metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊2D9, C8⋊2D18, D4⋊2D18, C72⋊4C22, D18.6D4, C24.33D6, C36.2C23, Dic9.8D4, D36.1C22, Dic18⋊1C22, D4⋊D9⋊2C2, (D4×D9)⋊2C2, (C9×D8)⋊4C2, C9⋊C8⋊1C22, C72⋊C2⋊3C2, C8⋊D9⋊3C2, C9⋊2(C8⋊C22), D4.D9⋊1C2, C3.(D8⋊S3), (C3×D8).6S3, (C3×D4).2D6, C6.90(S3×D4), C2.16(D4×D9), D4⋊2D9⋊1C2, C18.28(C2×D4), (D4×C9)⋊2C22, C4.2(C22×D9), (C4×D9).1C22, C12.41(C22×S3), SmallGroup(288,121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊D9
G = < a,b,c,d | a8=b2=c9=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >
Subgroups: 564 in 102 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, C9, Dic3, C12, D6, C2×C6, M4(2), D8, D8, SD16, C2×D4, C4○D4, D9, C18, C18, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C8⋊C22, Dic9, Dic9, C36, D18, D18, C2×C18, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C9⋊C8, C72, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, D4×C9, C22×D9, D8⋊S3, C8⋊D9, C72⋊C2, D4.D9, D4⋊D9, C9×D8, D4×D9, D4⋊2D9, D8⋊D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, C8⋊C22, D18, S3×D4, C22×D9, D8⋊S3, D4×D9, D8⋊D9
►On 72 points
(1 41 23 68 14 50 32 59)(2 42 24 69 15 51 33 60)(3 43 25 70 16 52 34 61)(4 44 26 71 17 53 35 62)(5 45 27 72 18 54 36 63)(6 37 19 64 10 46 28 55)(7 38 20 65 11 47 29 56)(8 39 21 66 12 48 30 57)(9 40 22 67 13 49 31 58)
(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(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)
(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)
G:=sub<Sym(72)| (1,41,23,68,14,50,32,59)(2,42,24,69,15,51,33,60)(3,43,25,70,16,52,34,61)(4,44,26,71,17,53,35,62)(5,45,27,72,18,54,36,63)(6,37,19,64,10,46,28,55)(7,38,20,65,11,47,29,56)(8,39,21,66,12,48,30,57)(9,40,22,67,13,49,31,58), (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72)>;
G:=Group( (1,41,23,68,14,50,32,59)(2,42,24,69,15,51,33,60)(3,43,25,70,16,52,34,61)(4,44,26,71,17,53,35,62)(5,45,27,72,18,54,36,63)(6,37,19,64,10,46,28,55)(7,38,20,65,11,47,29,56)(8,39,21,66,12,48,30,57)(9,40,22,67,13,49,31,58), (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72) );
G=PermutationGroup([[(1,41,23,68,14,50,32,59),(2,42,24,69,15,51,33,60),(3,43,25,70,16,52,34,61),(4,44,26,71,17,53,35,62),(5,45,27,72,18,54,36,63),(6,37,19,64,10,46,28,55),(7,38,20,65,11,47,29,56),(8,39,21,66,12,48,30,57),(9,40,22,67,13,49,31,58)], [(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,35),(29,34),(30,33),(31,32),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(45,54),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(63,72)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12 | 18A | 18B | 18C | 18D | ··· | 18I | 24A | 24B | 36A | 36B | 36C | 72A | ··· | 72F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 24 | 24 | 36 | 36 | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 4 | 4 | 18 | 36 | 2 | 2 | 18 | 36 | 2 | 8 | 8 | 4 | 36 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | D18 | D18 | C8⋊C22 | S3×D4 | D8⋊S3 | D4×D9 | D8⋊D9 |
| kernel | D8⋊D9 | C8⋊D9 | C72⋊C2 | D4.D9 | D4⋊D9 | C9×D8 | D4×D9 | D4⋊2D9 | C3×D8 | Dic9 | D18 | C24 | C3×D4 | D8 | C8 | D4 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 1 | 1 | 2 | 3 | 6 |
Matrix representation of D8⋊D9 ►in GL4(𝔽73) generated by
| 31 | 62 | 42 | 11 |
| 11 | 42 | 62 | 31 |
| 31 | 62 | 31 | 62 |
| 11 | 42 | 11 | 42 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 3 | 31 | 0 | 0 |
| 42 | 45 | 0 | 0 |
| 0 | 0 | 3 | 31 |
| 0 | 0 | 42 | 45 |
| 3 | 31 | 0 | 0 |
| 28 | 70 | 0 | 0 |
| 0 | 0 | 3 | 31 |
| 0 | 0 | 28 | 70 |
G:=sub<GL(4,GF(73))| [31,11,31,11,62,42,62,42,42,62,31,11,11,31,62,42],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[3,42,0,0,31,45,0,0,0,0,3,42,0,0,31,45],[3,28,0,0,31,70,0,0,0,0,3,28,0,0,31,70] >;
D8⋊D9 in GAP, Magma, Sage, TeX
D_8\rtimes D_9
% in TeX
G:=Group("D8:D9"); // GroupNames label
G:=SmallGroup(288,121);
// by ID
G=gap.SmallGroup(288,121);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,346,185,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations