D₃×#13076 #72270455

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

Aliases

direct product

D₃×#13076

Splittings

Fitting	C₃×#13076 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13076

Subgroups

	subgroup	G/H
center	C₂	D₃×#1893
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13076	C₂
socle	C₂×C₃	C₂×#1893
prime radical	D₃×#3060	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#13076	D₃
hypocenter	C₃	C₂×#13076
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13076

2596 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)(6,15,17,30)(7,18)(8,20)(10,21,24,22)(11,26)(12,13)(16,31,32,29)(19,33)(23,35,25,34)(27,28);(4,6,7,15,8,16,18,29,9,17,19,30,20,32,33,31)(5,10,11,22,13,25,26,34,14,24,28,21,12,23,27,35);(4,5)(6,10)(7,11)(8,13)(9,14)(12,20)(15,22)(16,25)(17,24)(18,26)(19,28)(21,30)(23,32)(27,33)(29,34)(31,35)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495573
maximal normal subgroup of	D₃×#495573

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

D3×#13076 #72270455