D₃×#11921467 #72241589

order	6912
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

D₃×#11921467

Splittings

Fitting	C₃×H(3,𝔽₄) ⋊ C₂×C₃⋊D₃
derived	C₃×#5450⋊C₃ ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×#5450⋊C₃	C₂²
Frattini	C₂²	D₃×A₄⋊S₄
Fitting	C₃×H(3,𝔽₄)	C₂×C₃⋊D₃
socle	C₂²×C₃	C₂×A₄⋊S₄
LCS	G ▷ C₃×#5450⋊C₃
UCS	C₁

1606 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 40 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:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

D₃×AΓL(2,𝔽₄)

GAP

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

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