Finite Group Databasebeta

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C7×#5822 #13061541

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

Aliases

direct productC7×#5822

Splittings

FittingC2×C3×C7×#41 ⋊ C2
HallC3×C7 ⋊ #575
SylowC3 ⋊ C7×#575

Subgroups

subgroupG/H
centerC4×C7C2×#67
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC2×C3×C7×#41C2
socleC2×C3×C7C22×D4
hypercenterC2×C7×#41D3
hypocenterC3C7×#575
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#41

524 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:

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

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, 10830);