#27590904

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #36708

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2848	C₂
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	#2848	D₅₃
hypocenter	C₅₃	#36708
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2848

2326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 5 generators on 81 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)(59,63)(62,65)(67,73)(71,74)(72,77)(80,81);(60,64)(62,65)(66,69,70,79)(67,72,73,77)(68,75,76,78)(71,80,74,81);(55,57)(60,64)(62,65);(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)(59,62)(61,64)(63,65)(66,68,70,76)(67,71,73,74)(69,78,79,75)(72,81,77,80);(54,55)(56,57)(58,59)(60,62)(61,63)(64,65)(66,67,70,73)(68,74,76,71)(69,77,79,72)(75,80,78,81)

Polycyclic presentation:

2^8,53|a^2=f,b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,i^b=i^52

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 376380);