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Cayley diagram

C5×C11 #281

order55
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C4×C5 (order 40) · Inn(G) = C1 · Out(G) = C2×C4×C5
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×AGL(1,𝔽11)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC55
abelianC5×C11

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 55

A minimal faithful permutation representation — 1 generator on 16 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16)

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

center ofC5×D3×C11, C5×D5×C11, C5×C11×A4, …
derived subgroup ofD55, Dic55, C11⋊AGL(1,𝔽5), …
Frattini subgroup ofC25×C121, C121⋊C25, C2×C3×C25×C121, …
Fitting subgroup ofD5×C11, C5×D11, D55, …
socle ofD5×C11, C5×D11, D55, …
maximal subgroup ofD5×C11, C5×D11, D55, …
maximal normal subgroup ofD5×C11, C5×D11, D55, …
maximal quotient ofC2×C5×C11, C3×C5×C11, C11×C25, …
outer automorphism group ofD121

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(55, 2);