#27542178

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2251 ⋊ C₂
Sylow	C₅₃ ⋊ #33969

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2251	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133158	C₂²
hypercenter	#2251	D₅₃
hypocenter	C₅₃	#33969
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2251

504 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)(54,57,58,56)(55,60,59,61)(62,65,68,67)(63,70,72,82)(64,75,74,83)(66,77,81,80)(69,86,85,87)(71,78,89,92)(73,91,90,88)(76,79,84,93);(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)(54,56,58,57)(55,60,59,61)(62,64)(63,69)(65,75)(66,79)(67,83)(68,74)(70,86)(71,88)(72,85)(73,78)(76,80)(77,84)(81,93)(82,87)(89,91)(90,92);(54,55,58,59)(56,60,57,61)(64,74)(69,85)(73,90)(75,83)(76,84)(79,93)(86,87)(88,91);(55,59)(60,61)(62,63,68,72)(64,73,74,90)(65,70,67,82)(66,78,81,92)(69,84,85,76)(71,77,89,80)(75,91,83,88)(79,86,93,87)

Polycyclic presentation:

2^8,53|a^2=h,b^2=e,c^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, 327654);