#27272004

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×D₆₄ ⋊ C₂
Sylow	C₅₃ ⋊ D₁₂₈

Subgroups

	subgroup	G/H
center	C₂	#20272927
commutator, cosocle	C₅₃×C₆₄	C₂²
Frattini	C₆₄	C₂×D₅₃
Fitting	C₅₃×D₆₄	C₂
socle	C₂×C₅₃	D₆₄
hypercenter	D₆₄	D₅₃
hypocenter	C₅₃	D₁₂₈
LCS	G ▷ C₅₃×C₆₄ ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄ ◁ D₆₄

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 181 points:

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

Polycyclic presentation:

2^8,53|c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,c^b=cd,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^52

Conjugacy classes & character table

977 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57480);