Finite Group Databasebeta

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D3×C72×AGL(1,𝔽11) #44568593

order32340
automorphism?
homologyAbelianization (H1) = C22×C5×C72 · Schur multiplier (H2) = C2×C7

Aliases

direct productD3×C72×AGL(1,𝔽11)
unambiguous semidirect productD11⋊(C5×D3×C72)
unambiguous semidirect product(D3×D11)⋊(C5×C72)
unambiguous semidirect product(C3×AGL(1,𝔽11))⋊(C2×C72)
unambiguous semidirect product(D3×C11⋊C5)⋊(C2×C72)
unambiguous semidirect product(C3⋊AGL(1,𝔽11))⋊(C2×C72)
unambiguous semidirect product(D3×C7×D11)⋊(C5×C7)
unambiguous semidirect product(D3×C72×D11)⋊C5

Splittings

Fitting, socle, HallC3×C72×C11 ⋊ C22×C5
derived, HallC3×C11 ⋊ C22×C5×C72
HallC3×C72×C11⋊C5 ⋊ C22
HallD3×C72×D11 ⋊ C5
HallC72×C11⋊C5 ⋊ C2×D3
HallC72×C11 ⋊ C2×C5×D3
HallC3×C11⋊C5 ⋊ C22×C72
HallC3×C72 ⋊ C2×AGL(1,𝔽11)
HallD3×D11 ⋊ C5×C72
HallC11⋊C5 ⋊ C2×D3×C72
SylowC11 ⋊ C2×C5×D3×C72
SylowC3 ⋊ C2×C72×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, hypercenterC72D3×AGL(1,𝔽11)
commutator, cosocle, hypocenterC3×C11C22×C5×C72
FrattiniC1G
Fitting, socleC3×C72×C11C22×C5
LCSG ▷ C3×C11
UCSC1 ◁ C72

400 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 1617

Conjugacy classes & character table

1617 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 295);