D₃×#12966 #72270747

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₃×#12966

Splittings

Fitting	C₃×#12966 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12966

Subgroups

	subgroup	G/H
center	C₂²	D₃×B(4,𝔽₂)
commutator	C₂×C₃×D₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12966	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	C₂×D₃×#127	C₂²
cosocle	C₂²×C₃×D₄	C₂⁴
hypercenter	#12966	D₃
hypocenter	C₃	C₂×#12966
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #12966

3184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

(2,3);(1,2,3)(8,11,12,10)(9,15,16,21)(13,18,20,19)(14,17,22,23);(4,6,7,5)(8,10)(9,14)(11,20)(12,18)(13,19)(15,16)(17,21)(22,23);(4,5,7,6)(8,9,13,17)(10,14,19,21)(11,15,18,22)(12,16,20,23)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500177, D₃×#501097, D₃×#512119
maximal normal subgroup of	D₃×#500177, D₃×#501097, D₃×#512119

GAP

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

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

D3×#12966 #72270747