C₃×#500168 #72270463

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#500168

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#24033
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴×C₃
prime radical	C₃×#61079	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#2932 ◁ G

1956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#500168
maximal subgroup of	D₃×#500168
maximal normal subgroup of	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)(4,5)(6,10)(7,20)(8,14)(9,13)(11,21)(12,23)(15,27)(16,25)(17,34)(18,24)(19,28)(22,30)(26,32)(29,31)(33,35), (4,6,7,17,23,15,31,33,9,19,21,32,8,18,22,16)(5,10,11,26,12,27,30,25,13,28,20,34,14,24,29,35), (6,15,19,18)(7,21)(10,24,28,27)(11,20)(16,32,33,17)(22,31)(25,34,35,26)(29,30), (6,16)(7,22)(8,23)(10,25)(11,29)(12,14)(15,32)(17,18)(19,33)(20,30)(21,31)(24,34)(26,27)(28,35));

C3×#500168 #72270463