Finite Group Databasebeta

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C3×#90274 #13060925

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

Aliases

direct productC3×#90274

Splittings

FittingC3×C4×C7×Q8 ⋊ C2
HallC3×C7 ⋊ #541
SylowC7 ⋊ C3×#541

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
FittingC3×C4×C7×Q8C2
socleC22×C3×C7C24
prime radicalC22×C3×D7C23
hypercenterC3×C4×Q8D7
hypocenterC7C3×#541
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2,3)(4,5,7,9,10,8,6)(11,14,15,13)(12,16,17,18)(20,26)(24,32)(25,34)(31,33);(11,13,15,14)(12,16,17,18)(19,22,23,30)(20,25,26,34)(21,27,28,29)(24,31,32,33);(19,21,23,28)(20,24,26,32)(22,29,30,27)(25,33,34,31);(5,6)(7,8)(9,10)(11,12)(13,18)(14,16)(15,17)(19,20)(21,24)(22,25)(23,26)(27,31)(28,32)(29,33)(30,34)

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, 10214);