D₃×#92392 #72243725

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

D₃×#92392

Splittings

Fitting, Sylow

C₃×H(5,𝔽₃) ⋊ C₂²

Subgroups

	subgroup	G/H
center, Frattini	C₃	C₃²×D₃×C₃⋊D₃
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂²×C₃²
Fitting	C₃×H(5,𝔽₃)	C₂²
socle	C₃²	C₂×C₃²×C₃⋊D₃
hypercenter	H(3,𝔽₃)	D₃×C₃⋊D₃
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₃ ◁ H(3,𝔽₃)

2016 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 2016 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)(5,9,18)(6,11)(8,15,17)(10,27,21,20,19,28)(12,23)(13,22)(16,29,26,25,24,30);(2,3)(4,5,8)(6,21,26)(7,9,17)(10,24,13)(11,27,29)(12,19,16)(14,18,15)(20,30,23)(22,28,25);(4,6,11)(5,10,20)(7,13,22)(8,16,25)(9,19,27)(12,23,14)(15,24,29)(17,26,30)(18,21,28)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243728, D₃×#13091645, D₃×#66432⋊C₃
maximal normal subgroup of	D₃×#13091645, 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)(5,9,18)(6,11)(8,15,17)(10,27,21,20,19,28)(12,23)(13,22)(16,29,26,25,24,30), (2,3)(4,5,8)(6,21,26)(7,9,17)(10,24,13)(11,27,29)(12,19,16)(14,18,15)(20,30,23)(22,28,25), (4,6,11)(5,10,20)(7,13,22)(8,16,25)(9,19,27)(12,23,14)(15,24,29)(17,26,30)(18,21,28));