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C3×#90339 #13060991

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

Aliases

direct productC3×#90339

Splittings

FittingC3×C7×#123 ⋊ C2
HallC3×C7 ⋊ #536
SylowC7 ⋊ C3×#536

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C7×#123C2
socleC22×C3×C7C24
prime radicalC2×C3×#1159C2
hypercenterC3×#123D7
hypocenterC7C3×#536
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×#123

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10280);