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C3×#90304 #13060957

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

Aliases

direct productC3×#90304

Splittings

FittingC3×C7×#121 ⋊ C2
HallC3×C7 ⋊ #538
SylowC7 ⋊ C3×#538

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C7×#121C2
socleC22×C3×C7C24
hypercenterC3×#121D7
hypocenterC7C3×#538
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×#121

500 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)(12,16)(15,18)(19,21)(20,24)(22,26)(23,25);(13,17)(15,18)(19,21,22,26)(20,23,25,24);(11,13)(12,15)(14,17)(16,18)(21,26)(23,24);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,23)(22,25)(24,26)

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, 10246);