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C3×#23915 #10642522

order768
nilpotency class3
isoclinismisoclinic class of #23912
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC3×#23915

Subgroups

subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1886 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)(9,13)(11,18)(14,22)(15,19)(17,20)(21,23);(4,5)(6,7)(10,16)(11,18)(13,19)(14,23);(5,7)(8,9)(10,14)(11,19)(12,15)(13,18)(16,23)(17,21)(20,22);(8,10,12,16)(9,14,15,23)(11,20,18,17)(13,21,19,22)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#6437449
maximal normal subgroup ofC3×#6437449

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 16903);