_3&&116280&&2^3\cdot3^2\cdot5\cdot17\cdot19\\ M_{24}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ McL&&27720&&2^3\cdot3^2\cdot5\cdot7\cdot11\\ He&&14280&&2^3\cdot3\cdot5\cdot7\cdot17\\ Ru&&633360&&2^4\cdot3\cdot5\cdot7\cdot13\cdot29\\ Suz&&360360&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ O'N&&10884720&&2^4\cdot3\cdot5\cdot7\cdot11\cdot19\cdot31\\ Co_3&&637560&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Co_2&&1275120&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Fi_{22}&&720720&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ HN&&2633400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot19\\ Ly&&10651271400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot31\cdot37\cdot67\\ Th&&57886920&&2^3\cdot3^3\cdot5\cdot7\cdot13\cdot19\cdot31\\ Fi_{23}&&845404560&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\\ Co_1&&16576560&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\cdot23\\ J_4&&607938537360&&2^4\cdot3\cdot5\cdot7\cdot11\cdot23\cdot29\cdot31\cdot37\cdot43\\ F_{3+}&&24516732240&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\cdot29\\ B&&234033344344800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot31\cdot47\\ M&&1165654792878376600800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\\ &&&&\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71\\ \end{align*} $$

Table-ized and factored (and fixed a typo)

Source Link

edit approved Dec 10, 2023 at 11:24

Steven Stadnicki

2k
17
24

I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:

[ [ "M11", 1320 ], 
  [ "M12", 1320 ], 
  [ "J1", 43890 ], 
  [ "M22", 9240 ], 
  [ "J2", 840 ], 
  [ "M23", 212520 ], 
  [ "HS", 9240 ], 
  [ "J3", 116280 ], 
  [ "M24", 212520 ], 
  [ "McL", 27720 ], 
  [ "He", 14280 ], 
  [ "Ru", 633360 ], 
  [ "Suz", 360360 ], 
  [ "ON", 10884720 ], 
  [ "Co3", 637560 ], 
  [ "Co2", 1275120 ], 
  [ "Fi22", 720720 ], 
  [ "HN", 2633400 ],
  [ "Ly", 10651271400 ], 
  [ "Th", 57886920 ], 
  [ "Fi23", 845404560 ], 
  [ "Co1", 16576560 ], 
  [ "J4", 607938537360 ], 
  [ "F3+", 24516732240 ], 
  [ "B", 234033344344800 ], 
  [ "M", 1165654792878376600800 ] ]

$$ \begin{align*} \mathbf{Group}&&\mathbf{Order}&&\mathbf{Factorization}\\ M_{11}&&1320&&2^3\cdot3\cdot5\cdot11\\ M_{12}&&1320&&2^3\cdot3\cdot5\cdot11\\ J_1&&43890&&2\cdot3\cdot5\cdot7\cdot11\cdot19\\ M_{22}&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_2&&840&&2^3\cdot3\cdot5\cdot7\\ M_{23}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ HS&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_3&&116280&&2^3\cdot3^2\cdot5\cdot17\cdot19\\ M_{24}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ McL&&27720&&2^3\cdot3^2\cdot5\cdot7\cdot11\\ He&&14280&&2^3\cdot3\cdot5\cdot7\cdot17\\ Ru&&633360&&2^4\cdot3\cdot5\cdot7\cdot13\cdot29\\ Suz&&360360&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ O'N&&10884720&&2^4\cdot3\cdot5\cdot7\cdot11\cdot19\cdot31\\ Co_3&&637560&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Co_2&&1275120&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Fi_{22}&&720720&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ HN&&2633400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot19\\ Ly&&10651271400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot31\cdot37\cdot67\\ Th&&57886920&&2^3\cdot3^3\cdot5\cdot7\cdot13\cdot19\cdot31\\ Fi_{23}&&845404560&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\\ Co_1&&16576560&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\cdot23\\ J_4&&607938537360&&2^4\cdot3\cdot5\cdot7\cdot11\cdot23\cdot29\cdot31\cdot37\cdot43\\ F_{3+}&&24516732240&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\cdot29\\ B&&234033344344800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot31\cdot47\\ M&&1165654792878376600800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\\ &&&&\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71\\ \end{align*} $$

I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:

[ [ "M11", 1320 ], 
  [ "M12", 1320 ], 
  [ "J1", 43890 ], 
  [ "M22", 9240 ], 
  [ "J2", 840 ], 
  [ "M23", 212520 ], 
  [ "HS", 9240 ], 
  [ "J3", 116280 ], 
  [ "M24", 212520 ], 
  [ "McL", 27720 ], 
  [ "He", 14280 ], 
  [ "Ru", 633360 ], 
  [ "Suz", 360360 ], 
  [ "ON", 10884720 ], 
  [ "Co3", 637560 ], 
  [ "Co2", 1275120 ], 
  [ "Fi22", 720720 ], 
  [ "HN", 2633400 ],
  [ "Ly", 10651271400 ], 
  [ "Th", 57886920 ], 
  [ "Fi23", 845404560 ], 
  [ "Co1", 16576560 ], 
  [ "J4", 607938537360 ], 
  [ "F3+", 24516732240 ], 
  [ "B", 234033344344800 ], 
  [ "M", 1165654792878376600800 ] ]

