metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C16⋊3D9, D18.C8, C144⋊4C2, C48.8S3, Dic9.C8, C9⋊1M5(2), C24.85D6, C8.19D18, C72.20C22, C9⋊C16⋊4C2, C9⋊C8.2C4, C6.7(S3×C8), C2.3(C8×D9), C18.2(C2×C8), (C8×D9).2C2, (C4×D9).2C4, C4.17(C4×D9), C3.(D6.C8), C12.66(C4×S3), C36.22(C2×C4), SmallGroup(288,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊D9
G = < a,b,c | a16=b9=c2=1, ab=ba, cac=a9, cbc=b-1 >
Smallest permutation representation of C16⋊D9
►On 144 points
Generators in S144
(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 47 144 96 121 56 104 27 71)(2 48 129 81 122 57 105 28 72)(3 33 130 82 123 58 106 29 73)(4 34 131 83 124 59 107 30 74)(5 35 132 84 125 60 108 31 75)(6 36 133 85 126 61 109 32 76)(7 37 134 86 127 62 110 17 77)(8 38 135 87 128 63 111 18 78)(9 39 136 88 113 64 112 19 79)(10 40 137 89 114 49 97 20 80)(11 41 138 90 115 50 98 21 65)(12 42 139 91 116 51 99 22 66)(13 43 140 92 117 52 100 23 67)(14 44 141 93 118 53 101 24 68)(15 45 142 94 119 54 102 25 69)(16 46 143 95 120 55 103 26 70)
(1 71)(2 80)(3 73)(4 66)(5 75)(6 68)(7 77)(8 70)(9 79)(10 72)(11 65)(12 74)(13 67)(14 76)(15 69)(16 78)(17 37)(18 46)(19 39)(20 48)(21 41)(22 34)(23 43)(24 36)(25 45)(26 38)(27 47)(28 40)(29 33)(30 42)(31 35)(32 44)(49 81)(50 90)(51 83)(52 92)(53 85)(54 94)(55 87)(56 96)(57 89)(58 82)(59 91)(60 84)(61 93)(62 86)(63 95)(64 88)(97 129)(98 138)(99 131)(100 140)(101 133)(102 142)(103 135)(104 144)(105 137)(106 130)(107 139)(108 132)(109 141)(110 134)(111 143)(112 136)(114 122)(116 124)(118 126)(120 128)
G:=sub<Sym(144)| (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), (1,47,144,96,121,56,104,27,71)(2,48,129,81,122,57,105,28,72)(3,33,130,82,123,58,106,29,73)(4,34,131,83,124,59,107,30,74)(5,35,132,84,125,60,108,31,75)(6,36,133,85,126,61,109,32,76)(7,37,134,86,127,62,110,17,77)(8,38,135,87,128,63,111,18,78)(9,39,136,88,113,64,112,19,79)(10,40,137,89,114,49,97,20,80)(11,41,138,90,115,50,98,21,65)(12,42,139,91,116,51,99,22,66)(13,43,140,92,117,52,100,23,67)(14,44,141,93,118,53,101,24,68)(15,45,142,94,119,54,102,25,69)(16,46,143,95,120,55,103,26,70), (1,71)(2,80)(3,73)(4,66)(5,75)(6,68)(7,77)(8,70)(9,79)(10,72)(11,65)(12,74)(13,67)(14,76)(15,69)(16,78)(17,37)(18,46)(19,39)(20,48)(21,41)(22,34)(23,43)(24,36)(25,45)(26,38)(27,47)(28,40)(29,33)(30,42)(31,35)(32,44)(49,81)(50,90)(51,83)(52,92)(53,85)(54,94)(55,87)(56,96)(57,89)(58,82)(59,91)(60,84)(61,93)(62,86)(63,95)(64,88)(97,129)(98,138)(99,131)(100,140)(101,133)(102,142)(103,135)(104,144)(105,137)(106,130)(107,139)(108,132)(109,141)(110,134)(111,143)(112,136)(114,122)(116,124)(118,126)(120,128)>;
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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(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), (1,47,144,96,121,56,104,27,71)(2,48,129,81,122,57,105,28,72)(3,33,130,82,123,58,106,29,73)(4,34,131,83,124,59,107,30,74)(5,35,132,84,125,60,108,31,75)(6,36,133,85,126,61,109,32,76)(7,37,134,86,127,62,110,17,77)(8,38,135,87,128,63,111,18,78)(9,39,136,88,113,64,112,19,79)(10,40,137,89,114,49,97,20,80)(11,41,138,90,115,50,98,21,65)(12,42,139,91,116,51,99,22,66)(13,43,140,92,117,52,100,23,67)(14,44,141,93,118,53,101,24,68)(15,45,142,94,119,54,102,25,69)(16