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Cayley diagram

C53 #264

order53
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C4×C13 (order 52) · Inn(G) = C1 · Out(G) = C4×C13
holomorphG ⋊ Aut(G) = AGL(1,𝔽53)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC53

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 53

A minimal faithful permutation representation — 1 generator on 53 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

center ofD3×C53, D5×C53, A4×C53, …
derived subgroup ofD53, Dic53, #6157, …
Frattini subgroup ofC2809, C5×C2809, C2×C3×C2809, …
Fitting subgroup ofD53, #6157, C53⋊C13, …
socle ofD53, #6157, C53⋊C13, …
maximal subgroup ofD53, C2×C53, C3×C53, …
maximal normal subgroup ofD53, C2×C53, C3×C53, …
maximal quotient ofC2×C53, C3×C53, C5×C53, …
outer automorphism group ofD107

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(53, 1);