#34116853

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	#123×C₁₅₇	C₄
prime radical	#23757781	C₂
cosocle	C₂×Dic₁₅₇	C₂⁴
hypercenter	#123	#10597845
hypocenter	C₁₅₇	#2929
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ #123

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:

(1,2,6,12,22,36,54,76,102,132,103,73,97,125,146,113,81,53,72,96,124,136,106,80,58,40,26,13,3,7,15,25,39,57,79,105,135,152,133,99,127,147,119,87,59,35,50,70,94,122,141,112,86,64,43,23,9,17,29,45,63,85,111,140,154,156,150,129,144,116,90,65,41,21,32,48,68,92,120,145,118,89,61,37,19,31,47,67,91,117,131,149,155,157,153,139,110,84,62,44,27,11,18,30,46,66,88,114,143,115,83,55,33,49,69,93,121,137,101,128,148,151,134,104,78,56,38,24,14,5,8,16,28,42,60,82,108,138,109,77,51,71,95,123,142,107,75,98,126,130,100,74,52,34,20,10,4)(163,168)(167,173)(169,175)(174,177);(2,5,4,3)(6,11,10,9)(7,8,14,13)(12,21,20,19)(15,18,24,23)(16,27,26,17)(22,35,34,33)(25,32,38,37)(28,41,40,31)(29,30,44,43)(36,53,52,51)(39,50,56,55)(42,59,58,49)(45,48,62,61)(46,65,64,47)(54,75,74,73)(57,72,78,77)(60,81,80,71)(63,70,84,83)(66,87,86,69)(67,68,90,89)(76,101,100,99)(79,98,104,103)(82,107,106,97)(85,96,110,109)(88,113,112,95)(91,94,116,115)(92,119,118,93)(102,131,130,129)(105,128,134,133)(108,137,136,127)(111,126,139,132)(114,142,141,125)(117,124,144,138)(120,146,145,123)(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);(2,4)(3,5)(6,10)(7,14)(8,13)(9,11)(12,20)(15,24)(16,26)(17,27)(18,23)(19,21)(22,34)(25,38)(28,40)(29,44)(30,43)(31,41)(32,37)(33,35)(36,52)(39,56)(42,58)(45,62)(46,64)(47,65)(48,61)(49,59)(50,55)(51,53)(54,74)(57,78)(60,80)(63,84)(66,86)(67,90)(68,89)(69,87)(70,83)(71,81)(72,77)(73,75)(76,100)(79,104)(82,106)(85,110)(88,112)(91,116)(92,118)(93,119)(94,115)(95,113)(96,109)(97,107)(98,103)(99,101)(102,130)(105,134)(108,136)(111,139)(114,141)(117,144)(120,145)(121,147)(122,143)(123,146)(124,138)(125,142)(126,132)(127,137)(128,133)(129,131)(135,151)(140,153)(148,152)(149,150)(154,157)(155,156)(159,161);(2,3,4,5)(6,9,10,11)(7,13,14,8)(12,19,20,21)(15,23,24,18)(16,17,26,27)(22,33,34,35)(25,37,38,32)(28,31,40,41)(29,43,44,30)(36,51,52,53)(39,55,56,50)(42,49,58,59)(45,61,62,48)(46,47,64,65)(54,73,74,75)(57,77,78,72)(60,71,80,81)(63,83,84,70)(66,69,86,87)(67,89,90,68)(76,99,100,101)(79,103,104,98)(82,97,106,107)(85,109,110,96)(88,95,112,113)(91,115,116,94)(92,93,118,119)(102,129,130,131)(105,133,134,128)(108,127,136,137)(111,132,139,126)(114,125,141,142)(117,138,144,124)(120,123,145,146)(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^b=h^156,h^c=h^129,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, 21227);