Finite Group Databasebeta

← Prev (#13060966) Next (#13060968) →

C3×#90314 #13060967

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

Aliases

direct productC3×#90314

Splittings

HallC3×C7 ⋊ #544
SylowC7 ⋊ C3×#544

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C24
prime radicalC23×C3×C7C23
hypercenterC3×C22⋊Q8D7
hypocenterC7C3×#544
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

412 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)(12,16)(15,18)(19,21,22,26)(20,23,24,25);(13,17)(15,18)(19,20,22,24)(21,25,26,23);(11,13)(12,15)(14,17)(16,18);(5,6)(7,8)(9,10)(11,12,14,16)(13,15,17,18)

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, 10256);