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Cayley diagram

C3×AGL(1,𝔽11) #66507

order330
isoclinismisoclinic class of AGL(1,𝔽11)
automorphismAut(G) = C2×AGL(1,𝔽11) (order 220) · Inn(G) = AGL(1,𝔽11) · Out(G) = C2
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC3×AGL(1,𝔽11)
unambiguous semidirect productD11⋊(C3×C5)
unambiguous semidirect product(C11⋊C5)⋊(C2×C3)
unambiguous semidirect product(C3×D11)⋊C5

Splittings

Fitting, socle, HallC3×C11 ⋊ C2×C5
derived, SylowC11 ⋊ C2×C3×C5
HallC3×C11⋊C5 ⋊ C2
HallC3×D11 ⋊ C5
HallC11⋊C5 ⋊ C2×C3
HallD11 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,𝔽11)
commutator, cosocle, hypocenterC11C2×C3×C5
FrattiniC1G
Fitting, socleC3×C11C2×C5
LCSG ▷ C11
UCSC1 ◁ C3

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 33

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽11), C2×C3×AGL(1,𝔽11), C9×AGL(1,𝔽11), …
maximal normal subgroup ofD3×AGL(1,𝔽11), C2×C3×AGL(1,𝔽11), C9×AGL(1,𝔽11), …
maximal quotient ofC3×Dic11⋊C5, C2×C3×AGL(1,𝔽11), C9×AGL(1,𝔽11), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(330, 1);