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C3×#90469 #13061120

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#90469

Splittings

FittingC3×C7×#137 ⋊ C2
HallC3×C7 ⋊ #577
SylowC7 ⋊ C3×#577

Subgroups

subgroupG/H
centerC2×C3C2×D4×D7
commutator, cosocleC4×C7C24×C3
FrattiniC4C23×C3×D7
FittingC3×C7×#137C2
socleC2×C3×C7C22×D4
hypercenterC3×#137D7
hypocenterC7C3×#577
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#137

516 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)(13,19)(14,22)(17,25)(20,21)(23,26);(11,13,15,21)(12,17,16,23)(14,19,22,20)(18,26,24,25);(11,12)(13,17)(14,18)(15,16)(19,26)(20,25)(21,23)(22,24);(5,6)(7,8)(9,10)(12,16)(17,23)(18,24)(25,26)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10409);