D₃×#27694 #72270330

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

Aliases

direct product

D₃×#27694

Splittings

Fitting	C₃×#27694 ⋊ C₂
Sylow	C₃ ⋊ C₂×#27694

Subgroups

	subgroup	G/H
center	C₄	D₃×#535
commutator, cosocle	C₂²×C₃×C₄	C₂⁵
Frattini	C₂²×C₄	C₂⁴×D₃
Fitting	C₃×#27694	C₂
socle	C₂×C₃	C₂×#2487
prime radical	D₃×#532	C₂²
hypercenter	#27694	D₃
hypocenter	C₃	C₂×#27694
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #27694

3564 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)(4,7,9,20,8,19,21,32)(5,12,13,29,11,26,27,33)(6,15,17,30,16,31,14,18)(10,23,24,34,25,35,22,28);(2,3);(5,11)(6,14,16,17)(7,19)(10,22,25,24)(13,27)(15,30,31,18)(20,32)(23,34,35,28);(4,6,8,16)(5,10)(7,18,19,30)(9,14,21,17)(11,25)(12,28)(13,22)(15,20,31,32)(23,33)(24,27)(26,34)(29,35);(4,5)(6,10)(7,12)(8,11)(9,13)(14,22)(15,23)(16,25)(17,24)(18,28)(19,26)(20,29)(21,27)(30,34)(31,35)(32,33)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500349, D₃×#500168
maximal normal subgroup of	D₃×#500349, D₃×#500168

GAP

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

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

D3×#27694 #72270330