#27434278

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2034 ⋊ C₂
Sylow	C₅₃ ⋊ #25344

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2034	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	#20279614	C₂
hypercenter	#2034	D₅₃
hypocenter	C₅₃	#25344
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #2034

882 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)(58,60,61,65)(59,63,62,64)(67,73)(71,79)(72,74)(80,81);(55,57)(67,72)(68,75)(69,78)(71,79)(73,74)(76,77);(54,55)(56,57)(59,62)(63,64)(66,68,69,76,70,77,78,75)(67,71,74,80,73,79,72,81);(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)(58,59,61,62)(60,64,65,63)(66,67,69,74,70,73,78,72)(68,71,76,80,77,79,75,81)

Polycyclic presentation:

2^8,53|a^2=ef,b^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,i^a=i^52,c^b=cg,f^c=fh,g^c=gh,i^d=i^52

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 219754);