Finite Group Databasebeta

← Prev (#13132424) Next (#13132426) →

C2×Dic5×AGL(1,𝔽7) #13132425

order1680
isoclinismisoclinic class of D5×AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×Dic5×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C3×Dic5)
unambiguous semidirect product(C2×D7×Dic5)⋊C3

Splittings

derived, HallC5×C7 ⋊ C22×C3×C4
HallC2×D7×Dic5 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C22×C4
HallΓL(1,𝔽8) ⋊ C22×Dic5
SylowC7 ⋊ C22×C3×Dic5
SylowC5 ⋊ C2×C4×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC22D5×AGL(1,𝔽7)
commutator, hypocenterC5×C7C22×C3×C4
FrattiniC2C2×D5×AGL(1,𝔽7)
Fitting, socleC22×C5×C7C22×C3
prime radicalC22×C5×AGL(1,𝔽7)C2
cosocleC2×C5×C7C23×C3
LCSG ▷ C5×C7
UCSC1 ◁ C22

216 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×D5×AGL(1,𝔽7), C2×C22⋊D5×AGL(1,𝔽7), AGL(1,𝔽7)×C22⋊Dic5, …
maximal normal subgroup ofC2×C4×D5×AGL(1,𝔽7), C2×C22⋊D5×AGL(1,𝔽7), AGL(1,𝔽7)×C22⋊Dic5, …
maximal quotient ofAGL(1,𝔽7)×#681, AGL(1,𝔽7)×C22⋊Dic5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 445);