#27537208

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2163 ⋊ C₂
Sylow	C₅₃ ⋊ #33423

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2163	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2163	D₅₃
hypocenter	C₅₃	#33423
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2163

590 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)(62,64)(63,68)(65,73)(66,74)(67,72)(69,79)(70,80)(71,78)(75,86)(76,84)(77,85)(81,90)(82,88)(83,89)(87,92)(91,93);(54,56,57,60)(55,59,58,61)(64,72)(68,78)(73,84)(74,85)(79,88)(80,89)(86,92)(90,93);(55,58)(59,61)(64,72)(68,78)(73,84)(74,85)(79,88)(80,89)(86,92)(90,93);(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)(62,63,65,69,66,70,75,81,67,71,76,82,77,83,87,91)(64,68,73,79,74,80,86,90,72,78,84,88,85,89,92,93)

Polycyclic presentation:

2^8,53|a^2=f,c^2=e,f^2=g,g^2=h,b^a=be,i^a=i^52,d^b=dh,d^c=dh,i^d=i^52

Conjugacy classes & character table

1556 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322684);