#27538989

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2239 ⋊ C₂
Sylow	C₅₃ ⋊ #33656

Subgroups

	subgroup	G/H
center	C₂	#481×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2239	C₂
socle	C₂×C₅₃	C₂×#481
prime radical	#20284326	C₂
hypercenter	#2239	D₅₃
hypocenter	C₅₃	#33656
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #2239

556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 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)(54,57,60,59)(55,62,64,74)(56,66,67,75)(58,69,71,73)(61,76,77,78)(63,70,79,85)(65,80,81,82)(68,83,84,72);(54,56,59,75,60,67,57,66)(55,61,62,76,64,77,74,78)(58,72,73,84,71,83,69,68)(63,80,70,81,79,82,85,65);(58,71)(63,79)(65,81)(68,84)(69,73)(70,85)(72,83)(80,82);(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)(56,65)(57,62)(58,70)(59,74)(60,64)(61,68)(63,73)(66,80)(67,81)(69,79)(71,85)(72,78)(75,82)(76,83)(77,84)

Polycyclic presentation:

2^8,53|c^2=eg,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,i^a=i^52,f^b=fh,e^c=eh,f^c=fgh,g^c=gh,i^d=i^52

Conjugacy classes & character table

560 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324465);