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D7×AGL(1,𝔽7)×C53 #44528400

order31164
automorphism?
homologyAbelianization (H1) = C22×C3×C53 · Schur multiplier (H2) = C2

Aliases

direct productD7×AGL(1,𝔽7)×C53
unambiguous semidirect productD7⋊(C3×D7×C53)
unambiguous semidirect product(C7×AGL(1,𝔽7))⋊(C2×C53)
unambiguous semidirect product(D7×ΓL(1,𝔽8))⋊(C2×C53)
unambiguous semidirect product(ΓL(1,𝔽8)⋊D7)⋊(C2×C53)

Splittings

Fitting, socle, HallC72×C53 ⋊ C22×C3
derived, SylowC72 ⋊ C22×C3×C53
HallD72×C53 ⋊ C3
HallC7×ΓL(1,𝔽8)×C53 ⋊ C22
HallD72 ⋊ C3×C53
HallC7×ΓL(1,𝔽8) ⋊ C22×C53

Subgroups

subgroupG/H
center, hypercenterC53D7×AGL(1,𝔽7)
commutator, cosocle, hypocenterC72C22×C3×C53
FrattiniC1G
Fitting, socleC72×C53C22×C3
LCSG ▷ C72
UCSC1 ◁ C53

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2226

Conjugacy classes & character table

1855 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31164, 92);