Finite Group Databasebeta

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C3×#90290 #13060941

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#90290

Splittings

FittingC3×C7×#121 ⋊ C2
HallC3×C7 ⋊ #550
SylowC7 ⋊ C3×#550

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C7×#121C2
socleC22×C3×C7C24
hypercenterC3×#121D7
hypocenterC7C3×#550
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×#121

624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

201 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10230);