#27428383

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #24930

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3013	C₂
socle	C₂²×C₅₃	D₄²
hypercenter	#3013	D₅₃
hypocenter	C₅₃	#24930
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #3013

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

Polycyclic presentation:

2^8,53|b^2=fh,d^2=h,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

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, 213859);