Finite Group Databasebeta

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C3×#90321 #13060976

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

Aliases

direct productC3×#90321

Splittings

FittingC3×C7×C22⋊Q8 ⋊ C2
HallC3×C7 ⋊ #546
SylowC7 ⋊ C3×#546

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C24
hypercenterC3×C22⋊Q8D7
hypocenterC7C3×#546
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 336

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

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

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, 10265);