Finite Group Databasebeta

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C7×#4699 #13055734

order1344
automorphism?
homologyAbelianization (H1) = C23×C7 · Schur multiplier (H2) = C2×C4

Aliases

direct productC7×#4699

Splittings

HallC3×C7 ⋊ #494
SylowC3 ⋊ C7×#494

Subgroups

subgroupG/H
centerC22×C7C2×D12
commutator, cosocleC2×C3×C4C23×C7
FrattiniC2×C4C22×D3×C7
FittingC3×C4×C7×C8C2
socle, prime radicalC22×C3×C7C2×D4
hypercenterC4×C7×C8D3
hypocenterC3C7×#494
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C42×C7 ◁ C4×C7×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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5023);