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Cayley diagram

C3×C11⋊C5 #4226

order165
isoclinismisoclinic class of C11⋊C5
automorphismAut(G) = C2×AGL(1,𝔽11) (order 220) · Inn(G) = C11⋊C5 · Out(G) = C22
holomorphG ⋊ Aut(G) = D3×(C11⋊C5)⋊AGL(1,𝔽11)
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC3×C11⋊C5
unambiguous semidirect productC11⋊(C3×C5)
unambiguous semidirect product(C3×C11)⋊C5

Splittings

Fitting, socle, HallC3×C11 ⋊ C5
derived, SylowC11 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC3C11⋊C5
commutator, cosocle, hypocenterC11C3×C5
FrattiniC1G
Fitting, socleC3×C11C5
LCSG ▷ C11
UCSC1 ◁ C3

8 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:

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

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽11), D3×C11⋊C5, C3⋊AGL(1,𝔽11), …
maximal normal subgroup ofC3×AGL(1,𝔽11), D3×C11⋊C5, C3⋊AGL(1,𝔽11), …
maximal quotient ofC2×C3×C11⋊C5, C9×C11⋊C5, C32×C11⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(165, 1);