D₃×#13416 #72270503

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₃×#13416

Splittings

Fitting	C₃×#13416 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13416

Subgroups

	subgroup	G/H
center	C₄	D₃×#486
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13416	C₂
socle	C₂×C₃	C₂×#2041
prime radical	D₃×#3150	C₂
hypercenter	#13416	D₃
hypocenter	C₃	C₂×#13416
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #13416

2420 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)(6,17,20,19)(7,23)(11,27,29,28)(12,31)(15,32,30,33)(16,24)(18,35,22,34)(21,26);(4,6,16,34,9,19,23,18,10,20,24,35,8,17,7,22)(5,11,26,33,13,27,12,30,14,29,21,32,25,28,31,15);(4,5)(6,15)(7,21)(8,13)(9,25)(10,14)(11,18)(12,16)(17,33)(19,32)(20,30)(22,29)(23,26)(24,31)(27,34)(28,35)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500168
maximal normal subgroup of	D₃×#500168

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

D3×#13416 #72270503