#27539128

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2237 ⋊ C₂
Sylow	C₅₃ ⋊ #33671

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	#20276640	C₂
hypercenter	#2237	D₅₃
hypocenter	C₅₃	#33671
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2237

554 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)(62,65,66,68)(63,71,72,82)(64,75,76,78)(67,79,80,81)(69,84,74,89)(70,86,87,73)(77,91,92,93)(83,90,88,85);(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)(63,70)(65,68)(67,81)(69,83)(71,73)(72,87)(74,88)(75,78)(77,93)(79,80)(82,86)(84,85)(89,90)(91,92);(55,58)(59,61)(62,64,66,76)(63,69,72,74)(65,75,68,78)(67,77,80,92)(70,83,87,88)(71,84,82,89)(73,85,86,90)(79,91,81,93);(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)(64,74)(65,71)(66,72)(67,70)(68,82)(69,76)(73,81)(75,89)(77,88)(78,84)(79,86)(80,87)(83,92)(85,91)(90,93)

Polycyclic presentation:

2^8,53|b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,i^a=i^52,d^c=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, 324604);