Finite Group Databasebeta

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C3×#89471 #13055132

order1344
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4

Aliases

direct productC3×#89471

Splittings

HallC3×C7 ⋊ #494
SylowC7 ⋊ C3×#494

Subgroups

subgroupG/H
centerC22×C3C2×D28
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C4×C7×C8C2
socle, prime radicalC22×C3×C7C2×D4
hypercenterC3×C4×C8D7
hypocenterC7C3×#494
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8

244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 1344

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(4,5,7,9,10,8,6)(15,17,24,28,19,26,18,25)(16,20,27,30,22,29,21,23);(11,12,13,14);(5,6)(7,8)(9,10)(12,14)(15,16,19,22)(17,23,26,30)(18,27,24,21)(20,28,29,25)

Conjugacy classes & character table

354 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4421);