C₃×#499752 #72270102

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₃×#499752

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#23848
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴×C₃
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#2932 ◁ G

2576 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)(7,11)(8,13)(9,14)(12,22)(15,26)(16,24)(17,25)(18,23)(19,27)(20,28)(21,29)(30,31)(32,35)(33,34);(4,6,7,17,22,15,31,33,9,19,20,32,8,18,21,16)(5,10,11,25,12,26,30,34,14,27,28,35,13,23,29,24);(6,15,19,18)(7,20)(10,23,27,26)(11,28)(16,32,33,17)(21,31)(24,25,34,35)(29,30);(6,16)(7,21)(8,22)(10,24)(11,29)(12,13)(15,32)(17,18)(19,33)(20,31)(23,25)(26,35)(27,34)(28,30)

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72270183
maximal subgroup of	#72270183
maximal normal subgroup of	#72270183

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

C3×#499752 #72270102