#27271977

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×C₆₄ ⋊ C₂
Sylow	C₅₃ ⋊ #7539

Subgroups

	subgroup	G/H
center	C₄	D₁₆₉₆
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
socle	C₂×C₅₃	#1415
prime radical	#20277024	C₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	#7539
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

88 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:

(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)(54,56)(55,63)(57,76)(58,71)(59,86)(60,91)(61,94)(62,74)(64,102)(65,96)(66,108)(67,110)(68,100)(69,75)(70,79)(72,77)(73,87)(78,127)(80,139)(81,142)(82,130)(83,144)(84,122)(85,97)(88,150)(89,148)(90,98)(92,152)(93,99)(95,101)(103,162)(104,165)(105,164)(106,167)(107,118)(109,169)(111,156)(112,123)(113,138)(114,141)(115,126)(116,172)(117,137)(119,124)(120,133)(121,125)(128,131)(129,135)(132,147)(134,178)(136,154)(140,181)(143,155)(145,168)(146,158)(149,157)(151,160)(153,163)(159,161)(166,173)(170,176)(171,180)(174,175)(177,179);(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)(54,55,58,65,62,68,61,93)(56,69,71,112,74,115,94,114)(57,75,78,123,82,126,81,141)(59,85,83,143,89,146,88,149)(60,90,84,111,92,151,87,67)(63,95,96,117,100,154,99,70)(64,101,103,137,105,136,153,79)(66,107,106,133,109,132,145,77)(72,119,118,173,120,174,147,165)(73,121,91,113,122,170,152,171)(76,108,127,167,130,169,142,168)(80,138,134,176,140,180,135,125)(86,102,144,162,148,164,150,163)(97,139,155,178,158,181,157,129)(98,159,156,177,160,116,110,128)(104,131,124,161,166,179,175,172)

Polycyclic presentation:

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

Conjugacy classes & character table

3398 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57453);