Finite Group Databasebeta

← Prev (#44568687) Next (#44568689) →

C7×AGL(1,𝔽7)×D55 #44568688

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×AGL(1,𝔽7)×D55
unambiguous semidirect productD7⋊(C3×C7×D55)
unambiguous semidirect product(C5×AGL(1,𝔽7))⋊(C7×D11)
unambiguous semidirect product(C11×AGL(1,𝔽7))⋊(C7×D5)
unambiguous semidirect product(D7×D55)⋊(C3×C7)

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C22×C3
derivedC5×C7×C11 ⋊ C22×C3×C7
HallC7×D7×D55 ⋊ C3
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C22
HallC7×C11×ΓL(1,𝔽8) ⋊ C2×D5
HallC5×C7×ΓL(1,𝔽8) ⋊ C2×D11
HallC72×C11 ⋊ C2×C3×D5
HallC5×C72 ⋊ C2×C3×D11
HallC7×ΓL(1,𝔽8) ⋊ C2×D55
HallC5×C11 ⋊ C2×C7×AGL(1,𝔽7)
SylowC72 ⋊ C2×C3×D55
SylowC11 ⋊ C7×D5×AGL(1,𝔽7)
SylowC5 ⋊ C7×D11×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC7AGL(1,𝔽7)×D55
commutator, cosocle, hypocenterC5×C7×C11C22×C3×C7
FrattiniC1G
Fitting, socleC5×C72×C11C22×C3
LCSG ▷ C5×C7×C11
UCSC1 ◁ C7

168 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 2310

Conjugacy classes & character table

1421 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 390);