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C3×#90327 #13060979

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

Aliases

direct productC3×#90327

Splittings

FittingC3×C7×C22⋊Q8 ⋊ C2
HallC3×C7 ⋊ #554
SylowC7 ⋊ C3×#554

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C24
prime radicalC2×C3×C22⋊D7C22
hypercenterC3×C22⋊Q8D7
hypocenterC7C3×#554
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 26 points:

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

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, 10268);