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(C5⋊AGL(1,𝔽7))⋊D79 #45684454

order33180
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product(C5⋊AGL(1,𝔽7))⋊D79
unambiguous semidirect product(ΓL(1,𝔽8)⋊D79)⋊D5

Splittings

Fitting, socle, derived, HallC5×C7×C79 ⋊ C22×C3
HallD35⋊D79 ⋊ C3
HallC5×ΓL(1,𝔽8)×C79 ⋊ C22
HallΓL(1,𝔽8)×C79 ⋊ C2×D5
HallC7×C79 ⋊ C2×C3×D5
HallC5×C79 ⋊ C2×AGL(1,𝔽7)
HallC5×ΓL(1,𝔽8) ⋊ C2×D79
SylowC79 ⋊ D5×AGL(1,𝔽7)
HallC5×C7 ⋊ C2×C3×D79
HallΓL(1,𝔽8) ⋊ D5×D79
SylowC7 ⋊ C3×D5×D79
SylowC5 ⋊ AGL(1,𝔽7)×D79

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C79C22×C3
LCSG ▷ C5×C7×C79
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2765

Polycyclic presentation:

2^2,3,5,7,79|e^a=e^6,f^a=f^78,d^b=d^4,e^b=e^6,e^c=e^2

Conjugacy classes & character table

691 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33180, 85);