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Cayley diagram

C22×AGL(1,𝔽11) #89189

order440
isoclinismisoclinic class of AGL(1,𝔽11)
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C23

Aliases

direct productC22×AGL(1,𝔽11)
unambiguous semidirect productD11⋊(C22×C5)
unambiguous semidirect product(C11⋊C5)⋊C23
unambiguous semidirect product(C22×D11)⋊C5
unambiguous semidirect product(C2×C11⋊C5)⋊C22

Splittings

Fitting, socleC22×C11 ⋊ C2×C5
derived, SylowC11 ⋊ C23×C5
HallC22×D11 ⋊ C5
HallC11⋊C5 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22AGL(1,𝔽11)
commutator, cosocle, hypocenterC11C23×C5
FrattiniC1G
Fitting, socleC22×C11C2×C5
LCSG ▷ C11
UCSC1 ◁ C22

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 44

A minimal faithful permutation representation — 3 generators on 15 points:

(6,10,11,9,14,7,8,15,12,13);(3,4)(5,6,9,14,8,11,15,10,13,12,7);(1,2)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of((C2×D11)⋊C4)⋊C5, C2×C4×AGL(1,𝔽11), C2×C4⋊AGL(1,𝔽11), …
maximal normal subgroup of((C2×D11)⋊C4)⋊C5, C2×C4×AGL(1,𝔽11), C2×C4⋊AGL(1,𝔽11), …
maximal quotient ofC2×(C11⋊C5)⋊Q8, C2×C4×AGL(1,𝔽11), C2×C4⋊AGL(1,𝔽11), …
automorphism group ofC11⋊C8, C4×D11, C22⋊D11, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(440, 42);