#27314949

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #14642

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1439	C₂
socle, cosocle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	#1439	D₅₃
hypocenter	C₅₃	#14642
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ #1439

1226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 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)(60,64)(62,65)(66,69,70,68)(67,71,72,73)(75,78)(76,80);(59,63)(62,65)(67,72)(68,69)(74,76)(75,79)(77,80)(78,81);(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)(66,68,70,69)(67,71,72,73)(74,75,77,78)(76,79,80,81);(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)(54,55,56,57)(58,59)(60,62)(61,63)(64,65)(66,67)(68,73)(69,71)(70,72)(75,78)(79,81)

Polycyclic presentation:

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

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, 100425);