,46,143,95,120,55,103,26,70), (1,71)(2,80)(3,73)(4,66)(5,75)(6,68)(7,77)(8,70)(9,79)(10,72)(11,65)(12,74)(13,67)(14,76)(15,69)(16,78)(17,37)(18,46)(19,39)(20,48)(21,41)(22,34)(23,43)(24,36)(25,45)(26,38)(27,47)(28,40)(29,33)(30,42)(31,35)(32,44)(49,81)(50,90)(51,83)(52,92)(53,85)(54,94)(55,87)(56,96)(57,89)(58,82)(59,91)(60,84)(61,93)(62,86)(63,95)(64,88)(97,129)(98,138)(99,131)(100,140)(101,133)(102,142)(103,135)(104,144)(105,137)(106,130)(107,139)(108,132)(109,141)(110,134)(111,143)(112,136)(114,122)(116,124)(118,126)(120,128) );
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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(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)], [(1,47,144,96,121,56,104,27,71),(2,48,129,81,122,57,105,28,72),(3,33,130,82,123,58,106,29,73),(4,34,131,83,124,59,107,30,74),(5,35,132,84,125,60,108,31,75),(6,36,133,85,126,61,109,32,76),(7,37,134,86,127,62,110,17,77),(8,38,135,87,128,63,111,18,78),(9,39,136,88,113,64,112,19,79),(10,40,137,89,114,49,97,20,80),(11,41,138,90,115,50,98,21,65),(12,42,139,91,116,51,99,22,66),(13,43,140,92,117,52,100,23,67),(14,44,141,93,118,53,101,24,68),(15,45,142,94,119,54,102,25,69),(16,46,143,95,120,55,103,26,70)], [(1,71),(2,80),(3,73),(4,66),(5,75),(6,68),(7,77),(8,70),(9,79),(10,72),(11,65),(12,74),(13,67),(14,76),(15,69),(16,78),(17,37),(18,46),(19,39),(20,48),(21,41),(22,34),(23,43),(24,36),(25,45),(26,38),(27,47),(28,40),(29,33),(30,42),(31,35),(32,44),(49,81),(50,90),(51,83),(52,92),(53,85),(54,94),(55,87),(56,96),(57,89),(58,82),(59,91),(60,84),(61,93),(62,86),(63,95),(64,88),(97,129),(98,138),(99,131),(100,140),(101,133),(102,142),(103,135),(104,144),(105,137),(106,130),(107,139),(108,132),(109,141),(110,134),(111,143),(112,136),(114,122),(116,124),(118,126),(120,128)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 9A | 9B | 9C | 12A | 12B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 48A | ··· | 48H | 72A | ··· | 72L | 144A | ··· | 144X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 18 | 2 | 1 | 1 | 18 | 2 | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D6 | D9 | C4×S3 | M5(2) | D18 | S3×C8 | C4×D9 | D6.C8 | C8×D9 | C16⋊D9 |
| kernel | C16⋊D9 | C9⋊C16 | C144 | C8×D9 | C9⋊C8 | C4×D9 | Dic9 | D18 | C48 | C24 | C16 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 3 | 2 | 4 | 3 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of C16⋊D9 ►in GL4(𝔽433) generated by
| 258 | 83 | 0 | 0 |
| 350 | 175 | 0 | 0 |
| 0 | 0 | 432 | 0 |
| 0 | 0 | 0 | 432 |
| 0 | 1 | 0 | 0 |
| 432 | 432 | 0 | 0 |
| 0 | 0 | 47 | 350 |
| 0 | 0 | 83 | 397 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 83 | 397 |
| 0 | 0 | 47 | 350 |
G:=sub<GL(4,GF(433))| [258,350,0,0,83,175,0,0,0,0,432,0,0,0,0,432],[0,432,0,0,1,432,0,0,0,0,47,83,0,0,350,397],[0,1,0,0,1,0,0,0,0,0,83,47,0,0,397,350] >;
C16⋊D9 in GAP, Magma, Sage, TeX
C_{16}\rtimes D_9 % in TeX
G:=Group("C16:D9"); // GroupNames label
G:=SmallGroup(288,5);
// by ID
G=gap.SmallGroup(288,5);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,36,58,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^16=b^9=c^2=1,a*b=b*a,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations
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