C₂²×C₅₃⋊#401 #27550386

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×C₅₃⋊#401

Splittings

Sylow

C₅₃ ⋊ C₂²×#401

Subgroups

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂²×#95×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂²×#95	D₅₃
hypocenter	C₅₃	C₂²×#401
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂²×#95

2942 conjugacy classes of subgroups; only the 1179 normal subgroups are shown.

normal subgroups only

This group has 2942 conjugacy classes of subgroups — too many to draw. Only its 1179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 13568

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

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 335862);