#27538910

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2237 ⋊ C₂
Sylow	C₅₃ ⋊ #33647

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2237	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20284298	C₂
hypercenter	#2237	D₅₃
hypocenter	C₅₃	#33647
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2237

584 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)(63,70)(68,81)(69,84)(71,85)(72,79)(80,91)(82,92)(83,89);(54,56,57,60)(55,59,58,61)(63,69)(65,76)(66,77)(68,80)(70,84)(71,72)(73,86)(74,87)(78,90)(79,85)(81,91)(82,83)(88,93)(89,92);(55,58)(59,61)(62,64,67,75)(63,68,70,81)(65,73,78,88)(66,74,77,87)(69,80,84,91)(71,82,85,92)(72,83,79,89)(76,86,90,93);(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)(54,55)(56,59)(57,58)(60,61)(62,63)(64,68)(65,69)(66,79)(67,70)(71,76)(72,77)(73,80)(74,89)(75,81)(78,84)(82,86)(83,87)(85,90)(88,91)(92,93)

Polycyclic presentation:

2^8,53|b^2=h,c^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,i^a=i^52,f^c=fg,g^c=gh,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, 324386);