D₃×#12953 #72270097

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₃×#12953

Splittings

Fitting	C₃×#12953 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12953

Subgroups

	subgroup	G/H
center	C₂	D₃×#1864
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12953	C₂
socle	C₂×C₃	C₂×#1864
prime radical	C₂²×D₃×D₈	C₂²
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#12953	D₃
hypocenter	C₃	C₂×#12953
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #12953

2264 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)(7,18)(8,19)(10,21)(11,23)(13,24)(14,26)(16,29)(17,31)(20,32)(22,34)(25,35)(27,33)(28,30);(4,6,7,15,19,28,32,31,9,16,20,29,8,17,18,30)(5,10,11,21,24,27,35,34,12,14,25,26,13,22,23,33);(4,5)(6,14,16,10)(7,11)(8,13)(9,12)(15,26,29,21)(17,27,28,22)(18,23)(19,24)(20,25)(30,34,31,33)(32,35)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495739, D₃×#500940
maximal normal subgroup of	D₃×#495739, D₃×#500940

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

D3×#12953 #72270097