C₃×#11714184 #72274999

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃×#11714184

Splittings

Fitting

C₃×#3027 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂×#1020⋊C₂
commutator, cosocle, hypocenter	#997	C₂³×C₃
Frattini	C₂³	C₂²×C₃×S₄
Fitting	C₃×#3027	D₃
hypercenter	C₃×C₄	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

1200 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#11714184, #72275031, C₃×#72217027
maximal normal subgroup of	D₃×#11714184, #72275031, C₃×#72217027

GAP

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

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