C₃×#483961 #72226049

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃×#483961

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#442
commutator	C₂×C₈	C₂³×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂⁴×C₃
prime radical	C₃×#33748	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#322 ◁ G

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 768

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

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72213975
maximal normal subgroup of	C₃×#72213975

GAP

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

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

C3×#483961 #72226049