#34115109

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	#106×C₁₅₇ ⋊ C₄
Sylow	C₁₅₇ ⋊ #1340

Subgroups

	subgroup	G/H
center	C₂	#23758125
commutator	C₂×C₈×C₁₅₇	C₂×C₄
Frattini	C₂×C₈	C₂×#10597845
Fitting	#106×C₁₅₇	C₄
socle	C₂×C₁₅₇	#327
prime radical	C₂²×D₁₅₇	D₈
cosocle	C₂×#14578937	C₂²
hypercenter	#106	#10597845
hypocenter	C₁₅₇	#1340
LCS	G ▷ C₂×C₈×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 189 points:

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

Polycyclic presentation:

2^7,157|a^2=d,b^2=g,c^2=fg,e^2=g,f^2=g,b^a=bc,c^a=ce,f^a=fg,h^a=h^129,c^b=cf,d^b=defg,e^b=eg,f^b=fg,h^b=h^156,d^c=dg,h^d=h^156

Conjugacy classes & character table

566 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 19483);