Finite Group Databasebeta

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C7×#5816 #13061532

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

Aliases

direct productC7×#5816

Splittings

FittingC22×C3×C7×Q8 ⋊ C2
HallC3×C7 ⋊ #536
SylowC3 ⋊ C7×#536

Subgroups

subgroupG/H
centerC22×C7C23×D3
commutator, cosocleC22×C3C24×C7
FrattiniC22C23×D3×C7
FittingC22×C3×C7×Q8C2
socleC22×C3×C7C24
prime radicalC2×C7×#235C2
hypercenterC22×C7×Q8D3
hypocenterC3C7×#536
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C22×C7×Q8

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10821);