metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D88, C11⋊1D8, C88⋊1C2, C8⋊1D11, D44⋊1C2, C4.9D22, C2.4D44, C22.2D4, C44.9C22, sometimes denoted D176 or Dih88 or Dih176, SmallGroup(176,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D88
G = < a,b | a88=b2=1, bab=a-1 >
Smallest permutation representation of D88
►On 88 points
Generators in S88
(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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88)
(1 88)(2 87)(3 86)(4 85)(5 84)(6 83)(7 82)(8 81)(9 80)(10 79)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)
G:=sub<Sym(88)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88), (1,88)(2,87)(3,86)(4,85)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88), (1,88)(2,87)(3,86)(4,85)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)], [(1,88),(2,87),(3,86),(4,85),(5,84),(6,83),(7,82),(8,81),(9,80),(10,79),(11,78),(12,77),(13,76),(14,75),(15,74),(16,73),(17,72),(18,71),(19,70),(20,69),(21,68),(22,67),(23,66),(24,65),(25,64),(26,63),(27,62),(28,61),(29,60),(30,59),(31,58),(32,57),(33,56),(34,55),(35,54),(36,53),(37,52),(38,51),(39,50),(40,49),(41,48),(42,47),(43,46),(44,45)]])
D88 is a maximal subgroup of
D176 C176⋊C2 C11⋊D16 C8.6D22 D88⋊7C2 C8⋊D22 D8×D11 D88⋊C2 D88⋊5C2
D88 is a maximal quotient of
D176 C176⋊C2 Dic88 C44.5Q8 C2.D88
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 44 | 44 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
47 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D8 | D11 | D22 | D44 | D88 |
| kernel | D88 | C88 | D44 | C22 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 5 | 5 | 10 | 20 |
Matrix representation of D88 ►in GL2(𝔽89) generated by
| 17 | 35 |
| 85 | 18 |
| 79 | 3 |
| 56 | 10 |
G:=sub<GL(2,GF(89))| [17,85,35,18],[79,56,3,10] >;
D88 in GAP, Magma, Sage, TeX
D_{88} % in TeX
G:=Group("D88"); // GroupNames label
G:=SmallGroup(176,6);
// by ID
G=gap.SmallGroup(176,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,61,66,182,42,4004]);
// Polycyclic
G:=Group<a,b|a^88=b^2=1,b*a*b=a^-1>;
// generators/relations
Export