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Cayley diagram

C2×AGL(1,𝔽11) #6352

order220
isoclinismisoclinic class of AGL(1,𝔽11)
automorphismAut(G) = C2×AGL(1,𝔽11) (order 220) · Inn(G) = AGL(1,𝔽11) · Out(G) = C2
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,𝔽11)
unambiguous semidirect productD11⋊(C2×C5)
unambiguous semidirect product(C2×D11)⋊C5
unambiguous semidirect product(C11⋊C5)⋊C22
unambiguous semidirect product(C2×C11⋊C5)⋊C2

Splittings

Fitting, socleC2×C11 ⋊ C2×C5
derived, SylowC11 ⋊ C22×C5
HallC11⋊C5 ⋊ C22
HallC2×D11 ⋊ C5

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽11)
commutator, cosocle, hypocenterC11C22×C5
FrattiniC1G
Fitting, socleC2×C11C2×C5
LCSG ▷ C11
UCSC1 ◁ C2

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 22

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×AGL(1,𝔽11), C4⋊AGL(1,𝔽11), C22⋊AGL(1,𝔽11), …
maximal normal subgroup ofC4×AGL(1,𝔽11), C4⋊AGL(1,𝔽11), C22⋊AGL(1,𝔽11), …
maximal quotient of(C11⋊C5)⋊Q8, C4×AGL(1,𝔽11), C4⋊AGL(1,𝔽11), …
automorphism group ofDic11, C2×D11, C3×D11, …
holomorph ofC2×C11

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(220, 7);