Finite Group Databasebeta

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C3×#90269 #13060920

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

Aliases

direct productC3×#90269

Splittings

FittingC3×C4×C7×Q8 ⋊ C2
HallC3×C7 ⋊ #533
SylowC7 ⋊ C3×#533

Subgroups

subgroupG/H
centerC2×C3×C4C22×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C4×C7×Q8C2
socleC22×C3×C7C24
prime radicalC22×C3×D7C23
hypercenterC3×C4×Q8D7
hypocenterC7C3×#533
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×Q8

444 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)(12,16)(15,18)(19,21,22,26)(20,23,24,25);(11,13,14,17)(12,15,16,18);(19,20,22,24)(21,25,26,23);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10209);