#27447580

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3082 ⋊ C₂
Sylow	C₅₃ ⋊ #26404

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3082	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#11719945	C₂³
hypercenter	#3082	D₅₃
hypocenter	C₅₃	#26404
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #3082

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(62,65)(63,70)(64,76)(66,80)(67,81)(68,85)(69,72)(71,90)(73,89)(74,88)(75,82)(77,92)(78,91)(79,83)(84,87)(86,93);(54,57,58,56)(55,59,60,61)(62,65)(63,69)(64,75)(66,79)(67,81)(68,84)(70,72)(71,89)(73,90)(74,88)(76,82)(77,91)(78,92)(80,83)(85,87)(86,93);(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)(54,56,58,57)(55,59,60,61)(62,64,67,78)(63,68,73,87)(65,76,81,91)(66,82,83,77)(69,88,90,93)(70,85,89,84)(71,86,72,74)(75,79,92,80);(54,55,58,60)(56,59,57,61)(62,63)(64,74)(65,69)(66,72)(67,73)(68,77)(70,79)(71,83)(75,88)(76,85)(78,86)(80,89)(81,90)(82,87)(84,91)(92,93)

Polycyclic presentation:

2^8,53|a^2=f,b^2=fh,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,i^b=i^52,g^c=gh,e^d=eh,i^d=i^52

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 233056);