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#10587898⋊C3 #13132058

order1680
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#10587898⋊C3

Splittings

Hall#10587898 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ M16
HallC5×C7 ⋊ C3×M16
HallΓL(1,𝔽8) ⋊ #681
SylowC7 ⋊ C3×#681
SylowC5 ⋊ #1136⋊C3

Subgroups

subgroupG/H
center, FrattiniC4C2×C5⋊AGL(1,𝔽7)
commutator, socleC2×C5×C7C2×C3×C4
FittingC2×C4×C5×C7C2×C3
prime radicalC22×C5×ΓL(1,𝔽8)C4
cosocleC4×C5×C7C22×C3
hypercenterC2×C4C5⋊AGL(1,𝔽7)
hypocenterC5×C7C3×M16
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C4 ◁ C2×C4

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#1136⋊C3, M16×C5⋊AGL(1,𝔽7), AGL(1,𝔽7)×#681, …
maximal normal subgroup ofD5×#1136⋊C3, M16×C5⋊AGL(1,𝔽7), AGL(1,𝔽7)×#681, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 78);