D₃×#13417 #72271631

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

Aliases

direct product

D₃×#13417

Splittings

Fitting	C₃×#13417 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13417

Subgroups

	subgroup	G/H
center	C₂	D₃×#2041
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13417	C₂
socle	C₂×C₃	C₂×#2041
prime radical	D₃×#3155	C₂
hypercenter	#13417	D₃
hypocenter	C₃	C₂×#13417
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #13417

2508 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)(7,19)(8,20)(10,22)(11,24)(13,25)(14,27)(16,17)(18,31)(21,32)(23,34)(26,35)(28,33)(29,30);(4,6,7,16,20,30,32,29,9,17,21,15,8,18,19,31)(5,10,11,22,25,28,35,34,12,14,26,27,13,23,24,33);(4,5)(6,14,17,10)(7,11)(8,13)(9,12)(15,22,16,27)(18,28,30,23)(19,24)(20,25)(21,26)(29,33,31,34)(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)(6,15)(7,19)(8,20)(10,22)(11,24)(13,25)(14,27)(16,17)(18,31)(21,32)(23,34)(26,35)(28,33)(29,30), (4,6,7,16,20,30,32,29,9,17,21,15,8,18,19,31)(5,10,11,22,25,28,35,34,12,14,26,27,13,23,24,33), (4,5)(6,14,17,10)(7,11)(8,13)(9,12)(15,22,16,27)(18,28,30,23)(19,24)(20,25)(21,26)(29,33,31,34)(32,35));

D3×#13417 #72271631