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C43×ΓL(1,𝔽8) #13054905

order1344
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C43 · Schur multiplier (H2) = C43

Aliases

direct productC43×ΓL(1,𝔽8)

Splittings

Fitting, HallC43×C7 ⋊ C3
derived, SylowC7 ⋊ C3×C43

Subgroups

subgroupG/H
center, hypercenterC43ΓL(1,𝔽8)
commutator, hypocenterC7C3×C43
FrattiniC23C23×ΓL(1,𝔽8)
FittingC43×C7C3
socle, cosocleC23×C7C23×C3
prime radicalC23×ΓL(1,𝔽8)C23
LCSG ▷ C7
UCSC1 ◁ C43

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 448

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4194);