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AGL(1,𝔽11)×#64424 #44568555

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productAGL(1,𝔽11)×#64424
unambiguous semidirect productD11⋊(C5×#64424)
unambiguous semidirect product(D11×#64424)⋊C5

Splittings

Fitting, socle, derived, HallC72×C11 ⋊ C22×C3×C5
HallC7⋊D7×AGL(1,𝔽11) ⋊ C3
HallC11⋊C5×#3861 ⋊ C22
HallD11×#64424 ⋊ C5
HallC72×C11⋊C5 ⋊ C22×C3
HallD11×C7⋊D7 ⋊ C3×C5
HallC11×#3861 ⋊ C22×C5
Hall#3861 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×#64424
SylowC72 ⋊ C2×C3×AGL(1,𝔽11)
SylowC11 ⋊ C2×C5×#64424

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C11C22×C3×C5
LCSG ▷ C72×C11
UCSC1

400 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 25
transitive degreeμt(G) = 539

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽11)×#64423⋊C2
maximal normal subgroup ofAGL(1,𝔽11)×#64423⋊C2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 257);