direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D22, C2×D11, C22⋊C2, C11⋊C22, sometimes denoted D44 or Dih22 or Dih44, SmallGroup(44,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — D22 |
Generators and relations for D22
G = < a,b | a22=b2=1, bab=a-1 >
Character table of D22
| class | 1 | 2A | 2B | 2C | 11A | 11B | 11C | 11D | 11E | 22A | 22B | 22C | 22D | 22E | |
| size | 1 | 1 | 11 | 11 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | orthogonal lifted from D11 |
| ρ6 | 2 | 2 | 0 | 0 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | orthogonal lifted from D11 |
| ρ7 | 2 | 2 | 0 | 0 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | orthogonal lifted from D11 |
| ρ8 | 2 | 2 | 0 | 0 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | orthogonal lifted from D11 |
| ρ9 | 2 | -2 | 0 | 0 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ1110-ζ11 | orthogonal faithful |
| ρ10 | 2 | -2 | 0 | 0 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ118-ζ113 | orthogonal faithful |
| ρ11 | 2 | -2 | 0 | 0 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ116-ζ115 | orthogonal faithful |
| ρ12 | 2 | 2 | 0 | 0 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | orthogonal lifted from D11 |
| ρ13 | 2 | -2 | 0 | 0 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ119-ζ112 | orthogonal faithful |
| ρ14 | 2 | -2 | 0 | 0 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ117-ζ114 | orthogonal faithful |
►On 22 points - transitive group 22T3
Generators in S22
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 22)(13 21)(14 20)(15 19)(16 18)
G:=sub<Sym(22)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22), (1,11)(2,10)(3,9)(4,8)(5,7)(12,22)(13,21)(14,20)(15,19)(16,18)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22), (1,11)(2,10)(3,9)(4,8)(5,7)(12,22)(13,21)(14,20)(15,19)(16,18) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,22),(13,21),(14,20),(15,19),(16,18)]])
G:=TransitiveGroup(22,3);
D22 is a maximal subgroup of
D44 C11⋊D4
D22 is a maximal quotient of Dic22 D44 C11⋊D4
Matrix representation of D22 ►in GL2(𝔽23) generated by
| 20 | 7 |
| 16 | 16 |
| 20 | 7 |
| 12 | 3 |
G:=sub<GL(2,GF(23))| [20,16,7,16],[20,12,7,3] >;
D22 in GAP, Magma, Sage, TeX
D_{22} % in TeX
G:=Group("D22"); // GroupNames label
G:=SmallGroup(44,3);
// by ID
G=gap.SmallGroup(44,3);
# by ID
G:=PCGroup([3,-2,-2,-11,362]);
// Polycyclic
G:=Group<a,b|a^22=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D22 in TeX
Character table of D22 in TeX