metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8:3D22, D88:6C2, Q8:2D22, C88:3C22, D22.7D4, D4.3D22, SD16:1D11, D44:2C22, C44.5C23, Dic11.9D4, D4:D11:3C2, (D4xD11):3C2, C11:C8:2C22, Q8:D11:2C2, C88:C2:1C2, C11:3(C8:C22), C22.31(C2xD4), C2.19(D4xD11), D44:C2:1C2, (C11xSD16):1C2, (Q8xC11):2C22, C4.5(C22xD11), (C4xD11).2C22, (D4xC11).3C22, SmallGroup(352,109)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D88:C2
G = < a,b,c | a88=b2=c2=1, bab=a-1, cac=a67, bc=cb >
Subgroups: 546 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2xC4, D4, D4, Q8, C23, C11, M4(2), D8, SD16, SD16, C2xD4, C4oD4, D11, C22, C22, C8:C22, Dic11, C44, C44, D22, D22, C2xC22, C11:C8, C88, C4xD11, C4xD11, D44, D44, C11:D4, D4xC11, Q8xC11, C22xD11, C88:C2, D88, D4:D11, Q8:D11, C11xSD16, D4xD11, D44:C2, D88:C2
Quotients: C1, C2, C22, D4, C23, C2xD4, D11, C8:C22, D22, C22xD11, D4xD11, D88:C2
►On 88 points
(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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(78 88)(79 87)(80 86)(81 85)(82 84)
(2 68)(3 47)(4 26)(6 72)(7 51)(8 30)(10 76)(11 55)(12 34)(14 80)(15 59)(16 38)(18 84)(19 63)(20 42)(22 88)(23 67)(24 46)(27 71)(28 50)(31 75)(32 54)(35 79)(36 58)(39 83)(40 62)(43 87)(44 66)(48 70)(52 74)(56 78)(60 82)(64 86)
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(78,88)(79,87)(80,86)(81,85)(82,84), (2,68)(3,47)(4,26)(6,72)(7,51)(8,30)(10,76)(11,55)(12,34)(14,80)(15,59)(16,38)(18,84)(19,63)(20,42)(22,88)(23,67)(24,46)(27,71)(28,50)(31,75)(32,54)(35,79)(36,58)(39,83)(40,62)(43,87)(44,66)(48,70)(52,74)(56,78)(60,82)(64,86)>;
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(78,88)(79,87)(80,86)(81,85)(82,84), (2,68)(3,47)(4,26)(6,72)(7,51)(8,30)(10,76)(11,55)(12,34)(14,80)(15,59)(16,38)(18,84)(19,63)(20,42)(22,88)(23,67)(24,46)(27,71)(28,50)(31,75)(32,54)(35,79)(36,58)(39,83)(40,62)(43,87)(44,66)(48,70)(52,74)(56,78)(60,82)(64,86) );
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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(78,88),(79,87),(80,86),(81,85),(82,84)], [(2,68),(3,47),(4,26),(6,72),(7,51),(8,30),(10,76),(11,55),(12,34),(14,80),(15,59),(16,38),(18,84),(19,63),(20,42),(22,88),(23,67),(24,46),(27,71),(28,50),(31,75),(32,54),(35,79),(36,58),(39,83),(40,62),(43,87),(44,66),(48,70),(52,74),(56,78),(60,82),(64,86)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22J | 44A | ··· | 44E | 44F | ··· | 44J | 88A | ··· | 88J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 4 | 22 | 44 | 44 | 2 | 4 | 22 | 4 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D11 | D22 | D22 | D22 | C8:C22 | D4xD11 | D88:C2 |
| kernel | D88:C2 | C88:C2 | D88 | D4:D11 | Q8:D11 | C11xSD16 | D4xD11 | D44:C2 | Dic11 | D22 | SD16 | C8 | D4 | Q8 | C11 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 5 | 10 |
Matrix representation of D88:C2 ►in GL4(F89) generated by
| 0 | 0 | 79 | 76 |
| 0 | 0 | 77 | 88 |
| 81 | 13 | 74 | 42 |
| 74 | 72 | 47 | 70 |
| 5 | 34 | 0 | 0 |
| 15 | 84 | 0 | 0 |
| 32 | 44 | 2 | 13 |
| 21 | 36 | 34 | 87 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 19 | 67 | 88 | 0 |
| 22 | 0 | 0 | 88 |
G:=sub<GL(4,GF(89))| [0,0,81,74,0,0,13,72,79,77,74,47,76,88,42,70],[5,15,32,21,34,84,44,36,0,0,2,34,0,0,13,87],[1,0,19,22,0,1,67,0,0,0,88,0,0,0,0,88] >;
D88:C2 in GAP, Magma, Sage, TeX
D_{88}\rtimes C_2 % in TeX
G:=Group("D88:C2"); // GroupNames label
G:=SmallGroup(352,109);
// by ID
G=gap.SmallGroup(352,109);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,362,116,86,297,159,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^88=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^67,b*c=c*b>;
// generators/relations