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#557⋊ΓL(1,𝔽8) #13060783

order1344
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#557⋊ΓL(1,𝔽8)

Splittings

Fitting, HallC7×#557 ⋊ C3
Sylow#557 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ #557⋊C3

Subgroups

subgroupG/H
center, FrattiniC22C22×C22⋊ΓL(1,𝔽8)
commutator, cosocleC2×C7×Q8C22×C3
FittingC7×#557C3
socleC22×C7C22×A4
prime radicalC2×C7⋊SL(2,𝔽3)C22
hypercenterC4⋊C4C22⋊ΓL(1,𝔽8)
hypocenterC7×Q8C3×Q8
LCSG ▷ C2×C7×Q8 ▷ C7×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 448

Polycyclic presentation:

2^2,3,2^3,7,2|a^2=d,b^2=d,e^2=h,f^2=h,b^a=bdh,e^c=ef,f^c=e,g^c=g^4,f^e=fh

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10072);