D₃×#33471 #72258472

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₃×#33471

Splittings

Fitting	C₃×#33471 ⋊ C₂
Sylow	C₃ ⋊ C₂×#33471

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃×#442
commutator	C₃×C₈	C₂⁴×C₄
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#33471	C₂
socle	C₂×C₃	C₂²×#442
prime radical	C₂×D₃×#509	C₂
cosocle	C₂×C₃×C₈	C₂⁵
hypercenter	#33471	D₃
hypocenter	C₃	C₂×#33471
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #33471

3972 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);(6,16)(7,21)(8,22)(10,24)(12,28)(13,27)(14,30)(15,33)(17,34)(18,19)(20,32)(23,25)(26,35)(29,31);(5,11)(6,15,17,19)(7,20)(10,23,14,29)(13,27)(16,33,34,18)(21,32)(24,25,30,31);(4,6,7,18,22,15,32,34,9,17,20,33,8,19,21,16)(5,10,12,25,27,29,35,30,11,14,26,31,13,23,28,24);(2,3)(4,5)(6,14)(7,12)(8,13)(9,11)(10,17)(15,23)(16,30)(18,31)(19,29)(20,26)(21,28)(22,27)(24,34)(25,33)(32,35)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#7622637, D₃×#500940
maximal normal subgroup of	D₃×#7622637, D₃×#500940

GAP

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

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

D3×#33471 #72258472