C₃×#11714175 #72275049

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#11714175

Splittings

Fitting

C₃×#3025 ⋊ 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×#3025	D₃
hypercenter	C₃×C₄	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

1206 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 1206 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)(4,13,7,6)(5,18,11,16)(8,14,15,12)(9,19,17,10);(6,15,13,12,7,14)(10,17,19,16,11,18);(4,5,8,9)(6,10,12,16)(7,11,15,17)(13,19,14,18)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72275057, D₃×#11714175, C₃×#72217027
maximal normal subgroup of	#72275057, D₃×#11714175, 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)(4,13,7,6)(5,18,11,16)(8,14,15,12)(9,19,17,10), (6,15,13,12,7,14)(10,17,19,16,11,18), (4,5,8,9)(6,10,12,16)(7,11,15,17)(13,19,14,18));