D₃×#29271 #72271576

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

Aliases

direct product

D₃×#29271

Splittings

Fitting	C₃×#29271 ⋊ C₂
Sylow	C₃ ⋊ C₂×#29271

Subgroups

	subgroup	G/H
center	C₄	D₃×#535
commutator, cosocle	C₂²×C₃×C₄	C₂⁵
Frattini	C₂²×C₄	C₂⁴×D₃
Fitting	C₃×#29271	C₂
socle	C₂×C₃	C₂×#2580
prime radical	C₂×D₃×#135	C₂²
hypercenter	#29271	D₃
hypocenter	C₃	C₂×#29271
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #29271

3400 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)(4,7,8,20,9,19,22,32)(5,11,13,28,12,27,29,35)(6,16,15,31,17,33,21,18)(10,24,25,34,14,30,23,26);(2,3);(6,15,17,21)(7,19)(10,23,14,25)(11,27)(16,18,33,31)(20,32)(24,34,30,26)(28,35);(4,6)(5,10)(7,18)(8,21)(9,17)(11,26)(12,14)(13,23)(15,22)(16,32)(19,31)(20,33)(24,35)(25,29)(27,34)(28,30);(4,5)(6,14,17,10)(7,11)(8,13)(9,12)(15,23,21,25)(16,30,33,24)(18,34,31,26)(19,27)(20,28)(22,29)(32,35)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500940
maximal normal subgroup of	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)(4,7,8,20,9,19,22,32)(5,11,13,28,12,27,29,35)(6,16,15,31,17,33,21,18)(10,24,25,34,14,30,23,26), (2,3), (6,15,17,21)(7,19)(10,23,14,25)(11,27)(16,18,33,31)(20,32)(24,34,30,26)(28,35), (4,6)(5,10)(7,18)(8,21)(9,17)(11,26)(12,14)(13,23)(15,22)(16,32)(19,31)(20,33)(24,35)(25,29)(27,34)(28,30), (4,5)(6,14,17,10)(7,11)(8,13)(9,12)(15,23,21,25)(16,30,33,24)(18,34,31,26)(19,27)(20,28)(22,29)(32,35));

D3×#29271 #72271576