#34116852

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₁₅₇ ⋊ #2929

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597845
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	C₂²⋊Q₈×C₁₅₇	C₄
prime radical	#23757786	C₂
cosocle	C₂×Dic₁₅₇	C₂⁴
hypercenter	C₂²⋊Q₈	#10597845
hypocenter	C₁₅₇	#2929
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 177
transitive degree	μ_t(G) = 5024

A minimal faithful permutation representation — 4 generators on 177 points:

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

Polycyclic presentation:

2^7,157|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,h^a=h^28,h^c=h^129,h^d=h^156,h^f=h^156

Conjugacy classes & character table

602 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 21226);