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C4×C22⋊(C22⋊ΓL(1,𝔽8)) #13062323

order1344
isoclinismisoclinic class of C22⋊(C22⋊ΓL(1,𝔽8))
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productC4×C22⋊(C22⋊ΓL(1,𝔽8))
unambiguous semidirect product(C4×C7)⋊(C22⋊A4)

Splittings

Fitting, HallC24×C4×C7 ⋊ C3
derivedC24×C7 ⋊ C3×C4
SylowC24×C4 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ C4×C22⋊A4

Subgroups

subgroupG/H
center, hypercenterC4C22⋊(C22⋊ΓL(1,𝔽8))
commutator, hypocenterC24×C7C3×C4
FrattiniC2C2×C22⋊(C22⋊ΓL(1,𝔽8))
FittingC24×C4×C7C3
socle, cosocleC25×C7C2×C3
prime radicalC2×C22⋊(C22⋊ΓL(1,𝔽8))C2
LCSG ▷ C24×C7
UCSC1 ◁ C4

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11612);