metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D44, C4⋊D11, C11⋊1D4, C44⋊1C2, D22⋊1C2, C2.4D22, C22.3C22, sometimes denoted D88 or Dih44 or Dih88, SmallGroup(88,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D44
G = < a,b | a44=b2=1, bab=a-1 >
Character table of D44
| class | 1 | 2A | 2B | 2C | 4 | 11A | 11B | 11C | 11D | 11E | 22A | 22B | 22C | 22D | 22E | 44A | 44B | 44C | 44D | 44E | 44F | 44G | 44H | 44I | 44J | |
| size | 1 | 1 | 22 | 22 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | 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 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | ζ119+ζ112 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ118+ζ113 | ζ119+ζ112 | ζ117+ζ114 | ζ1110+ζ11 | ζ116+ζ115 | ζ116+ζ115 | ζ1110+ζ11 | orthogonal lifted from D11 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | ζ117+ζ114 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ116+ζ115 | ζ117+ζ114 | ζ118+ζ113 | ζ119+ζ112 | ζ1110+ζ11 | ζ1110+ζ11 | ζ119+ζ112 | orthogonal lifted from D11 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | ζ116+ζ115 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ119+ζ112 | ζ116+ζ115 | ζ1110+ζ11 | ζ118+ζ113 | ζ117+ζ114 | ζ117+ζ114 | ζ118+ζ113 | orthogonal lifted from D11 |
| ρ9 | 2 | 2 | 0 | 0 | -2 | ζ117+ζ114 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ1110+ζ11 | ζ119+ζ112 | ζ118+ζ113 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ116-ζ115 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ1110-ζ11 | -ζ119-ζ112 | orthogonal lifted from D22 |
| ρ10 | 2 | 2 | 0 | 0 | -2 | ζ119+ζ112 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ116+ζ115 | ζ1110+ζ11 | ζ117+ζ114 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ118-ζ113 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ116-ζ115 | -ζ1110-ζ11 | orthogonal lifted from D22 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | ζ118+ζ113 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ1110+ζ11 | ζ118+ζ113 | ζ116+ζ115 | ζ117+ζ114 | ζ119+ζ112 | ζ119+ζ112 | ζ117+ζ114 | orthogonal lifted from D11 |
| ρ12 | 2 | 2 | 0 | 0 | -2 | ζ1110+ζ11 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | -ζ119-ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ117-ζ114 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ118-ζ113 | -ζ116-ζ115 | orthogonal lifted from D22 |
| ρ13 | 2 | 2 | 0 | 0 | -2 | ζ118+ζ113 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ119+ζ112 | ζ117+ζ114 | ζ116+ζ115 | -ζ116-ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ1110-ζ11 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ117-ζ114 | -ζ119-ζ112 | -ζ119-ζ112 | -ζ117-ζ114 | orthogonal lifted from D22 |
