Finite Group Databasebeta

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C3×#454694 #72253302

order1536
nilpotency class3
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4

Aliases

direct productC3×#454694

Subgroups

subgroupG/H
center, socleC22×C3#2668
commutator, Frattini, cosocleC23×C4C24×C3
prime radicalC2×C3×#3003C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

4706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2)(3,4)(5,6,10,14,9,12,8,13)(7,15,18,19,16,20,17,11)(21,22,23);(2,4)(5,7,8,17,9,16,10,18)(6,11,14,15,12,19,13,20);(6,12)(7,16)(11,19)(13,14)(15,20)(17,18);(6,13)(7,16)(8,10)(11,20)(12,14)(15,19)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253334, D3×#454694
maximal subgroup of#72253334, D3×#454694, C3×#72215520
maximal normal subgroup of#72253334, D3×#454694, C3×#72215520

GAP

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

gap> G := Group((1,2)(3,4)(5,6,10,14,9,12,8,13)(7,15,18,19,16,20,17,11)(21,22,23), (2,4)(5,7,8,17,9,16,10,18)(6,11,14,15,12,19,13,20), (6,12)(7,16)(11,19)(13,14)(15,20)(17,18), (6,13)(7,16)(8,10)(11,20)(12,14)(15,19));