metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.D9, C9⋊2SD32, C24.7D6, C8.5D18, C36.4D4, C18.9D8, Dic36⋊3C2, C72.3C22, C9⋊C16⋊2C2, (C9×D8).1C2, (C3×D8).2S3, C3.(D8.S3), C2.5(D4⋊D9), C4.2(C9⋊D4), C6.16(D4⋊S3), C12.2(C3⋊D4), SmallGroup(288,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.D9
G = < a,b,c,d | a8=b2=c9=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >
Smallest permutation representation of D8.D9
►On 144 points
Generators in S144
(1 59 23 50 14 68 32 41)(2 60 24 51 15 69 33 42)(3 61 25 52 16 70 34 43)(4 62 26 53 17 71 35 44)(5 63 27 54 18 72 36 45)(6 55 19 46 10 64 28 37)(7 56 20 47 11 65 29 38)(8 57 21 48 12 66 30 39)(9 58 22 49 13 67 31 40)(73 109 100 136 82 118 91 127)(74 110 101 137 83 119 92 128)(75 111 102 138 84 120 93 129)(76 112 103 139 85 121 94 130)(77 113 104 140 86 122 95 131)(78 114 105 141 87 123 96 132)(79 115 106 142 88 124 97 133)(80 116 107 143 89 125 98 134)(81 117 108 144 90 126 99 135)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 37)(7 38)(8 39)(9 40)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(109 118)(110 119)(111 120)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)
(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 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144)
(1 84 14 75)(2 83 15 74)(3 82 16 73)(4 90 17 81)(5 89 18 80)(6 88 10 79)(7 87 11 78)(8 86 12 77)(9 85 13 76)(19 106 28 97)(20 105 29 96)(21 104 30 95)(22 103 31 94)(23 102 32 93)(24 101 33 92)(25 100 34 91)(26 108 35 99)(27 107 36 98)(37 124 46 115)(38 123 47 114)(39 122 48 113)(40 121 49 112)(41 120 50 111)(42 119 51 110)(43 118 52 109)(44 126 53 117)(45 125 54 116)(55 142 64 133)(56 141 65 132)(57 140 66 131)(58 139 67 130)(59 138 68 129)(60 137 69 128)(61 136 70 127)(62 144 71 135)(63 143 72 134)
G:=sub<Sym(144)| (1,59,23,50,14,68,32,41)(2,60,24,51,15,69,33,42)(3,61,25,52,16,70,34,43)(4,62,26,53,17,71,35,44)(5,63,27,54,18,72,36,45)(6,55,19,46,10,64,28,37)(7,56,20,47,11,65,29,38)(8,57,21,48,12,66,30,39)(9,58,22,49,13,67,31,40)(73,109,100,136,82,118,91,127)(74,110,101,137,83,119,92,128)(75,111,102,138,84,120,93,129)(76,112,103,139,85,121,94,130)(77,113,104,140,86,122,95,131)(78,114,105,141,87,123,96,132)(79,115,106,142,88,124,97,133)(80,116,107,143,89,125,98,134)(81,117,108,144,90,126,99,135), (1,41)(2,42)(3,43)(4,44)(5,45)(6,37)(7,38)(8,39)(9,40)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126), (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)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,84,14,75)(2,83,15,74)(3,82,16,73)(4,90,17,81)(5,89,18,80)(6,88,10,79)(7,87,11,78)(8,86,12,77)(9,85,13,76)(19,106,28,97)(20,105,29,96)(21,104,30,95)(22,103,31,94)(23,102,32,93)(24,101,33,92)(25,100,34,91)(26,108,35,99)(27,107,36,98)(37,124,46,115)(38,123,47,114)(39,122,48,113)(40,121,49,112)(41,120,50,111)(42,119,51,110)(43,118,52,109)(44,126,53,117)(45,125,54,116)(55,142,64,133)(56,141,65,132)(57,140,66,131)(58,139,67,130)(59,138,68,129)(60,137,69,128)(61,136,70,127)(62,144,71,135)(63,143,72,134)>;
G:=Group( (1,59,23,50,14,68,32,41)(2,60,24,51,15,69,33,42)(3,61,25,52,16,70,34,43)(4,62,26,53,17,71,35,44)(5,63,27,54,18,72,36,45)(6,55,19,46,10,64,28,37)(7,56,20,47,11,65,29,38)(8,57,21,48,12,66,30,39)(9,58,22,49,13,67,31,40)(73,109,100,136,82,118,91,127)(74,110,101,137,83,119,92,128)(75,111,102,138,84,120,93,129)(76,112,103,139,85,121,94,130)(77,113,104,140,86,122,95,131)(78,114,105,141,87,123,96,132)(79,115,106,142,88,124,97,133)(80,116,107,143,89,125,98,134)(81,117,108,144,90,126,99,135), (1,41)(2,42)(3,43)(4,44)(5,45)(6,37)(7,38)(8,39)(9,40)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