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Cayley diagram

C3×C11 #145

order33
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C5 (order 20) · Inn(G) = C1 · Out(G) = C22×C5
holomorphG ⋊ Aut(G) = D3×AGL(1,𝔽11)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC33
abelianC3×C11

Subgroups

subgroupG/H
centerGC1
commutator, FrattiniC1G
LCSG ▷ C1
UCSC1 ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 1 generator on 14 points:

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

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

center ofC3×D3×C11, C11×H(3,𝔽3), C11×3−1+2, …
derived subgroup ofD33, Dic33, D3×D11, …
Frattini subgroup ofC9×C121, C2×C9×C121, D9×C121, …
Fitting subgroup ofD3×C11, C3×D11, D33, …
socle ofD3×C11, C3×D11, D33, …
maximal subgroup ofD3×C11, C3×D11, D33, …
maximal normal subgroup ofD3×C11, C3×D11, D33, …
maximal quotient ofC2×C3×C11, C9×C11, C32×C11, …
outer automorphism group ofD67

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33, 1);