Finite Group Databasebeta

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C3×#23407 #10642014

order768
nilpotency class3
isoclinismisoclinic class of #23400
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC3×#23407

Subgroups

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

2106 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)(9,14)(10,17)(11,12)(16,19)(18,20)(21,23);(4,5)(6,7)(8,9)(10,15)(11,19)(12,14)(13,16)(17,23)(18,22)(20,21);(5,7)(8,10)(9,15)(11,17)(12,20)(13,18)(14,21)(16,22)(19,23);(9,16)(14,19)(15,22)(21,23)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72248558, #72248566, D3×#23407, …
maximal subgroup of#72248558, #72248566, C3×#417774, …
maximal normal subgroup of#72248558, #72248566, C3×#417774, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 16395);