C₃×#500650 #72271677

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

Aliases

direct product

C₃×#500650

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#24196
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴×C₃
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#2961 ◁ G

1558 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)(15,24,34,32)(16,26,17,25)(19,33,35,27)(20,28)(21,29)(22,30)(23,31);(4,6,7,17,22,35,21,15,9,18,23,16,8,19,20,34)(5,10,11,26,30,33,29,32,12,14,31,25,13,27,28,24);(6,15)(7,20)(8,22)(10,24)(11,28)(13,30)(14,32)(16,19)(17,35)(18,34)(21,23)(25,33)(26,27)(29,31);(6,16,18,17)(7,21)(8,22)(10,25,14,26)(11,29)(13,30)(15,19,34,35)(20,23)(24,33,32,27)(28,31)

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72271714
maximal subgroup of	#72271714
maximal normal subgroup of	#72271714

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

C3×#500650 #72271677