Finite Group Databasebeta

← Prev (#13061118) Next (#13061120) →

C3×#90468 #13061119

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

Aliases

direct productC3×#90468

Splittings

FittingC3×C7×#137 ⋊ C2
HallC3×C7 ⋊ #575
SylowC7 ⋊ C3×#575

Subgroups

subgroupG/H
centerC2×C3C2×D4×D7
commutator, cosocleC4×C7C24×C3
FrattiniC4C23×C3×D7
FittingC3×C7×#137C2
socleC2×C3×C7C22×D4
hypercenterC3×#137D7
hypocenterC7C3×#575
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#137

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 336

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

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10408);