D₃×#13005 #72270140

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

Aliases

direct product

D₃×#13005

Splittings

Fitting	C₃×#13005 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13005

Subgroups

	subgroup	G/H
center	C₂	D₃×#1874
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13005	C₂
socle	C₂×C₃	C₂×#1874
prime radical	D₃×#3067	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#13005	D₃
hypocenter	C₃	C₂×#13005
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13005

2784 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)(6,16,19,33)(7,21)(8,23)(10,24,26,31)(11,28)(12,13)(15,30,27,25)(17,34,32,18)(20,29)(22,35);(4,6,7,17,8,18,21,16,9,19,22,32,23,34,35,33)(5,10,11,25,13,27,28,24,14,26,20,30,12,15,29,31);(4,5)(6,15)(7,20)(8,13)(9,14)(10,18)(11,22)(12,23)(16,30)(17,24)(19,27)(21,29)(25,33)(26,34)(28,35)(31,32)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495739
maximal normal subgroup of	D₃×#495739

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

D3×#13005 #72270140