D₃×#33659 #72270486

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

Aliases

direct product

D₃×#33659

Splittings

Fitting	C₃×#33659 ⋊ C₂
Sylow	C₃ ⋊ C₂×#33659

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃×#481
commutator, cosocle	C₂×C₃×C₈	C₂⁵
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#33659	C₂
socle	C₂×C₃	C₂²×#481
prime radical	D₃×#3060	C₂
hypercenter	#33659	D₃
hypocenter	C₃	C₂×#33659
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #33659

3164 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)(6,15)(12,18)(13,26)(14,16)(17,28)(27,31)(29,30);(2,3);(6,14,15,16)(7,19)(9,21)(10,22,25,23)(12,27)(13,26)(17,30,28,29)(18,31)(20,32)(24,35,33,34);(4,6,7,16,9,17,19,30,8,15,20,14,21,28,32,29)(5,10,12,23,13,24,27,35,11,25,18,22,26,33,31,34);(4,5)(6,10)(7,18)(8,11)(9,13)(12,20)(14,23)(15,25)(16,22)(17,24)(19,31)(21,26)(27,32)(28,33)(29,35)(30,34)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500168, D₃×#7623148
maximal normal subgroup of	D₃×#500168, D₃×#7623148

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

D3×#33659 #72270486