#27411199

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2016 ⋊ C₂
Sylow	C₅₃ ⋊ #23425

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2016	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×#11719949	C₂²
hypercenter	#2016	D₅₃
hypocenter	C₅₃	#23425
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2016

1938 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)(64,76)(69,84)(74,85)(75,90)(77,78)(83,89)(88,93)(91,92);(59,61)(62,65,68,82)(63,70,73,87)(64,75,76,90)(66,81,67,80)(69,83,84,89)(71,79,72,86)(74,88,85,93)(77,92,78,91);(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)(55,57)(58,59)(60,61)(62,64)(63,69)(65,75)(66,78)(67,77)(68,76)(70,83)(71,85)(72,74)(73,84)(79,93)(80,92)(81,91)(82,90)(86,88)(87,89);(54,55)(56,57)(62,63,68,73)(64,74,76,85)(65,79,82,86)(66,72,67,71)(69,78,84,77)(70,80,87,81)(75,89,90,83)(88,91,93,92)

Polycyclic presentation:

2^8,53|a^2=h,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,i^b=i^52,e^c=eh,f^c=fh,i^d=i^52

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196675);