#27447098

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2085 ⋊ C₂
Sylow	C₅₃ ⋊ #26372

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2085	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×D₂₁₂	C₂³
hypercenter	#2085	D₅₃
hypocenter	C₅₃	#26372
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2085

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 81 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)(67,72)(69,79)(71,75)(73,74)(76,78)(80,81);(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)(58,61,62,60)(59,63,64,65);(54,55)(56,57)(58,60,62,61)(59,63,64,65)(66,68,70,77)(67,71,74,80)(69,78,79,76)(72,75,73,81);(58,59,62,64)(60,63,61,65)(66,67)(68,75)(69,72)(70,74)(71,76)(73,79)(77,81)(78,80)

Polycyclic presentation:

2^8,53|a^2=f,b^2=fh,c^2=f,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,i^c=i^52,e^d=eh,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, 232574);