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C2×C4×ΓL(1,𝔽8)⋊Q8 #13057201

order1344
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×C4×ΓL(1,𝔽8)⋊Q8
unambiguous semidirect product(C2×C4×Dic14)⋊C3

Splittings

HallC2×C4×Dic14 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C4×Q8
SylowC7 ⋊ C2×C3×C4×Q8

Subgroups

subgroupG/H
centerC22×C4C2×AGL(1,𝔽7)
commutatorC2×C7C23×C3×C4
FrattiniC22C23×AGL(1,𝔽7)
FittingC2×C42×C7C2×C3
socleC23×C7C23×C3
prime radicalC23×ΓL(1,𝔽8)C23
cosocleC22×C7C24×C3
hypercenterC2×C42AGL(1,𝔽7)
hypocenterC7C2×C3×C4×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C2×C42

596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 448

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6490);