Finite Group Databasebeta

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C3×#90263 #13060914

order1344
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productC3×#90263

Splittings

FittingC3×C4×C7×Q8 ⋊ C2
HallC3×C7 ⋊ #519
SylowC7 ⋊ C3×#519

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutatorC2×C7C23×C3×C4
FrattiniC22C23×C3×D7
FittingC3×C4×C7×Q8C2
socleC22×C3×C7C24
prime radicalC22×C3×D7C23
cosocleC22×C7C24×C3
hypercenterC3×C4×Q8D7
hypocenterC7C3×#519
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

532 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)(20,28)(21,30)(27,29);(15,18,19,26)(16,21,22,30)(17,23,24,25)(20,27,28,29);(15,17,19,24)(16,20,22,28)(18,25,26,23)(21,29,30,27);(5,6)(7,8)(9,10)(15,16)(17,20)(18,21)(19,22)(23,27)(24,28)(25,29)(26,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10203);