Finite Group Databasebeta

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C3×#90275 #13060926

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

Aliases

direct productC3×#90275

Splittings

FittingC3×C4×C7×Q8 ⋊ C2
HallC3×C7 ⋊ #556
SylowC7 ⋊ C3×#556

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C4×C7×Q8C2
socleC22×C3×C7C24
prime radicalC2×C3×D28C22
hypercenterC3×C4×Q8D7
hypocenterC7C3×#556
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10215);