D₃×#92391 #72243797

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product

D₃×#92391

Splittings

Fitting, Sylow	C₃×H(5,𝔽₃) ⋊ C₂²
derived	C₃⁴ ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃⁴	C₂²×C₃²
Frattini	C₃	C₃²×D₃×C₃⋊D₃
Fitting	C₃×H(5,𝔽₃)	C₂²
socle	C₃²	C₂×C₃²×C₃⋊D₃
LCS	G ▷ C₃⁴
UCS	C₁

1897 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1897 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 30
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(5,14,17,8,9,16)(6,11)(7,13)(10,27,26,19,18,29)(12,21)(15,28,25,20,24,30)(22,23);(2,3)(4,6,22)(5,10,28)(7,12,21)(8,15,29)(9,18,30)(11,13,23)(14,24,19)(16,25,27)(17,26,20);(4,5,8)(6,10,15)(7,9,16)(11,20,19)(12,18,25)(13,17,14)(21,30,27)(22,28,29)(23,26,24)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243811, D₃×#13091647, D₃×#13091644, …
maximal normal subgroup of	D₃×#66432⋊C₃

GAP

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

gap> G := Group((1,2,3)(5,14,17,8,9,16)(6,11)(7,13)(10,27,26,19,18,29)(12,21)(15,28,25,20,24,30)(22,23), (2,3)(4,6,22)(5,10,28)(7,12,21)(8,15,29)(9,18,30)(11,13,23)(14,24,19)(16,25,27)(17,26,20), (4,5,8)(6,10,15)(7,9,16)(11,20,19)(12,18,25)(13,17,14)(21,30,27)(22,28,29)(23,26,24));