#34116539

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	Q₃₂×C₁₅₇ ⋊ C₄
Sylow	C₁₅₇ ⋊ #2179

Subgroups

	subgroup	G/H
center	C₂	D₈×#10597845
commutator	C₈×C₁₅₇	C₂²×C₄
Frattini	C₈	C₂²×#10597845
Fitting	Q₃₂×C₁₅₇	C₄
socle	C₂×C₁₅₇	C₄×D₈
prime radical	#23757706	C₂
cosocle	C₈×D₁₅₇	C₂³
hypercenter	Q₃₂	#10597845
hypocenter	C₁₅₇	#2179
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 189
transitive degree	μ_t(G) = 5024

A minimal faithful permutation representation — 3 generators on 189 points:

(1,2,7,3)(4,11,14,8)(5,16,12,6)(9,24,23,17)(10,29,25,15)(13,20,27,21)(18,41,36,30)(19,46,42,28)(22,37,40,34)(26,33,44,38)(31,62,53,47)(32,67,63,45)(35,58,57,51)(39,54,61,55)(43,50,65,59)(48,87,74,68)(49,92,88,66)(52,83,78,72)(56,79,82,76)(60,75,86,80)(64,71,90,84)(69,116,99,93)(70,121,117,91)(73,112,103,97)(77,108,107,101)(81,104,111,105)(85,100,115,109)(89,96,119,113)(94,142,128,122)(95,147,143,120)(98,140,132,126)(102,138,135,130)(106,136,131,134)(110,133,127,137)(114,129,123,139)(118,125,145,141)(124,149,152,146)(144,148,154,150)(151,155,156,153)(158,161,163,176,164,177,180,162)(159,166,168,178,170,167,169,185)(160,172,173,181,175,189,174,179)(165,182,183,171,184,186,187,188);(2,4)(3,6)(5,9)(7,13)(8,15)(10,18)(11,12)(14,22)(16,26)(17,28)(19,31)(20,25)(21,24)(23,35)(27,39)(29,43)(30,45)(32,48)(33,42)(34,41)(36,52)(37,38)(40,56)(44,60)(46,64)(47,66)(49,69)(50,63)(51,62)(53,73)(54,59)(55,58)(57,77)(61,81)(65,85)(67,89)(68,91)(70,94)(71,88)(72,87)(74,98)(75,84)(76,83)(78,102)(79,80)(82,106)(86,110)(90,114)(92,118)(93,120)(95,123)(96,117)(97,116)(99,127)(100,113)(101,112)(103,131)(104,109)(105,108)(107,135)(111,132)(115,128)(119,124)(121,144)(122,146)(125,143)(126,142)(129,141)(130,140)(133,139)(134,138)(136,137)(145,151)(147,153)(148,152)(149,150)(154,157)(155,156)(158,160)(159,165)(161,172)(162,179)(163,173)(164,175)(166,182)(167,186)(168,183)(169,187)(170,184)(171,178)(174,180)(176,181)(177,189)(185,188);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,91)(53,72,73,87)(57,76,77,83)(61,80,81,79)(65,84,85,75)(67,88,89,71)(70,93,94,120)(74,97,98,116)(78,101,102,112)(82,105,106,108)(86,109,110,104)(90,113,114,100)(92,117,118,96)(95,122,123,146)(99,126,127,142)(103,130,131,140)(107,134,135,138)(111,137,132,136)(115,139,128,133)(119,141,124,129)(121,143,144,125)(145,150,151,149)(147,152,153,148)(154,156,157,155)(158,159,163,168,164,170,180,169)(160,171,173,186,175,188,174,182)(161,167,176,185,177,166,162,178)(165,181,183,189,184,179,187,172)

Polycyclic presentation:

2^7,157|a^2=e,c^2=e,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,h^a=h^129,d^b=df,f^b=fg,h^b=h^156,h^c=h^129,h^e=h^156

Conjugacy classes & character table

461 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 20913);