#27547283

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2318 ⋊ C₂
Sylow	C₅₃ ⋊ #34395

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₅₃×#2318	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂×#13133188	C₂²
hypercenter	#2318	D₅₃
hypocenter	C₅₃	#34395
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2318

1990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 332759);