D₃×#12981 #72270375

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₃×#12981

Splittings

Fitting	C₃×#12981 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12981

Subgroups

	subgroup	G/H
center	C₈	D₃×#120
commutator	C₂³×C₃	C₂³×C₈
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12981	C₂
socle	C₂×C₃	C₂×#1869
prime radical	D₃×#525	C₄
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#12981	D₃
hypocenter	C₃	C₂×#12981
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #12981

2732 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:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495505
maximal normal subgroup of	D₃×#495505

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

D3×#12981 #72270375