D₃×#13318 #72271647

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₃×#13318

Splittings

Fitting	C₃×#13318 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13318

Subgroups

	subgroup	G/H
center	C₂	D₃×#1999
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13318	C₂
socle	C₂×C₃	C₂×#1999
hypercenter	#13318	D₃
hypocenter	C₃	C₂×#13318
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13318

3040 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)(6,16,18,33)(7,15)(8,21)(10,26,27,32)(11,25)(13,24)(14,28,31,30)(17,35,34,19)(20,29)(22,23);(4,6,15,33,8,19,7,17,9,18,22,16,21,35,23,34)(5,10,25,26,13,28,11,14,12,27,20,32,24,30,29,31);(4,5)(6,14)(7,20)(8,24)(9,12)(10,17)(11,22)(13,21)(15,29)(16,30)(18,31)(19,26)(23,25)(27,34)(28,33)(32,35)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#499269
maximal normal subgroup of	D₃×#499269

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

D3×#13318 #72271647