I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:

$$ \begin{align*} \mathbf{Group}&&\mathbf{Order}&&\mathbf{Factorization}\\ M_{11}&&1320&&2^3\cdot3\cdot5\cdot11\\ M_{12}&&1320&&2^3\cdot3\cdot5\cdot11\\ J_1&&43890&&2\cdot3\cdot5\cdot7\cdot11\cdot19\\ M_{22}&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_2&&840&&2^3\cdot3\cdot5\cdot7\\ M_{23}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ HS&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_3&&116280&&2^3\cdot3^2\cdot5\cdot17\cdot19\\ M_{24}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ McL&&27720&&2^3\cdot3^2\cdot5\cdot7\cdot11\\ He&&14280&&2^3\cdot3\cdot5\cdot7\cdot17\\ Ru&&633360&&2^4\cdot3\cdot5\cdot7\cdot13\cdot29\\ Suz&&360360&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ O'N&&10884720&&2^4\cdot3\cdot5\cdot7\cdot11\cdot19\cdot31\\ Co_3&&637560&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Co_2&&1275120&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Fi_{22}&&720720&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ HN&&2633400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot19\\ Ly&&10651271400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot31\cdot37\cdot67\\ Th&&57886920&&2^3\cdot3^3\cdot5\cdot7\cdot13\cdot19\cdot31\\ Fi_{23}&&845404560&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\\ Co_1&&16576560&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\cdot23\\ J_4&&607938537360&&2^4\cdot3\cdot5\cdot7\cdot11\cdot23\cdot29\cdot31\cdot37\cdot43\\ F_{3+}&&24516732240&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\cdot29\\ B&&234033344344800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot31\cdot47\\ M&&1165654792878376600800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\\ &&&&\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71\\ \end{align*} $$

added 20 characters in body

Source Link

edited Dec 9, 2023 at 23:37

Ghoster

161
1
3

Here is a completeI couldn't find an online table of exponents for sporadic groups, produced byso I used GAP. to produce one:

[ [ "M11", 1320 ], 
  [ "M12", 1320 ], 
  [ "J1", 43890 ], 
  [ "M22", 9240 ], 
  [ "J2", 840 ], 
  [ "M23", 212520 ], 
  [ "HS", 9240 ], 
  [ "J3", 116280 ], 
  [ "M24", 212520 ], 
  [ "McL", 27720 ], 
  [ "He", 14280 ], 
  [ "Ru", 633360 ], 
  [ "Suz", 360360 ], 
  [ "ON", 10884720 ], 
  [ "Co3", 637560 ], 
  [ "Co2", 1275120 ], 
  [ "Fi22", 720720 ], 
  [ "HN", 2633400 ],
  [ "Ly", 10651271400 ], 
  [ "Th", 57886920 ], 
  [ "Fi23", 845404560 ], 
  [ "Co1", 16576560 ], 
  [ "J4", 607938537360 ], 
  [ "F3+", 24516732240 ], 
  [ "B", 234033344344800 ], 
  [ "M", 1165654792878376600800 ] ]

Here is a complete table of exponents for sporadic groups, produced by GAP.

[ [ "M11", 1320 ], 
  [ "M12", 1320 ], 
  [ "J1", 43890 ], 
  [ "M22", 9240 ], 
  [ "J2", 840 ], 
  [ "M23", 212520 ], 
  [ "HS", 9240 ], 
  [ "J3", 116280 ], 
  [ "M24", 212520 ], 
  [ "McL", 27720 ], 
  [ "He", 14280 ], 
  [ "Ru", 633360 ], 
  [ "Suz", 360360 ], 
  [ "ON", 10884720 ], 
  [ "Co3", 637560 ], 
  [ "Co2", 1275120 ], 
  [ "Fi22", 720720 ], 
  [ "HN", 2633400 ],
  [ "Ly", 10651271400 ], 
  [ "Th", 57886920 ], 
  [ "Fi23", 845404560 ], 
  [ "Co1", 16576560 ], 
  [ "J4", 607938537360 ], 
  [ "F3+", 24516732240 ], 
  [ "B", 234033344344800 ], 
  [ "M", 1165654792878376600800 ] ]

I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:

[ [ "M11", 1320 ], 
  [ "M12", 1320 ], 
  [ "J1", 43890 ], 
  [ "M22", 9240 ], 
  [ "J2", 840 ], 
  [ "M23", 212520 ], 
  [ "HS", 9240 ], 
  [ "J3", 116280 ], 
  [ "M24", 212520 ], 
  [ "McL", 27720 ], 
  [ "He", 14280 ], 
  [ "Ru", 633360 ], 
  [ "Suz", 360360 ], 
  [ "ON", 10884720 ], 
  [ "Co3", 637560 ], 
  [ "Co2", 1275120 ], 
  [ "Fi22", 720720 ], 
  [ "HN", 2633400 ],
  [ "Ly", 10651271400 ], 
  [ "Th", 57886920 ], 
  [ "Fi23", 845404560 ], 
  [ "Co1", 16576560 ], 
  [ "J4", 607938537360 ], 
  [ "F3+", 24516732240 ], 
  [ "B", 234033344344800 ], 
  [ "M", 1165654792878376600800 ] ]

added 1 character in body

Source Link

edited Dec 9, 2023 at 23:36

Ghoster

161
1
3

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created Dec 9, 2023 at 23:35

Ghoster

161
1
3

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