Finite Group Databasebeta

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C3×#401626 #72270034

order1536
nilpotency class3
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC3×#401626

Subgroups

subgroupG/H
center, socleC22×C3C2×#397
commutator, Frattini, cosocleC23×C4C24×C3
prime radicalC24×C3×D4C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3×C4 ◁ G

4688 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)(8,9,12,16,13,17,11,15)(10,18,20,14,19,23,21,22);(4,5,6,7)(8,10,11,20,13,19,12,21)(9,14,15,23,17,22,16,18);(5,7)(10,19)(14,22)(18,23)(20,21);(9,15)(10,19)(11,12)(14,23)(16,17)(18,22)

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270113, D3×#401626
maximal subgroup of#72270113, D3×#401626, C3×#72215520
maximal normal subgroup of#72270113, D3×#401626, 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)(8,9,12,16,13,17,11,15)(10,18,20,14,19,23,21,22), (4,5,6,7)(8,10,11,20,13,19,12,21)(9,14,15,23,17,22,16,18), (5,7)(10,19)(14,22)(18,23)(20,21), (9,15)(10,19)(11,12)(14,23)(16,17)(18,22));