Finite Group Databasebeta

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C3×#60373 #10678980

order768
nilpotency class3
isoclinismisoclinic class of #60372
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C210

Aliases

direct productC3×#60373

Subgroups

subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×#41 ◁ G

5058 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)(4,5)(6,7)(8,9,12,16)(10,14,17,21)(11,13,19,15)(18,20,23,22);(9,13)(11,19)(14,20)(15,16)(18,23)(21,22);(5,7);(8,10)(9,14)(11,18)(12,17)(13,20)(15,22)(16,21)(19,23);(10,17)(14,21)(18,23)(20,22)

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#60373, #72272146
maximal subgroup ofC3×#10574840, D3×#60373, #72272146
maximal normal subgroup ofC3×#10574840, D3×#60373, #72272146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53361);