C₃×#11923785 #72245669

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃×#11923785

Splittings

Fitting

C₃²×#585 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	C₃⋊#1020
commutator, cosocle, hypocenter	C₃×#997	C₂²×C₃
Frattini	C₂	C₂×C₃×C₃⋊#1020
Fitting	C₃²×#585	D₃
socle	C₂×C₃²	C₂×#1020
LCS	G ▷ C₃×#997
UCS	C₁ ◁ C₃×C₄

1477 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 1477 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 22
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 22 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245689, C₃×#72215523
maximal normal subgroup of	C₃×#72215523

GAP

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

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