D₃×#12949 #72270087

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

Aliases

direct product

D₃×#12949

Splittings

Fitting	C₃×#12949 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12949

Subgroups

	subgroup	G/H
center	C₂	D₃×#1864
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12949	C₂
socle	C₂×C₃	C₂×#1864
prime radical	C₂×D₃×#135	C₂²
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#12949	D₃
hypocenter	C₃	C₂×#12949
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #12949

2124 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)(6,15,17,16)(7,19)(8,21)(10,22,14,23)(11,25)(13,27)(18,31,29,30)(20,32)(24,34,28,33)(26,35);(4,6,7,16,8,18,19,31,9,17,20,15,21,29,32,30)(5,10,11,23,13,24,25,34,12,14,26,22,27,28,35,33);(4,5)(6,14,17,10)(7,11)(8,13)(9,12)(15,23,16,22)(18,28,29,24)(19,25)(20,26)(21,27)(30,34,31,33)(32,35)

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

D3×#12949 #72270087