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G = D23  order 46 = 2·23

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D23, C23⋊C2, sometimes denoted D46 or Dih23 or Dih46, SmallGroup(46,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — D23
Chief series
C1 — C23 — D23
Lower central
C23 — D23
Upper central
C1

Generators and relations for D23
 G = < a,b | a23=b2=1, bab=a-1 >

C1
23C2
C23
D23

Character table of D23

 class 1223A23B23C23D23E23F23G23H23I23J23K
 size 12322222222222
ρ11111111111111    trivial
ρ21-111111111111    linear of order 2
ρ320ζ2316+ζ237ζ2322+ζ23ζ2314+ζ239ζ2317+ζ236ζ2321+ζ232ζ2313+ζ2310ζ2318+ζ235ζ2320+ζ233ζ2312+ζ2311ζ2319+ζ234ζ2315+ζ238    orthogonal faithful
ρ420ζ2318+ζ235ζ2319+ζ234ζ2313+ζ2310ζ2322+ζ23ζ2315+ζ238ζ2317+ζ236ζ2320+ζ233ζ2312+ζ2311ζ2321+ζ232ζ2316+ζ237ζ2314+ζ239    orthogonal faithful
ρ520ζ2319+ζ234ζ2317+ζ236ζ2315+ζ238ζ2313+ζ2310ζ2312+ζ2311ζ2314+ζ239ζ2316+ζ237ζ2318+ζ235ζ2320+ζ233ζ2322+ζ23ζ2321+ζ232    orthogonal faithful
ρ620ζ2321+ζ232ζ2320+ζ233ζ2319+ζ234ζ2318+ζ235ζ2317+ζ236ζ2316+ζ237ζ2315+ζ238ζ2314+ζ239ζ2313+ζ2310ζ2312+ζ2311ζ2322+ζ23    orthogonal faithful
ρ720ζ2317+ζ236ζ2314+ζ239ζ2312+ζ2311ζ2315+ζ238ζ2318+ζ235ζ2321+ζ232ζ2322+ζ23ζ2319+ζ234ζ2316+ζ237ζ2313+ζ2310ζ2320+ζ233    orthogonal faithful
ρ820ζ2322+ζ23ζ2313+ζ2310ζ2321+ζ232ζ2314+ζ239ζ2320+ζ233ζ2315+ζ238ζ2319+ζ234ζ2316+ζ237ζ2318+ζ235ζ2317+ζ236ζ2312+ζ2311    orthogonal faithful
ρ920ζ2313+ζ2310ζ2315+ζ238ζ2320+ζ233ζ2321+ζ232ζ2316+ζ237ζ2312+ζ2311ζ2317+ζ236ζ2322+ζ23ζ2319+ζ234ζ2314+ζ239ζ2318+ζ235    orthogonal faithful
ρ1020ζ2315+ζ238ζ2312+ζ2311ζ2316+ζ237ζ2320+ζ233ζ2322+ζ23ζ2318+ζ235ζ2314+ζ239ζ2313+ζ2310ζ2317+ζ236ζ2321+ζ232ζ2319+ζ234    orthogonal faithful
ρ1120ζ2320+ζ233ζ2316+ζ237ζ2317+ζ236ζ2319+ζ234ζ2314+ζ239ζ2322+ζ23ζ2312+ζ2311ζ2321+ζ232ζ2315+ζ238ζ2318+ζ235ζ2313+ζ2310    orthogonal faithful
ρ1220ζ2314+ζ239ζ2321+ζ232ζ2318+ζ235ζ2312+ζ2311ζ2319+ζ234ζ2320+ζ233ζ2313+ζ2310ζ2317+ζ236ζ2322+ζ23ζ2315+ζ238ζ2316+ζ237    orthogonal faithful
ρ1320ζ2312+ζ2311ζ2318+ζ235ζ2322+ζ23ζ2316+ζ237ζ2313+ζ2310ζ2319+ζ234ζ2321+ζ232ζ2315+ζ238ζ2314+ζ239ζ2320+ζ233ζ2317+ζ236    orthogonal faithful

Permutation representations of D23
►On 23 points: primitive - transitive group 23T2
Generators in S23
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23)

(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)
 
G:=sub<Sym(23)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13)]])
 
G:=TransitiveGroup(23,2);
 

D23 is a maximal subgroup of   D69  D115  D161
D23 is a maximal quotient of   Dic23  D69  D115  D161

Matrix representation of D23 ►in GL2(𝔽47) generated by

646
10
,
646
3541
G:=sub<GL(2,GF(47))| [6,1,46,0],[6,35,46,41] >;
 

D23 in GAP, Magma, Sage, TeX 

D_{23}
 
% in TeX
 
G:=Group("D23");
 
// GroupNames label
 
G:=SmallGroup(46,1);
 
// by ID
 
G=gap.SmallGroup(46,1);
 
# by ID
 
G:=PCGroup([2,-2,-23,177]);
 
// Polycyclic
 
G:=Group<a,b|a^23=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D23 in TeX
Character table of D23 in TeX

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