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D7×C3⋊(C7⋊AGL(1,𝔽11)) #44568599

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

Aliases

direct productD7×C3⋊(C7⋊AGL(1,𝔽11))
unambiguous semidirect product(C3×D7)⋊(C7⋊AGL(1,𝔽11))
unambiguous semidirect product(C7×D7)⋊(C3⋊AGL(1,𝔽11))
unambiguous semidirect product(D7×C11⋊C5)⋊D21
unambiguous semidirect product(D7×D231)⋊C5

Splittings

Fitting, socle, derived, HallC3×C72×C11 ⋊ C22×C5
HallC3×C72×C11⋊C5 ⋊ C22
HallD7×D231 ⋊ C5
HallC72×C11⋊C5 ⋊ C2×D3
HallC72×C11 ⋊ C2×C5×D3
HallC3×C11⋊C5 ⋊ D72
HallC3×C72 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ D7×D21
SylowC72 ⋊ C2×C3⋊AGL(1,𝔽11)
HallC3×C11 ⋊ C5×D72
SylowC11 ⋊ C5×D7×D21
SylowC3 ⋊ D7×C7⋊AGL(1,𝔽11)

Subgroups

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

208 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 462

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 301);