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Cayley diagram

C3×C53 #3925

order159
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C4×C13 (order 104) · Inn(G) = C1 · Out(G) = C2×C4×C13
holomorphG ⋊ Aut(G) = D3×AGL(1,𝔽53)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC159
abelianC3×C53

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 56
transitive degreeμt(G) = 159

A minimal faithful permutation representation — 1 generator on 56 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

center ofC3×D3×C53, H(3,𝔽3)×C53, 3−1+2×C53, …
derived subgroup ofD159, Dic159, C3⋊#6157, …
Frattini subgroup ofC9×C2809, D25281
Fitting subgroup ofD3×C53, C3×D53, D159, …
socle ofD3×C53, C3×D53, D159, …
maximal subgroup ofD3×C53, C3×D53, D159, …
maximal normal subgroup ofD3×C53, C3×D53, D159, …
maximal quotient ofC2×C3×C53, C9×C53, C32×C53, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(159, 1);