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ΓL(1,𝔽8)×#546 #13060707

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

Aliases

direct productΓL(1,𝔽8)×#546
unambiguous semidirect product(C7×#546)⋊C3

Splittings

Fitting, HallC7×#546 ⋊ C3
SylowC7 ⋊ C3×#546

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#546C3
hypercenter#546ΓL(1,𝔽8)
hypocenterC7C3×#546
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #546

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#3532, ΓL(1,𝔽8)×#3484, AGL(1,𝔽7)×#546, …
maximal normal subgroup ofΓL(1,𝔽8)×#3532, ΓL(1,𝔽8)×#3484, AGL(1,𝔽7)×#546, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 9996);