Finite Group Databasebeta

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C3×#89430 #13055088

order1344
isoclinismisoclinic class of #89428
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#89430

Splittings

HallC3×C7 ⋊ #400
SylowC7 ⋊ C3×#400

Subgroups

subgroupG/H
centerC23×C3C22×D7
commutator, cosocleC23×C7C23×C3
FrattiniC23C22×C3×D7
FittingC3×C7×#95C2
socle, prime radicalC23×C3×C7C23
hypercenterC3×#95D7
hypocenterC7C3×#400
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23×C3 ◁ C3×#95

268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 1344

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(4,5,7,9,10,8,6)(11,12)(13,14)(16,20)(19,22)(23,25,26,30)(24,28,27,29);(11,12,13,14)(15,17,18,21)(16,19,20,22)(24,27)(25,30);(5,6)(7,8)(9,10)(11,12,13,14)(15,16)(17,19)(18,20)(21,22)(23,24,26,27)(25,29,30,28)

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4377);