D₃×#12873 #72269911

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

Aliases

direct product

D₃×#12873

Splittings

Fitting	C₃×#12873 ⋊ C₂
Sylow	C₃ ⋊ C₂×#12873

Subgroups

	subgroup	G/H
center	C₂	D₃×#1823
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12873	C₂
socle	C₂×C₃	C₂×#1823
prime radical	C₂×D₃×#41	D₄
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#12873	D₃
hypocenter	C₃	C₂×#12873
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #12873

2044 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)(6,15,17,31)(7,20)(9,22)(10,23,14,28)(11,26)(12,27)(16,18,32,33)(19,34)(21,35)(24,30,29,25);(4,6,7,16,9,18,20,31,8,17,21,32,22,33,35,15)(5,10,11,24,12,25,26,23,13,14,19,29,27,30,34,28);(4,5)(6,14)(7,19)(8,13)(9,12)(10,17)(11,21)(15,28)(16,24)(18,30)(20,34)(22,27)(23,31)(25,33)(26,35)(29,32)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#494058, D₃×#500349
maximal normal subgroup of	D₃×#494058, D₃×#500349

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

D3×#12873 #72269911