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C7×#5777 #13061493

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

Aliases

direct productC7×#5777

Splittings

FittingC3×C7×#137 ⋊ C2
HallC3×C7 ⋊ #575
SylowC3 ⋊ C7×#575

Subgroups

subgroupG/H
centerC2×C7C2×D3×D4
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×#137C2
socleC2×C3×C7C22×D4
hypercenterC7×#137D3
hypocenterC3C7×#575
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#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:

(1,2,3,4,5,6,7)(8,9,10)(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);(9,10)(13,19)(17,23)(20,22)(25,26)

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, 10782);