#27550212

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #34526

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₄×#125×C₅₃	C₂
socle, prime radical, cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#34526
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄×#125

1800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 5 generators on 89 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,57,55)(75,80)(79,82)(81,85)(88,89);(59,65)(63,72)(64,68)(71,73)(74,77,78,87)(75,81,80,85)(76,84,83,86)(79,88,82,89);(54,55,57,56)(58,61,62,70)(59,64,65,68)(60,66,67,69)(63,71,72,73)(75,80)(76,83)(81,85)(84,86);(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)(58,60,62,67)(59,63,65,72)(61,69,70,66)(64,73,68,71)(74,76)(75,79)(77,86)(78,83)(80,82)(81,89)(84,87)(85,88);(54,55,57,56)(58,59)(60,63)(61,68)(62,65)(64,70)(66,73)(67,72)(69,71)(74,75)(76,82)(77,85)(78,80)(79,83)(81,87)(84,88)(86,89)

Polycyclic presentation:

2^8,53|a^2=h,b^2=fg,c^2=fgh,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,i^b=i^52

Conjugacy classes & character table

1532 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 335688);