Finite Group Databasebeta

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C3×#89724 #13055383

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

Aliases

direct productC3×#89724

Splittings

HallC3×C7 ⋊ #397
SylowC7 ⋊ C3×#397

Subgroups

subgroupG/H
centerC23×C3C22×D7
commutator, cosocleC23×C7C23×C3
FrattiniC23C22×C3×D7
FittingC2×C3×C7×C22⋊C4C2
socleC23×C3×C7C23
prime radicalC24×C3×C7C22
hypercenterC2×C3×C22⋊C4D7
hypocenterC7C3×#397
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 3 generators on 26 points:

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

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, 4672);