Finite Group Databasebeta

← Prev (#13060714) Next (#13060716) →

ΓL(1,𝔽8)×#554 #13060715

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

Aliases

direct productΓL(1,𝔽8)×#554
unambiguous semidirect product(C7×#554)⋊C3

Splittings

Fitting, HallC7×#554 ⋊ C3
SylowC7 ⋊ C3×#554

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#554C3
prime radicalC2×D4×ΓL(1,𝔽8)C22
hypercenter#554ΓL(1,𝔽8)
hypocenterC7C3×#554
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #554

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 224

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

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

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, 10004);