D₃×#92300 #72224525

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

Aliases

direct product

D₃×#92300

Splittings

Fitting, Sylow	C₃×C₃²⋊H(3,𝔽₃) ⋊ C₂²
derived	C₃²×H(3,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₃×C₃⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle, hypocenter	C₃²×H(3,𝔽₃)	C₂²×C₃
Frattini	C₃²	C₃×D₃×C₃⋊D₃
Fitting	C₃×C₃²⋊H(3,𝔽₃)	C₂²
socle	C₃³	C₂×C₃×C₃⋊D₃
LCS	G ▷ C₃²×H(3,𝔽₃)
UCS	C₁ ◁ C₃

1547 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 1547 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 21
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#11757749, D₃×#11757730, D₃×(((C₃⋊D₃)⋊C₃)⋊D₃)⋊C₃, …
maximal normal subgroup of	D₃×#11757749, D₃×#11757730, D₃×(((C₃⋊D₃)⋊C₃)⋊D₃)⋊C₃

GAP

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

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