#27539180

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2226 ⋊ C₂
Sylow	C₅₃ ⋊ #33676

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2226	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133236	C₂²
hypercenter	#2226	D₅₃
hypocenter	C₅₃	#33676
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

548 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)(62,65,68,67)(63,71,73,81)(64,74,77,76)(66,78,80,79)(69,83,85,91)(70,86,87,72)(75,88,90,89)(82,92,93,84);(63,70)(65,67)(66,79)(69,82)(71,72)(73,87)(74,76)(75,89)(78,80)(81,86)(83,84)(85,93)(88,90)(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)(55,58)(59,61)(62,64,68,77)(63,69,73,85)(65,74,67,76)(66,75,80,90)(70,82,87,93)(71,83,81,91)(72,84,86,92)(78,88,79,89);(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)(62,63,68,73)(64,69,77,85)(65,71,67,81)(66,70,80,87)(72,78,86,79)(74,83,76,91)(75,82,90,93)(84,88,92,89)

Polycyclic presentation:

2^8,53|a^2=h,b^2=h,d^2=eh,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,i^a=i^52,i^b=i^52,d^c=dh,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, 324656);