Finite Group Databasebeta

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C3×#90360 #13061011

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

Aliases

direct productC3×#90360

Splittings

FittingC3×C7×#124 ⋊ C2
HallC3×C7 ⋊ #542
SylowC7 ⋊ C3×#542

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C7×#124C2
socleC22×C3×C7C24
prime radicalC3×#6502C2
hypercenterC3×#124D7
hypocenterC7C3×#542
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×#124

432 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)(11,14,15,13)(12,16,17,18)(20,24)(23,26);(11,13,15,14)(12,16,17,18)(19,21,22,25)(20,23,24,26);(13,14)(16,18)(19,20)(21,23)(22,24)(25,26);(5,6)(7,8)(9,10)(11,12,15,17)(13,16,14,18)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10300);