#27420056

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3013 ⋊ C₂
Sylow	C₅₃ ⋊ #24290

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3013	C₂
socle	C₂²×C₅₃	D₄²
hypercenter	#3013	D₅₃
hypocenter	C₅₃	#24290
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #3013

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(58,61,76,89,63,78,62,77)(59,66,69,83,68,82,67,75)(60,71,74,88,73,87,72,86)(64,79,65,81,80,85,70,84);(55,57)(58,60)(59,65)(61,71)(62,72)(63,73)(64,67)(66,81)(68,70)(69,80)(74,76)(75,79)(77,86)(78,87)(82,84)(83,85)(88,89);(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)(58,60,62,72,63,73,76,74)(59,64,67,70,68,80,69,65)(61,71,77,86,78,87,89,88)(66,79,75,84,82,85,83,81);(54,55)(56,57)(58,59)(60,70)(61,75)(62,69)(63,68)(64,72)(65,73)(66,77)(67,76)(71,85)(74,80)(78,83)(79,87)(81,88)(82,89)(84,86)

Polycyclic presentation:

2^8,53|b^2=e,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,i^b=i^52,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, 205532);