#27422218

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2047 ⋊ C₂
Sylow	C₅₃ ⋊ #24462

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2047	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#16811436	C₂
hypercenter	#2047	D₅₃
hypocenter	C₅₃	#24462
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2047

1302 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)(58,62)(59,60)(61,76)(63,70)(64,65)(66,71)(67,81)(68,73)(69,72)(74,89)(75,85)(77,84)(78,83)(79,87)(80,82)(86,88);(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)(58,61,64,78)(59,66,68,74)(60,71,73,89)(62,76,65,83)(63,77,82,79)(67,85,69,86)(70,84,80,87)(72,88,81,75);(58,60,64,73)(59,65,68,62)(61,75,78,88)(63,81,82,72)(66,84,74,87)(67,80,69,70)(71,77,89,79)(76,85,83,86);(54,55,56,57)(58,59,64,68)(60,70,73,80)(61,74,78,66)(62,72,65,81)(63,69,82,67)(71,87,89,84)(75,83,88,76)(77,85,79,86)

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 207694);