C₃²×D₃×#64424 #72230474

order	15876
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product

C₃²×D₃×#64424

Splittings

Fitting, socle	C₃³×C₇² ⋊ C₂²×C₃
derived	C₃×C₇² ⋊ C₂²×C₃³
Hall	C₃³×#3861 ⋊ C₂²
Sylow	C₇² ⋊ C₂×C₃³×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	D₃×#64424
commutator, cosocle, hypocenter	C₃×C₇²	C₂²×C₃³
Frattini	C₁	G
Fitting, socle	C₃³×C₇²	C₂²×C₃
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₃²

4360 conjugacy classes of subgroups; only the 358 normal subgroups are shown.

normal subgroups only

This group has 4360 conjugacy classes of subgroups — too many to draw. Only its 358 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) = 23
transitive degree	μ_t(G) = 882

A minimal faithful permutation representation — 3 generators on 23 points:

(4,5,6)(7,8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,23);(1,2,3)(8,9)(10,11,13,16,15,14,12);(11,13,15)(12,14,16)(18,20,22)(19,21,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₃×C₃⋊D₃×#64424, C₃×D₃×#13183634
maximal normal subgroup of	D₃×C₃⋊D₃×#64424, C₃×D₃×#13183634

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,5,6)(7,8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,23), (1,2,3)(8,9)(10,11,13,16,15,14,12), (11,13,15)(12,14,16)(18,20,22)(19,21,23));