#27271980

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×M₁₂₈ ⋊ C₂
Sylow	C₅₃ ⋊ #7540

Subgroups

	subgroup	G/H
center	C₂	#20272894
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×M₁₂₈	C₂
socle	C₂×C₅₃	#1415
prime radical	#20277024	C₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	M₁₂₈	D₅₃
hypocenter	C₅₃	#7540
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 117 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)(54,55)(56,67)(57,73)(58,77)(59,80)(60,62)(61,81)(63,72)(64,74)(65,78)(66,83)(68,87)(69,102)(70,106)(71,84)(75,112)(76,82)(79,113)(85,116)(86,95)(88,111)(89,114)(90,100)(91,99)(92,105)(93,97)(94,96)(98,103)(101,115)(104,117)(107,110)(108,109);(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)(55,61,62,83)(56,66)(57,72)(58,74)(59,78)(63,89,87,97)(64,91,88,96)(65,92,80,86)(67,73,84,82)(68,76)(69,101)(70,105)(71,81)(75,100)(77,85,113,115)(79,111)(90,102,109,117)(93,94)(95,103)(98,112,106,110)(99,114)(104,116)(107,108)

Polycyclic presentation:

2^8,53|a^2=d,c^2=efh,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=ceh,d^b=dh,e^b=ef,f^b=fg,g^b=gh,i^b=i^52

Conjugacy classes & character table

2126 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57456);