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C2×ΓL(1,𝔽8)⋊(D71⋊C5) #44447169

order29820
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×ΓL(1,𝔽8)⋊(D71⋊C5)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(D71⋊C5)
unambiguous semidirect product(C2×C71⋊C5)⋊AGL(1,𝔽7)
unambiguous semidirect product(C2×ΓL(1,𝔽8)⋊D71)⋊C5
unambiguous semidirect product(C2×C7⋊(D71⋊C5))⋊C3

Splittings

Fitting, socleC2×C7×C71 ⋊ C2×C3×C5
derived, HallC7×C71 ⋊ C22×C3×C5
HallC2×C7⋊(D71⋊C5) ⋊ C3
HallΓL(1,𝔽8)×C71⋊C5 ⋊ C22
HallC2×ΓL(1,𝔽8)⋊D71 ⋊ C5
HallC7×C71⋊C5 ⋊ C22×C3
HallC2×D497 ⋊ C3×C5
HallΓL(1,𝔽8)×C71 ⋊ C22×C5
HallC71⋊C5 ⋊ C2×AGL(1,𝔽7)
SylowC71 ⋊ C2×C5×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C2×D71⋊C5
SylowC7 ⋊ C2×C3×D71⋊C5

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)⋊(D71⋊C5)
commutator, cosocle, hypocenterC7×C71C22×C3×C5
FrattiniC1G
Fitting, socleC2×C7×C71C2×C3×C5
LCSG ▷ C7×C71
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 994

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 57);