Finite Group Databasebeta

← Prev (#13051298) Next (#13051300) →

C7×#4517 #13051299

order1344
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#4517

Splittings

HallC3×C7 ⋊ #362
SylowC3 ⋊ C7×#362

Subgroups

subgroupG/H
centerC2×C7#821
commutatorC3×C8C2×C4×C7
FrattiniC2×C8C2×D3×C7
FittingC3×C7×M32C2
socleC2×C3×C7#102
prime radicalC22×C3×C7SD16
cosocleC2×C3×C8C22×C7
hypercenterC7×M32D3
hypocenterC3C7×#362
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×C8 ◁ C7×M32

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:

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

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 588);