#27428342

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2898 ⋊ C₂
Sylow	C₅₃ ⋊ #24927

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2898	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20284030	C₂
hypercenter	#2898	D₅₃
hypocenter	C₅₃	#24927
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2898

862 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(62,65)(63,70)(64,74)(66,79)(67,80)(68,77)(69,83)(71,87)(72,86)(73,88)(75,91)(76,90)(78,81)(82,89)(84,92)(85,93);(55,59)(58,61)(62,64,67,76)(63,69,72,85)(65,74,80,90)(66,75,81,73)(68,82,71,84)(70,83,86,93)(77,89,87,92)(78,88,79,91);(54,56,57,60)(55,58,59,61)(63,68)(64,73)(66,81)(70,77)(71,72)(74,88)(75,76)(78,79)(82,84)(86,87)(89,92)(90,91);(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)(56,58)(57,59)(60,61)(62,63)(64,69)(65,77)(66,68)(67,72)(70,78)(71,81)(73,84)(74,89)(75,82)(76,85)(79,86)(80,87)(83,88)(90,92)(91,93)

Polycyclic presentation:

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

Conjugacy classes & character table

1187 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 213818);