D₃×#12011 #72226257

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₃×#12011

Splittings

Fitting	C₃×#12011 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12011

Subgroups

	subgroup	G/H
center	C₈	C₂×D₃×D₈
commutator	C₃×C₈	C₂³×C₈
Frattini	C₄×C₈	C₂³×D₃
Fitting	C₃×#12011	C₂
socle	C₂×C₃	C₂²×#335
prime radical	C₂³×D₃	#108
cosocle	C₃×C₄×C₈	C₂⁴
hypercenter	#12011	D₃
hypocenter	C₃	C₂×#12011
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #12011

1032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3)(4,6,19,33,9,22,25,34,10,18,7,20,24,35,8,21)(5,12,23,32,16,29,13,27,11,26,30,31,15,28,14,17);(5,11)(6,18)(13,14)(15,16)(20,33)(21,34)(22,35)(23,30);(2,3);(4,5,7,13,9,15,8,23,10,11,19,14,24,16,25,30)(6,17,20,29,22,31,21,12,18,27,33,28,35,32,34,26)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#483512
maximal normal subgroup of	D₃×#483512

GAP

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

gap> G := Group((1,2,3)(4,6,19,33,9,22,25,34,10,18,7,20,24,35,8,21)(5,12,23,32,16,29,13,27,11,26,30,31,15,28,14,17), (5,11)(6,18)(13,14)(15,16)(20,33)(21,34)(22,35)(23,30), (2,3), (4,5,7,13,9,15,8,23,10,11,19,14,24,16,25,30)(6,17,20,29,22,31,21,12,18,27,33,28,35,32,34,26));

D3×#12011 #72226257