Finite Group Databasebeta

← Prev (#57) Next (#59) →

Cayley diagram

C2×C11 #58

order22
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C5 (order 10) · Inn(G) = C1 · Out(G) = C2×C5
holomorphG ⋊ Aut(G) = C2×AGL(1,𝔽11)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC22
abelianC2×C11
multiplicative(ℤ/23)ˣ, (ℤ/46)ˣ

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 22

A minimal faithful permutation representation — 1 generator on 13 points:

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

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

center ofD4×C11, Q8×C11, C11×Dic3, …
derived subgroup ofDic22, D44, C22⋊D11, …
Frattini subgroup ofDic121, C4×C121, Dic242, …
Fitting subgroup ofDic11, C2×D11, C2×C11⋊C5, …
socle ofDic11, C4×C11, C2×D11, …
maximal subgroup ofDic11, C4×C11, C2×D11, …
maximal normal subgroup ofDic11, C4×C11, C2×D11, …
maximal quotient ofC4×C11, C22×C11, D3×C11, …
automorphism group ofC23, C2×C23
outer automorphism group ofC23, C2×C23, Dic23, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22, 2);