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Cayley diagram

C31 #93

order31
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C3×C5 (order 30) · Inn(G) = C1 · Out(G) = C2×C3×C5
holomorphG ⋊ Aut(G) = AGL(1,𝔽31)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC31

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 31

A minimal faithful permutation representation — 1 generator on 31 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)

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

center ofD3×C31, D5×C31, A4×C31, …
derived subgroup ofD31, C31⋊C3, Dic31, …
Frattini subgroup ofC961, D961, C2×C961, …
Fitting subgroup ofD31, C31⋊C3, ΓL(1,𝔽32), …
socle ofD31, C31⋊C3, ΓL(1,𝔽32), …
maximal subgroup ofD31, C2×C31, C31⋊C3, …
maximal normal subgroup ofD31, C2×C31, C31⋊C3, …
maximal quotient ofC2×C31, C3×C31, C5×C31, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31, 1);