C₃×#499269 #72271618

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

Aliases

direct product

C₃×#499269

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#23573
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴×C₃
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#2875 ◁ G

1638 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)(6,14,18,10)(7,11)(8,13)(9,12)(15,23,32,30)(16,26,19,24)(17,31,34,25)(20,27)(21,28)(22,29)(33,35);(4,6,7,17,22,16,33,15,9,18,21,34,8,19,20,32)(5,10,11,25,29,24,35,30,12,14,28,31,13,26,27,23);(6,15)(7,20)(8,22)(10,23)(11,27)(13,29)(14,30)(16,17)(18,32)(19,34)(21,33)(24,31)(25,26)(28,35);(6,16,18,19)(7,21)(10,24,14,26)(11,28)(15,17,32,34)(20,33)(23,31,30,25)(27,35)

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#499269, #72271782
maximal subgroup of	D₃×#499269, #72271782
maximal normal subgroup of	D₃×#499269, #72271782

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

C3×#499269 #72271618