Finite Group Databasebeta

← Prev (#13051155) Next (#13051157) →

C3×#89291 #13051156

order1344
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#89291

Splittings

HallC3×C7 ⋊ #362
SylowC7 ⋊ C3×#362

Subgroups

subgroupG/H
centerC2×C3#6383
commutatorC7×C8C2×C3×C4
FrattiniC2×C8C2×C3×D7
FittingC3×C7×#108C2
socleC2×C3×C7#102
prime radicalC22×C3×C7SD16
cosocleC2×C7×C8C22×C3
hypercenterC3×#108D7
hypocenterC7C3×#362
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C3×#108

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3)(4,5,7,9,10,8,6)(11,13,14,22,16,25,28,24)(12,17,18,33,20,35,26,34)(15,30,27,32,31,41,36,23)(19,37,29,40,38,39,42,21);(5,6)(7,8)(9,10)(11,12,16,20)(13,21,25,40)(14,26,28,18)(15,29,31,42)(17,32,35,23)(19,36,38,27)(22,39,24,37)(30,34,41,33)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 445);