D₃×#33595 #72270645

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₃×#33595

Splittings

Fitting	C₃×#33595 ⋊ C₂
Sylow	C₃ ⋊ C₂×#33595

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃×#459
commutator, cosocle	C₂×C₃×C₈	C₂⁵
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#33595	C₂
socle	C₂×C₃	C₂²×#459
hypercenter	#33595	D₃
hypocenter	C₃	C₂×#33595
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #33595

4216 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)(10,24)(12,27)(13,28)(14,26)(23,32)(25,31)(29,30);(2,3);(6,15)(7,19)(8,20)(10,23)(12,29)(13,28)(14,31)(16,18)(17,33)(21,35)(22,34)(24,32)(25,26)(27,30);(4,6,7,16,20,22,35,33,9,17,21,34,8,18,19,15)(5,10,12,25,28,14,30,32,11,24,27,31,13,26,29,23);(4,5)(6,14,17,26)(7,12)(8,13)(9,11)(10,22,24,18)(15,25,33,31)(16,32,34,23)(19,29)(20,28)(21,27)(30,35)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#7623148
maximal normal subgroup of	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)(10,24)(12,27)(13,28)(14,26)(23,32)(25,31)(29,30), (2,3), (6,15)(7,19)(8,20)(10,23)(12,29)(13,28)(14,31)(16,18)(17,33)(21,35)(22,34)(24,32)(25,26)(27,30), (4,6,7,16,20,22,35,33,9,17,21,34,8,18,19,15)(5,10,12,25,28,14,30,32,11,24,27,31,13,26,29,23), (4,5)(6,14,17,26)(7,12)(8,13)(9,11)(10,22,24,18)(15,25,33,31)(16,32,34,23)(19,29)(20,28)(21,27)(30,35));

D3×#33595 #72270645