126), (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)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,84,14,75)(2,83,15,74)(3,82,16,73)(4,90,17,81)(5,89,18,80)(6,88,10,79)(7,87,11,78)(8,86,12,77)(9,85,13,76)(19,106,28,97)(20,105,29,96)(21,104,30,95)(22,103,31,94)(23,102,32,93)(24,101,33,92)(25,100,34,91)(26,108,35,99)(27,107,36,98)(37,124,46,115)(38,123,47,114)(39,122,48,113)(40,121,49,112)(41,120,50,111)(42,119,51,110)(43,118,52,109)(44,126,53,117)(45,125,54,116)(55,142,64,133)(56,141,65,132)(57,140,66,131)(58,139,67,130)(59,138,68,129)(60,137,69,128)(61,136,70,127)(62,144,71,135)(63,143,72,134) );
G=PermutationGroup([[(1,59,23,50,14,68,32,41),(2,60,24,51,15,69,33,42),(3,61,25,52,16,70,34,43),(4,62,26,53,17,71,35,44),(5,63,27,54,18,72,36,45),(6,55,19,46,10,64,28,37),(7,56,20,47,11,65,29,38),(8,57,21,48,12,66,30,39),(9,58,22,49,13,67,31,40),(73,109,100,136,82,118,91,127),(74,110,101,137,83,119,92,128),(75,111,102,138,84,120,93,129),(76,112,103,139,85,121,94,130),(77,113,104,140,86,122,95,131),(78,114,105,141,87,123,96,132),(79,115,106,142,88,124,97,133),(80,116,107,143,89,125,98,134),(81,117,108,144,90,126,99,135)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,37),(7,38),(8,39),(9,40),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(109,118),(110,119),(111,120),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126)], [(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),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144)], [(1,84,14,75),(2,83,15,74),(3,82,16,73),(4,90,17,81),(5,89,18,80),(6,88,10,79),(7,87,11,78),(8,86,12,77),(9,85,13,76),(19,106,28,97),(20,105,29,96),(21,104,30,95),(22,103,31,94),(23,102,32,93),(24,101,33,92),(25,100,34,91),(26,108,35,99),(27,107,36,98),(37,124,46,115),(38,123,47,114),(39,122,48,113),(40,121,49,112),(41,120,50,111),(42,119,51,110),(43,118,52,109),(44,126,53,117),(45,125,54,116),(55,142,64,133),(56,141,65,132),(57,140,66,131),(58,139,67,130),(59,138,68,129),(60,137,69,128),(61,136,70,127),(62,144,71,135),(63,143,72,134)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12 | 16A | 16B | 16C | 16D | 18A | 18B | 18C | 18D | ··· | 18I | 24A | 24B | 36A | 36B | 36C | 72A | ··· | 72F |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 16 | 16 | 16 | 16 | 18 | 18 | 18 | 18 | ··· | 18 | 24 | 24 | 36 | 36 | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 8 | 2 | 2 | 72 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D8 | D9 | C3⋊D4 | SD32 | D18 | C9⋊D4 | D4⋊S3 | D8.S3 | D4⋊D9 | D8.D9 |
| kernel | D8.D9 | C9⋊C16 | Dic36 | C9×D8 | C3×D8 | C36 | C24 | C18 | D8 | C12 | C9 | C8 | C4 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 4 | 3 | 6 | 1 | 2 | 3 | 6 |
Matrix representation of D8.D9 ►in GL4(𝔽433) generated by
| 432 | 0 | 0 | 0 |
| 0 | 432 | 0 | 0 |
| 0 | 0 | 0 | 213 |
| 0 | 0 | 124 | 206 |
| 432 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 213 |
| 0 | 0 | 309 | 0 |
| 296 | 0 | 0 | 0 |
| 0 | 256 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 256 | 0 | 0 |
| 296 | 0 | 0 | 0 |
| 0 | 0 | 187 | 159 |
| 0 | 0 | 194 | 246 |
G:=sub<GL(4,GF(433))| [432,0,0,0,0,432,0,0,0,0,0,124,0,0,213,206],[432,0,0,0,0,1,0,0,0,0,0,309,0,0,213,0],[296,0,0,0,0,256,0,0,0,0,1,0,0,0,0,1],[0,296,0,0,256,0,0,0,0,0,187,194,0,0,159,246] >;
D8.D9 in GAP, Magma, Sage, TeX
D_8.D_9
% in TeX
G:=Group("D8.D9"); // GroupNames label
G:=SmallGroup(288,34);
// by ID
G=gap.SmallGroup(288,34);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,254,135,142,675,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^9=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations
Export