#27539303

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2234 ⋊ C₂
Sylow	C₅₃ ⋊ #33686

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2234	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20285358	C₂
hypercenter	#2234	D₅₃
hypocenter	C₅₃	#33686
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2234

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|c^2=h,d^2=eh,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^c=fg,g^c=gh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324779);