D₃×#33329 #72228221

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₃×#33329

Splittings

Fitting	C₃×#33329 ⋊ C₂
Sylow	C₃ ⋊ C₂×#33329

Subgroups

	subgroup	G/H
center	C₄	C₂²×C₄²×D₃
commutator	C₃×C₄	C₂³×C₄²
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#33329	C₂
socle	C₂×C₃	C₂³×#97
prime radical	C₂×D₃×#41	C₂×C₄
cosocle	C₂×C₃×C₈	C₂⁵
hypercenter	#33329	D₃
hypocenter	C₃	C₂×#33329
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄ ◁ #33329

4292 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);(6,15,16,28)(7,18)(10,21,14,26)(12,23)(17,31,30,29)(19,32)(22,34,27,33)(25,35);(5,11)(10,14)(12,23)(13,24)(21,26)(22,27)(25,35)(33,34);(4,6,7,17,8,15,19,29,9,16,18,30,20,28,32,31)(5,10,12,22,13,21,25,33,11,14,23,27,24,26,35,34);(2,3)(4,5)(6,14)(7,12)(8,13)(9,11)(10,16)(15,26)(17,27)(18,23)(19,25)(20,24)(21,28)(22,30)(29,34)(31,33)(32,35)

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228223, D₃×#7622637, D₃×#495505, …
maximal normal subgroup of	D₃×#7622637, D₃×#495505, D₃×#495573
maximal quotient of	#72228223

GAP

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

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

D3×#33329 #72228221