D₃×#13077 #72270472

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₃×#13077

Splittings

Fitting	C₃×#13077 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13077

Subgroups

	subgroup	G/H
center	C₂	D₃×#1893
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13077	C₂
socle	C₂×C₃	C₂×#1893
prime radical	D₃×#3060	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#13077	D₃
hypocenter	C₃	C₂×#13077
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13077

2612 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)(6,14,16,29)(7,18)(9,20)(10,21,23,28)(11,25)(12,27)(15,17,30,31)(19,32)(22,34,33,24)(26,35);(4,6,7,15,9,17,18,29,8,16,19,30,20,31,32,14)(5,10,11,22,12,24,25,21,13,23,26,33,27,34,35,28);(4,5)(6,10)(7,11)(8,13)(9,12)(14,28)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,29)(30,33)(31,34)(32,35)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500168, D₃×#495573
maximal normal subgroup of	D₃×#500168, D₃×#495573

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

D3×#13077 #72270472