#27541850

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2244 ⋊ C₂
Sylow	C₅₃ ⋊ #33937

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2244	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20276747	C₂
hypercenter	#2244	D₅₃
hypocenter	C₅₃	#33937
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2244

494 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)(54,56,57,61)(55,59,58,60)(64,74)(68,73)(75,89)(76,90)(81,84)(82,87)(83,88)(91,92);(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)(63,69)(64,74)(66,79)(67,78)(68,81)(70,85)(71,72)(73,84)(75,91)(77,93)(80,86)(82,88)(83,87)(89,92);(54,55,57,58)(56,60,61,59)(62,64,65,74)(63,68,70,73)(66,75,77,89)(67,76,78,90)(69,84,85,81)(71,82,86,87)(72,83,80,88)(79,92,93,91);(55,58)(59,60)(62,63)(64,73)(65,70)(66,69)(67,80)(68,74)(71,79)(72,78)(75,81)(76,83)(77,85)(82,91)(84,89)(86,93)(87,92)(88,90)

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327326);