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C2×(C11⋊C5)⋊#64425 #44568582

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2×C7

Aliases

direct productC2×(C11⋊C5)⋊#64425
unambiguous semidirect product(C2×C11⋊C5)⋊#64425
unambiguous semidirect product(C2×C11⋊#64425)⋊C5

Splittings

Fitting, socleC2×C72×C11 ⋊ C2×C3×C5
derived, HallC72×C11 ⋊ C22×C3×C5
HallC2×C7⋊(C7⋊AGL(1,𝔽11)) ⋊ C3
HallC11⋊C5×#3862 ⋊ C22
HallC2×C11⋊#64425 ⋊ C5
HallC72×C11⋊C5 ⋊ C22×C3
HallC2×C7⋊D77 ⋊ C3×C5
HallC11×#3862 ⋊ C22×C5
Hall#3862 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×#64425
SylowC72 ⋊ C2×C3×AGL(1,𝔽11)
SylowC11 ⋊ C2×C5×#64425

Subgroups

subgroupG/H
center, hypercenterC2(C11⋊C5)⋊#64425
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC2×C72×C11C2×C3×C5
LCSG ▷ C72×C11
UCSC1 ◁ C2

200 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 27
transitive degreeμt(G) = 462

A minimal faithful permutation representation — 2 generators on 27 points:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 284);