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Cayley diagram

C22⋊AGL(1,𝔽11) #89158

order440
isoclinismisoclinic class of (C11⋊C5)⋊Q8
automorphismAut(G) = C22×AGL(1,𝔽11) (order 440) · Inn(G) = C2×AGL(1,𝔽11) · Out(G) = C2
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC22⋊AGL(1,𝔽11)
unambiguous semidirect product(C22⋊D11)⋊C5

Splittings

FittingC22×C11 ⋊ C2×C5
HallC22⋊D11 ⋊ C5
HallC11⋊C5 ⋊ D4
SylowC11 ⋊ C5×D4

Subgroups

subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽11)
commutator, socle, cosocleC2×C11C22×C5
FittingC22×C11C2×C5
hypercenterC22AGL(1,𝔽11)
hypocenterC11C5×D4
LCSG ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C22

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

2^2,5,2,11|b^a=bd,e^a=e^10,e^c=e^9

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4339⋊C5, D4×AGL(1,𝔽11), #4341⋊C5, …
maximal normal subgroup of#4339⋊C5, D4×AGL(1,𝔽11), #4341⋊C5, …
maximal quotient of(Dic11⋊C4)⋊C5, ((C2×D11)⋊C4)⋊C5, #4323⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(440, 11);