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AGL(1,𝔽11)×C3⋊(C7⋊D7) #44568606

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

Aliases

direct productAGL(1,𝔽11)×C3⋊(C7⋊D7)
unambiguous semidirect productD11⋊(C5×C3⋊(C7⋊D7))
unambiguous semidirect product(C3×AGL(1,𝔽11))⋊(C7⋊D7)
unambiguous semidirect product(C7×AGL(1,𝔽11))⋊D21
unambiguous semidirect product(D11×C3⋊(C7⋊D7))⋊C5

Splittings

Fitting, socle, derived, HallC3×C72×C11 ⋊ C22×C5
HallC3×C72×C11⋊C5 ⋊ C22
HallD11×C3⋊(C7⋊D7) ⋊ C5
HallC72×C11⋊C5 ⋊ C2×D3
HallC72×C11 ⋊ C2×C5×D3
HallC3×C11⋊C5 ⋊ C2×C7⋊D7
HallC3×C72 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×C3⋊(C7⋊D7)
SylowC72 ⋊ D3×AGL(1,𝔽11)
HallC3×C11 ⋊ C2×C5×C7⋊D7
SylowC11 ⋊ C2×C5×C3⋊(C7⋊D7)
SylowC3 ⋊ C7⋊D7×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C72×C11C22×C5
LCSG ▷ C3×C72×C11
UCSC1

400 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 1617

A minimal faithful permutation representation — 3 generators on 28 points:

(2,3)(4,5,10,6,12)(7,14,13,8,11)(16,17)(18,19)(20,21)(23,24)(25,26)(27,28);(1,2,3)(5,6)(7,9)(8,11)(10,14)(12,13)(22,23,25,27,28,26,24);(15,16,18,20,21,19,17)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 308);