#27413889

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2093 ⋊ C₂
Sylow	C₅₃ ⋊ #23730

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2093	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×#11719947	C₂²
hypercenter	#2093	D₅₃
hypocenter	C₅₃	#23730
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2093

1126 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)(66,68,71,78)(67,74,75,81)(69,76,70,77)(72,79,73,80);(55,57)(67,73)(69,70)(72,75)(74,80)(76,77)(79,81);(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)(54,55)(56,57)(58,60,61,65)(59,62,63,64)(67,72)(69,70)(73,75)(74,79)(76,77)(80,81);(58,59,61,63)(60,64,65,62)(66,67,69,73,71,75,70,72)(68,74,76,80,78,81,77,79)

Polycyclic presentation:

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

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 199365);