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C3×#90217 #13060871

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

Aliases

direct productC3×#90217

Splittings

FittingC3×C7×#117 ⋊ C2
HallC3×C7 ⋊ #536
SylowC7 ⋊ C3×#536

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C7×#117C2
socleC22×C3×C7C24
prime radicalC2×C3×#1157C2
hypercenterC3×#117D7
hypocenterC7C3×#536
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×#117

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10160);