#27305132

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×M₁₂₈ ⋊ C₂
Sylow	C₅₃ ⋊ #13737

Subgroups

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₅₃
commutator, socle	C₂×C₅₃	C₂²×C₃₂
Fitting	C₅₃×M₁₂₈	C₂
prime radical	#11719945	C₁₆
cosocle	C₃₂×C₅₃	C₂³
hypercenter	M₁₂₈	D₅₃
hypocenter	C₅₃	#13737
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₃₂ ◁ M₁₂₈

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 181
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 181 points:

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

Polycyclic presentation:

2^8,53|a^2=d,d^2=e,e^2=f,f^2=g,g^2=h,i^a=i^52,c^b=ch,i^c=i^52

Conjugacy classes & character table

2240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90608);