Finite Group Databasebeta

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C3×#34963 #10653570

order768
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC3×#34963

Subgroups

subgroupG/H
centerC22×C3×C4C24
commutator, Frattini, cosocleC23C25×C3
socleC23×C3C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

3270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 5 generators on 23 points:

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

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#34963, #72270054
maximal subgroup ofC3×#6453969, D3×#34963, #72270054
maximal normal subgroup ofC3×#6453969, D3×#34963, #72270054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 27951);