D₃×#33638 #72271298

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

D₃×#33638

Splittings

Fitting	C₃×#33638 ⋊ C₂
Sylow	C₃ ⋊ C₂×#33638

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃×#476
commutator, cosocle	C₂×C₃×C₈	C₂⁵
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#33638	C₂
socle	C₂×C₃	C₂²×#476
prime radical	C₂×D₃×#135	C₂²
hypercenter	#33638	D₃
hypocenter	C₃	C₂×#33638
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #33638

2944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 35 points:

(1,2,3)(5,11)(10,26)(12,28)(13,23)(14,29)(20,35)(25,34)(27,30);(2,3);(6,16,18,31)(7,15)(8,24)(10,25,26,34)(12,20)(13,23)(14,30,29,27)(17,33,21,19)(22,32)(28,35);(4,6,15,31,24,33,7,21,9,18,32,16,8,19,22,17)(5,10,20,34,23,30,12,29,11,26,35,25,13,27,28,14);(4,5)(6,14,18,29)(7,20)(8,23)(9,11)(10,17,26,21)(12,15)(13,24)(16,27,31,30)(19,34,33,25)(22,35)(28,32)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#7623148, D₃×#500940
maximal normal subgroup of	D₃×#7623148, D₃×#500940

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3)(5,11)(10,26)(12,28)(13,23)(14,29)(20,35)(25,34)(27,30), (2,3), (6,16,18,31)(7,15)(8,24)(10,25,26,34)(12,20)(13,23)(14,30,29,27)(17,33,21,19)(22,32)(28,35), (4,6,15,31,24,33,7,21,9,18,32,16,8,19,22,17)(5,10,20,34,23,30,12,29,11,26,35,25,13,27,28,14), (4,5)(6,14,18,29)(7,20)(8,23)(9,11)(10,17,26,21)(12,15)(13,24)(16,27,31,30)(19,34,33,25)(22,35)(28,32));

D3×#33638 #72271298