#44542154

order	31902
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₁₃×C₄₀₉ ⋊ C₂×C₃
Hall	#30630086 ⋊ C₂
Hall	D₅₃₁₇ ⋊ C₃
Sylow	C₄₀₉ ⋊ D₁₃⋊C₃
Sylow	C₁₃ ⋊ D₄₀₉⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₄₀₉	C₂×C₃
LCS	G ▷ C₁₃×C₄₀₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 422
transitive degree	μ_t(G) = 5317

A minimal faithful permutation representation — 2 generators on 422 points:

(2,6,7,3,4,8)(5,12,11,9,10,13)(15,19,20,16,17,21)(18,27,28,22,23,37)(24,40,41,30,31,54)(25,39,35,33,29,26)(32,59,60,43,44,77)(34,57,58,46,42,51)(36,38,56,48,49,52)(45,82,83,62,63,106)(47,80,81,65,61,74)(50,55,79,67,68,75)(53,95,93,71,72,70)(64,111,112,85,86,141)(66,109,110,88,84,103)(69,78,108,90,91,104)(73,124,125,96,97,99)(76,130,122,100,101,94)(87,146,147,114,115,182)(89,144,145,117,113,138)(92,107,143,119,120,139)(98,159,160,126,127,134)(102,166,164,131,132,129)(105,171,157,135,136,123)(116,187,188,149,150,229)(118,185,186,152,148,179)(121,142,184,154,155,180)(128,200,201,161,162,175)(133,207,208,167,168,170)(137,213,205,172,173,165)(140,218,198,176,177,158)(151,234,235,190,191,282)(153,232,233,193,189,226)(156,183,231,195,196,227)(163,247,248,202,203,222)(169,254,255,209,210,217)(174,261,259,214,215,212)(178,266,252,219,220,206)(181,271,245,223,224,199)(192,287,288,237,238,341)(194,285,286,240,236,279)(197,230,284,242,243,280)(204,300,301,249,250,275)(211,307,308,256,257,270)(216,314,315,262,263,265)(221,320,312,267,268,260)(225,325,305,272,273,253)(228,330,298,276,277,246)(239,346,347,290,291,395)(241,344,345,293,289,338)(244,283,343,295,296,339)(251,357,358,302,303,334)(258,364,365,309,310,329)(264,371,372,316,317,324)(269,377,375,321,322,319)(274,380,369,326,327,313)(278,383,362,331,332,306)(281,386,355,335,336,299)(292,387,363,337,349,407)(294,398,399,350,348,392)(297,342,397,352,353,393)(304,404,405,359,360,388)(311,409,410,366,367,385)(318,413,414,373,374,382)(323,411,416,368,378,379)(328,406,415,361,381,376)(333,400,412,351,384,370)(340,417,402,389,390,356)(354,396,408,391,401,421)(394,422,420,418,419,403);(1,2,5,4,8,11,10,12,13,7,6,9,3)(14,15,18,24,32,45,64,87,116,151,192,239,292,335,243,150,189,203,210,215,220,224,227,229,279,334,385,414,365,301,233,147,184,218,213,207,200,185,146,183,228,278,333,350,293,240,193,152,117,88,65,46,33,17,21,26,36,50,69,92,121,156,197,244,297,354,389,296,191,236,250,257,263,268,273,277,280,282,338,388,416,372,308,248,186,112,143,171,166,159,144,111,142,181,225,274,328,359,302,249,202,161,126,96,71,49,23,29,37,51,70,94,123,158,199,246,299,356,403,418,353,238,289,303,310,317,322,327,332,336,339,341,392,415,375,315,255,201,145,83,108,130,124,109,82,107,140,178,221,269,323,366,309,256,209,167,131,100,68,31,42,52,54,74,99,129,165,206,253,306,363,408,422,401,291,348,360,367,374,378,381,384,387,390,393,395,412,369,312,259,208,160,110,60,79,95,80,59,78,105,137,174,216,264,318,373,316,262,214,172,135,91,44,61,72,75,77,103,134,170,212,260,313,370,347,397,417,349,400,406,411,413,409,404,398,346,396,419,421,407,362,305,252,205,164,125,81,41,56,57,40,55,76,102,133,169,211,258,311,368,321,267,219,176,120,63,84,97,101,104,106,138,175,217,265,319,376,399,288,343,386,383,380,377,371,364,357,344,287,342,394,420,402,355,298,245,198,157,122,93,58,28,39,27,38,53,73,98,128,163,204,251,304,361,326,272,223,155,86,113,127,132,136,139,141,179,222,270,324,379,405,345,235,284,330,325,320,314,307,300,285,234,283,340,391,352,295,242,195,154,119,90,67,48,35,20,19,25,34,47,66,89,118,153,194,241,294,351,331,276,196,115,148,162,168,173,177,180,182,226,275,329,382,410,358,286,188,231,271,266,261,254,247,232,187,230,281,337,290,237,190,149,114,85,62,43,30,22,16)

Polycyclic presentation:

2,3,13,409|c^a=c^12,d^a=d^408,c^b=c^3,d^b=d^53

Conjugacy classes & character table

892 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31902, 24);