Finite Group Databasebeta

← Prev (#13061488) Next (#13061490) →

C7×#5773 #13061489

order1344
automorphism?
homologyAbelianization (H1) = C24×C7 · Schur multiplier (H2) = C25

Aliases

direct productC7×#5773

Splittings

FittingC3×C7×AGL(1,ℤ/8) ⋊ C2
HallC3×C7 ⋊ #575
SylowC3 ⋊ C7×#575

Subgroups

subgroupG/H
centerC2×C7C2×D3×D4
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×AGL(1,ℤ/8)C2
socleC2×C3×C7C22×D4
hypercenterC7×AGL(1,ℤ/8)D3
hypocenterC3C7×#575
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×AGL(1,ℤ/8)

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:

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

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10778);