#27552674

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2407 ⋊ C₂
Sylow	C₅₃ ⋊ #34727

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2407	C₂
hypercenter	#2407	D₅₃
hypocenter	C₅₃	#34727
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2407

2736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 73 points:

(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)(58,60,61,64)(59,63,62,65)(66,69,70,68)(67,71,72,73);(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)(55,57)(66,68,70,69)(67,73,72,71);(55,57)(67,72)(71,73);(54,55)(56,57)(59,62)(63,65)(66,68,70,69)(67,71,72,73);(58,59)(60,63)(61,62)(64,65)(66,67,70,72)(68,73,69,71)

Polycyclic presentation:

2^8,53|a^2=g,b^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^b=dh,i^e=i^52

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 338150);