#72210876

order	4368
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

non-solvable

Aliases

No aliases for this group.

Splittings

derived

Sp(2,𝔽₁₃) ⋊ C₂

Subgroups

	subgroup	G/H
center, Fitting, hypercenter	C₄	PSL(2,𝔽₁₃)
commutator, hypocenter	Sp(2,𝔽₁₃)	C₂
Frattini, socle, cosocle	C₂	C₂×PSL(2,𝔽₁₃)
LCS	G ▷ Sp(2,𝔽₁₃)
UCS	C₁ ◁ C₄

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(1,111,141,153,99,13,57,55,15,97,155,139,113,12,71,41,29,83,156,125,127,167,85,27,43,69)(2,40,100,124,16,26,114,110,30,25,128,96,44,11,142,82,58,166,143,68,72,152,157,54,86,138)(3,151,59,95,115,39,158,165,45,109,101,53,144,10,31,123,87,67,130,24,17,137,73,81,129,38)(4,80,18,66,32,52,46,51,60,37,74,23,88,9,102,164,116,150,117,136,131,122,145,108,159,94)(5,22,146,50,118,65,103,93,75,121,47,149,19,8,160,36,132,64,104,79,89,107,61,135,33,163)(6,120,105,21,48,78,147,148,90,49,20,106,119,7,62,77,161,134,91,35,34,92,133,162,76,63)(14,168)(28,154)(42,140)(56,126)(70,112)(84,98);(1,107,136,154,102,19,55,53,21,100,143,134,109,12,75,46,28,80,163,127,129,161,82,26,48,73)(2,45,103,126,22,38,110,106,29,18,117,99,49,11,137,79,56,160,144,72,76,153,164,52,83,133)(3,152,57,98,111,44,165,159,50,105,91,64,145,10,30,125,84,71,138,17,23,132,77,78,118,37)(4,90,24,70,31,63,51,43,58,36,65,16,85,9,92,158,112,151,119,131,139,124,146,104,166,97)(5,15,147,42,120,69,93,96,66,123,39,150,25,8,167,35,140,62,113,89,86,116,59,130,32,157)(6,122,114,14,40,88,148,149,87,41,13,115,121,7,60,68,168,142,94,34,33,95,141,156,67,61)(20,162)(27,155)(47,135)(54,128)(74,108)(81,101);(13,160)(14,161)(15,162)(16,163)(17,164)(18,165)(19,166)(20,167)(21,168)(22,156)(23,157)(24,158)(25,159)(26,151)(27,152)(28,153)(29,154)(30,155)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,142)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,120)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,129)(62,117)(63,118)(64,119)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,102)(79,103)(80,91)(81,92)(82,93)(83,94)(84,95)(85,96)(86,97)(87,98)(88,99)(89,100)(90,101)

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72210876, #72210882
maximal normal subgroup of	C₃×#72210876, #72210882
maximal quotient of	C₃×#72210876

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((1,111,141,153,99,13,57,55,15,97,155,139,113,12,71,41,29,83,156,125,127,167,85,27,43,69)(2,40,100,124,16,26,114,110,30,25,128,96,44,11,142,82,58,166,143,68,72,152,157,54,86,138)(3,151,59,95,115,39,158,165,45,109,101,53,144,10,31,123,87,67,130,24,17,137,73,81,129,38)(4,80,18,66,32,52,46,51,60,37,74,23,88,9,102,164,116,150,117,136,131,122,145,108,159,94)(5,22,146,50,118,65,103,93,75,121,47,149,19,8,160,36,132,64,104,79,89,107,61,135,33,163)(6,120,105,21,48,78,147,148,90,49,20,106,119,7,62,77,161,134,91,35,34,92,133,162,76,63)(14,168)(28,154)(42,140)(56,126)(70,112)(84,98), (1,107,136,154,102,19,55,53,21,100,143,134,109,12,75,46,28,80,163,127,129,161,82,26,48,73)(2,45,103,126,22,38,110,106,29,18,117,99,49,11,137,79,56,160,144,72,76,153,164,52,83,133)(3,152,57,98,111,44,165,159,50,105,91,64,145,10,30,125,84,71,138,17,23,132,77,78,118,37)(4,90,24,70,31,63,51,43,58,36,65,16,85,9,92,158,112,151,119,131,139,124,146,104,166,97)(5,15,147,42,120,69,93,96,66,123,39,150,25,8,167,35,140,62,113,89,86,116,59,130,32,157)(6,122,114,14,40,88,148,149,87,41,13,115,121,7,60,68,168,142,94,34,33,95,141,156,67,61)(20,162)(27,155)(47,135)(54,128)(74,108)(81,101), (13,160)(14,161)(15,162)(16,163)(17,164)(18,165)(19,166)(20,167)(21,168)(22,156)(23,157)(24,158)(25,159)(26,151)(27,152)(28,153)(29,154)(30,155)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,142)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,120)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,129)(62,117)(63,118)(64,119)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,102)(79,103)(80,91)(81,92)(82,93)(83,94)(84,95)(85,96)(86,97)(87,98)(88,99)(89,100)(90,101));