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C71×C22⋊ΓL(1,𝔽8) #20238558

order5964
isoclinismisoclinic class of C22⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C71 · Schur multiplier (H2) = C2

Aliases

direct productC71×C22⋊ΓL(1,𝔽8)
unambiguous semidirect productC22⋊(ΓL(1,𝔽8)×C71)
unambiguous semidirect productC7⋊(A4×C71)
unambiguous semidirect product(C22×C71)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C7×C71)⋊A4

Splittings

Fitting, socle, HallC22×C7×C71 ⋊ C3
derived, HallC22×C7 ⋊ C3×C71
HallC7×C71 ⋊ A4
HallC22×C71 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ A4×C71
SylowC22 ⋊ ΓL(1,𝔽8)×C71

Subgroups

subgroupG/H
center, hypercenterC71C22⋊ΓL(1,𝔽8)
commutator, hypocenterC22×C7C3×C71
FrattiniC1G
Fitting, socleC22×C7×C71C3
LCSG ▷ C22×C7
UCSC1 ◁ C71

20 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

852 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4×ΓL(1,𝔽8)×C71, C3×C71×C22⋊ΓL(1,𝔽8), C22⋊ΓL(1,𝔽8)×C71⋊C5, …
maximal normal subgroup ofA4×ΓL(1,𝔽8)×C71, C3×C71×C22⋊ΓL(1,𝔽8), C22⋊ΓL(1,𝔽8)×C71⋊C5, …
maximal quotient ofC71×C22⋊(C7⋊C9), C3×C71×C22⋊ΓL(1,𝔽8), C5×C71×C22⋊ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(5964, 37);