| ρ14 | 2 | 2 | 0 | 0 | -2 | ζ116+ζ115 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ117+ζ114 | ζ118+ζ113 | ζ1110+ζ11 | -ζ1110-ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ119-ζ112 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ118-ζ113 | -ζ117-ζ114 | -ζ117-ζ114 | -ζ118-ζ113 | orthogonal lifted from D22 |
| ρ15 | 2 | 2 | 0 | 0 | 2 | ζ1110+ζ11 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ118+ζ113 | ζ116+ζ115 | ζ119+ζ112 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ117+ζ114 | ζ1110+ζ11 | ζ119+ζ112 | ζ116+ζ115 | ζ118+ζ113 | ζ118+ζ113 | ζ116+ζ115 | orthogonal lifted from D11 |
| ρ16 | 2 | -2 | 0 | 0 | 0 | ζ1110+ζ11 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ4ζ119+ζ4ζ112 | ζ4ζ1110-ζ4ζ11 | ζ4ζ117-ζ4ζ114 | -ζ4ζ117+ζ4ζ114 | -ζ4ζ1110+ζ4ζ11 | ζ4ζ119-ζ4ζ112 | -ζ43ζ116+ζ43ζ115 | ζ43ζ118-ζ43ζ113 | -ζ43ζ118+ζ43ζ113 | ζ43ζ116-ζ43ζ115 | orthogonal faithful |
| ρ17 | 2 | -2 | 0 | 0 | 0 | ζ1110+ζ11 | ζ118+ζ113 | ζ117+ζ114 | ζ116+ζ115 | ζ119+ζ112 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ119-ζ112 | ζ4ζ119-ζ4ζ112 | -ζ4ζ1110+ζ4ζ11 | -ζ4ζ117+ζ4ζ114 | ζ4ζ117-ζ4ζ114 | ζ4ζ1110-ζ4ζ11 | -ζ4ζ119+ζ4ζ112 | ζ43ζ116-ζ43ζ115 | -ζ43ζ118+ζ43ζ113 | ζ43ζ118-ζ43ζ113 | -ζ43ζ116+ζ43ζ115 | orthogonal faithful |
| ρ18 | 2 | -2 | 0 | 0 | 0 | ζ118+ζ113 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ116-ζ115 | ζ43ζ116-ζ43ζ115 | ζ43ζ118-ζ43ζ113 | -ζ4ζ1110+ζ4ζ11 | ζ4ζ1110-ζ4ζ11 | -ζ43ζ118+ζ43ζ113 | -ζ43ζ116+ζ43ζ115 | -ζ4ζ117+ζ4ζ114 | -ζ4ζ119+ζ4ζ112 | ζ4ζ119-ζ4ζ112 | ζ4ζ117-ζ4ζ114 | orthogonal faithful |
| ρ19 | 2 | -2 | 0 | 0 | 0 | ζ116+ζ115 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ4ζ1110+ζ4ζ11 | ζ43ζ116-ζ43ζ115 | ζ4ζ119-ζ4ζ112 | -ζ4ζ119+ζ4ζ112 | -ζ43ζ116+ζ43ζ115 | ζ4ζ1110-ζ4ζ11 | ζ43ζ118-ζ43ζ113 | ζ4ζ117-ζ4ζ114 | -ζ4ζ117+ζ4ζ114 | -ζ43ζ118+ζ43ζ113 | orthogonal faithful |
| ρ20 | 2 | -2 | 0 | 0 | 0 | ζ117+ζ114 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ43ζ118+ζ43ζ113 | ζ4ζ117-ζ4ζ114 | -ζ43ζ116+ζ43ζ115 | ζ43ζ116-ζ43ζ115 | -ζ4ζ117+ζ4ζ114 | ζ43ζ118-ζ43ζ113 | -ζ4ζ119+ζ4ζ112 | -ζ4ζ1110+ζ4ζ11 | ζ4ζ1110-ζ4ζ11 | ζ4ζ119-ζ4ζ112 | orthogonal faithful |
| ρ21 | 2 | -2 | 0 | 0 | 0 | ζ119+ζ112 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ117-ζ114 | ζ4ζ117-ζ4ζ114 | -ζ4ζ119+ζ4ζ112 | -ζ43ζ118+ζ43ζ113 | ζ43ζ118-ζ43ζ113 | ζ4ζ119-ζ4ζ112 | -ζ4ζ117+ζ4ζ114 | ζ4ζ1110-ζ4ζ11 | ζ43ζ116-ζ43ζ115 | -ζ43ζ116+ζ43ζ115 | -ζ4ζ1110+ζ4ζ11 | orthogonal faithful |
