Finite Group Databasebeta

← Prev (#13060737) Next (#13060739) →

ΓL(1,𝔽8)×#577 #13060738

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productΓL(1,𝔽8)×#577

Splittings

Fitting, HallC7×#577 ⋊ C3
SylowC7 ⋊ C3×#577

Subgroups

subgroupG/H
centerC2C22×D4×ΓL(1,𝔽8)
commutator, cosocleC4×C7C24×C3
FrattiniC4C24×ΓL(1,𝔽8)
FittingC7×#577C3
socleC2×C7C22×C3×D4
hypercenter#577ΓL(1,𝔽8)
hypocenterC7C3×#577
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ #41 ◁ #577

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

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

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#3585, ΓL(1,𝔽8)×#3291, AGL(1,𝔽7)×#577, …
maximal normal subgroup ofΓL(1,𝔽8)×#3585, ΓL(1,𝔽8)×#3291, AGL(1,𝔽7)×#577, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10027);