Finite Group Databasebeta

← Prev (#13060882) Next (#13060884) →

C3×#90231 #13060883

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

Aliases

direct productC3×#90231

Splittings

FittingC3×C4×C7×D4 ⋊ C2
HallC3×C7 ⋊ #553
SylowC7 ⋊ C3×#553

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C4×C7×D4C2
socleC22×C3×C7C24
hypercenterC3×C4×D4D7
hypocenterC7C3×#553
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C4×D4

476 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)(11,13,14,17)(12,16,15,18)(19,22,23,21)(20,24,25,26);(12,15)(16,18)(19,21,23,22)(20,24,25,26);(13,17)(16,18);(5,6)(7,8)(9,10)(11,12,14,15)(13,16,17,18)(19,20,23,25)(21,26,22,24)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10172);