| ρ22 | 2 | -2 | 0 | 0 | 0 | ζ116+ζ115 | ζ117+ζ114 | ζ119+ζ112 | ζ118+ζ113 | ζ1110+ζ11 | -ζ116-ζ115 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ118-ζ113 | -ζ1110-ζ11 | ζ4ζ1110-ζ4ζ11 | -ζ43ζ116+ζ43ζ115 | -ζ4ζ119+ζ4ζ112 | ζ4ζ119-ζ4ζ112 | ζ43ζ116-ζ43ζ115 | -ζ4ζ1110+ζ4ζ11 | -ζ43ζ118+ζ43ζ113 | -ζ4ζ117+ζ4ζ114 | ζ4ζ117-ζ4ζ114 | ζ43ζ118-ζ43ζ113 | orthogonal faithful |
| ρ23 | 2 | -2 | 0 | 0 | 0 | ζ117+ζ114 | ζ1110+ζ11 | ζ116+ζ115 | ζ119+ζ112 | ζ118+ζ113 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ118-ζ113 | ζ43ζ118-ζ43ζ113 | -ζ4ζ117+ζ4ζ114 | ζ43ζ116-ζ43ζ115 | -ζ43ζ116+ζ43ζ115 | ζ4ζ117-ζ4ζ114 | -ζ43ζ118+ζ43ζ113 | ζ4ζ119-ζ4ζ112 | ζ4ζ1110-ζ4ζ11 | -ζ4ζ1110+ζ4ζ11 | -ζ4ζ119+ζ4ζ112 | orthogonal faithful |
| ρ24 | 2 | -2 | 0 | 0 | 0 | ζ119+ζ112 | ζ116+ζ115 | ζ118+ζ113 | ζ1110+ζ11 | ζ117+ζ114 | -ζ119-ζ112 | -ζ118-ζ113 | -ζ116-ζ115 | -ζ1110-ζ11 | -ζ117-ζ114 | -ζ4ζ117+ζ4ζ114 | ζ4ζ119-ζ4ζ112 | ζ43ζ118-ζ43ζ113 | -ζ43ζ118+ζ43ζ113 | -ζ4ζ119+ζ4ζ112 | ζ4ζ117-ζ4ζ114 | -ζ4ζ1110+ζ4ζ11 | -ζ43ζ116+ζ43ζ115 | ζ43ζ116-ζ43ζ115 | ζ4ζ1110-ζ4ζ11 | orthogonal faithful |
| ρ25 | 2 | -2 | 0 | 0 | 0 | ζ118+ζ113 | ζ119+ζ112 | ζ1110+ζ11 | ζ117+ζ114 | ζ116+ζ115 | -ζ118-ζ113 | -ζ1110-ζ11 | -ζ119-ζ112 | -ζ117-ζ114 | -ζ116-ζ115 | -ζ43ζ116+ζ43ζ115 | -ζ43ζ118+ζ43ζ113 | ζ4ζ1110-ζ4ζ11 | -ζ4ζ1110+ζ4ζ11 | ζ43ζ118-ζ43ζ113 | ζ43ζ116-ζ43ζ115 | ζ4ζ117-ζ4ζ114 | ζ4ζ119-ζ4ζ112 | -ζ4ζ119+ζ4ζ112 | -ζ4ζ117+ζ4ζ114 | orthogonal faithful |
►On 44 points
Generators in S44
(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 32 33 34 35 36 37 38 39 40 41 42 43 44)
(1 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)
G:=sub<Sym(44)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)], [(1,44),(2,43),(3,42),(4,41),(5,40),(6,39),(7,38),(8,37),(9,36),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,24),(22,23)]])
D44 is a maximal subgroup of
C8⋊D11 D88 D4⋊D11 Q8⋊D11 D44⋊5C2 D4×D11 D44⋊C2 C3⋊D44 D132 D44⋊C5 C5⋊D44 D220
D44 is a maximal quotient of
C8⋊D11 D88 Dic44 C44⋊C4 D22⋊C4 C3⋊D44 D132 C5⋊D44 D220
Matrix representation of D44 ►in GL2(𝔽43) generated by
| 40 | 1 |
| 42 | 0 |
| 38 | 29 |
| 14 | 5 |
G:=sub<GL(2,GF(43))| [40,42,1,0],[38,14,29,5] >;
D44 in GAP, Magma, Sage, TeX
D_{44} % in TeX
G:=Group("D44"); // GroupNames label
G:=SmallGroup(88,5);
// by ID
G=gap.SmallGroup(88,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-11,49,21,1283]);
// Polycyclic
G:=Group<a,b|a^44=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D44 in TeX
Character table of D44 in TeX