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Cayley diagram

C23 #59

order23
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C11 (order 22) · Inn(G) = C1 · Out(G) = C2×C11
holomorphG ⋊ Aut(G) = AGL(1,𝔽23)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC23

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 23

A minimal faithful permutation representation — 1 generator on 23 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)

Conjugacy classes & character table

23 conjugacy classes — view the full character table.

Relationships

center ofD3×C23, D5×C23, A4×C23, …
derived subgroup ofD23, Dic23, C2×D23, …
Frattini subgroup ofC529, D529, C2×C529, …
Fitting subgroup ofD23, C23⋊C11, AGL(1,𝔽23), …
socle ofD23, C23⋊C11, AGL(1,𝔽23), …
maximal subgroup ofD23, C2×C23, C3×C23, …
maximal normal subgroup ofD23, C2×C23, C3×C23, …
maximal quotient ofC2×C23, C3×C23, C5×C23, …
outer automorphism group ofD47

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23, 1);