Finite Group Databasebeta

← Prev (#13060718) Next (#13060720) →

Q82×ΓL(1,𝔽8) #13060719

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

Aliases

direct productQ82×ΓL(1,𝔽8)

Splittings

Fitting, HallC7×Q82 ⋊ C3
SylowC7 ⋊ C3×Q82

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×Q82C3
prime radicalC22×ΓL(1,𝔽8)C24
hypercenterQ82ΓL(1,𝔽8)
hypocenterC7C3×Q82
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ Q82

424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 448

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10008);