Finite Group Databasebeta

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C3×#60423 #10679030

order768
nilpotency class3
isoclinismisoclinic class of #60367
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC3×#60423

Subgroups

subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G

2890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#60423, #72248817, #72253830
maximal subgroup ofD3×#60423, #72248817, C3×#394249, …
maximal normal subgroup ofD3×#60423, #72248817, C3×#394249, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53411);