metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C9⋊3SD32, Q16⋊1D9, C36.5D4, C24.8D6, C8.6D18, D72.2C2, C18.10D8, C72.4C22, C9⋊C16⋊3C2, (C9×Q16)⋊1C2, C2.6(D4⋊D9), C4.3(C9⋊D4), C3.(C8.6D6), (C3×Q16).1S3, C6.17(D4⋊S3), C12.3(C3⋊D4), SmallGroup(288,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9⋊SD32
G = < a,b,c | a9=b16=c2=1, bab-1=cac=a-1, cbc=b7 >
Smallest permutation representation of C9⋊SD32
►On 144 points
Generators in S144
(1 141 52 102 21 75 48 93 127)(2 128 94 33 76 22 103 53 142)(3 143 54 104 23 77 34 95 113)(4 114 96 35 78 24 105 55 144)(5 129 56 106 25 79 36 81 115)(6 116 82 37 80 26 107 57 130)(7 131 58 108 27 65 38 83 117)(8 118 84 39 66 28 109 59 132)(9 133 60 110 29 67 40 85 119)(10 120 86 41 68 30 111 61 134)(11 135 62 112 31 69 42 87 121)(12 122 88 43 70 32 97 63 136)(13 137 64 98 17 71 44 89 123)(14 124 90 45 72 18 99 49 138)(15 139 50 100 19 73 46 91 125)(16 126 92 47 74 20 101 51 140)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 79)(18 70)(19 77)(20 68)(21 75)(22 66)(23 73)(24 80)(25 71)(26 78)(27 69)(28 76)(29 67)(30 74)(31 65)(32 72)(33 109)(34 100)(35 107)(36 98)(37 105)(38 112)(39 103)(40 110)(41 101)(42 108)(43 99)(44 106)(45 97)(46 104)(47 111)(48 102)(49 88)(50 95)(51 86)(52 93)(53 84)(54 91)(55 82)(56 89)(57 96)(58 87)(59 94)(60 85)(61 92)(62 83)(63 90)(64 81)(113 139)(114 130)(115 137)(116 144)(117 135)(118 142)(119 133)(120 140)(121 131)(122 138)(123 129)(124 136)(125 143)(126 134)(127 141)(128 132)
G:=sub<Sym(144)| (1,141,52,102,21,75,48,93,127)(2,128,94,33,76,22,103,53,142)(3,143,54,104,23,77,34,95,113)(4,114,96,35,78,24,105,55,144)(5,129,56,106,25,79,36,81,115)(6,116,82,37,80,26,107,57,130)(7,131,58,108,27,65,38,83,117)(8,118,84,39,66,28,109,59,132)(9,133,60,110,29,67,40,85,119)(10,120,86,41,68,30,111,61,134)(11,135,62,112,31,69,42,87,121)(12,122,88,43,70,32,97,63,136)(13,137,64,98,17,71,44,89,123)(14,124,90,45,72,18,99,49,138)(15,139,50,100,19,73,46,91,125)(16,126,92,47,74,20,101,51,140), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,79)(18,70)(19,77)(20,68)(21,75)(22,66)(23,73)(24,80)(25,71)(26,78)(27,69)(28,76)(29,67)(30,74)(31,65)(32,72)(33,109)(34,100)(35,107)(36,98)(37,105)(38,112)(39,103)(40,110)(41,101)(42,108)(43,99)(44,106)(45,97)(46,104)(47,111)(48,102)(49,88)(50,95)(51,86)(52,93)(53,84)(54,91)(55,82)(56,89)(57,96)(58,87)(59,94)(60,85)(61,92)(62,83)(63,90)(64,81)(113,139)(114,130)(115,137)(116,144)(117,135)(118,142)(119,133)(120,140)(121,131)(122,138)(123,129)(124,136)(125,143)(126,134)(127,141)(128,132)>;
G:=Group( (1,141,52,102,21,75,48,93,127)(2,128,94,33,76,22,103,53,142)(3,143,54,104,23,77,34,95,113)(4,114,96,35,78,24,105,55,144)(5,129,56,106,25,79,36,81,115)(6,116,82,37,80,26,107,57,130)(7,131,58,108,27,65,38,83,117)(8,118,84,39,66,28,109,59,132)(9,133,60,110,29,67,40,85,119)(10,120,86,41,68,30,111,61,134)(11,135,62,112,31,69,42,87,121)(12,122,88,43,70,32,97,63,136)(13,137,64,98,17,71,44,89,123)(14,124,90,45,72,18,99,49,138)(15,139,50,100,19,73,46,91,125)(16,126,92,47,74,20,101,51,140), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,79)(18,70)(19,77)(20,68)(21,75)(22,66)(23,73)(24,80)(25,71)(26,78)(27,69)(28,76)(29,67)(30,74)(31,65)(32,72)(33,109)(34,100)(35,107)(36,98)(37,105)(38,112)(39,103)(40,110)(41,101)(42,108)(43,99)(44,106)(45,97)(46,104)(47,111)(48,102)(49,88)(50,95)(51,86)(52,93)(53,84)(54,91)(55,82)(56,89)(57,96)(58,87)(59,94)(60,85)(61,92)(62,83)(63,90)(64,81)(113,139)(114,130)(115,137)(116,144)(117,135)(118,142)(119,133)(120,140)(121,131)(122,138)(123,129)(124,136)(125,143)(126,134)(127,141)(128,132) );
G=PermutationGroup([(1,141,52,102,21,75,48,93,127),(2,128,94,33,76,22,103,53,142),(3,143,54,104,23,77,34,95,113),(4,114,96,35,78,24,105,55,144),(5,129,56,106,25,79,36,81,115),(6,116,82,37,80,26,107,57,130),(7,131,58,108,27,65,38,83,117),(8,118,84,39,66,28,109,59,132),(9,133,60,110,29,67,40,85,119),(10,120,86,41,68,30,111,61,134),(11,135,62,112,31,69,42,87,121),(12,122,88,43,70,32,97,63,136),(13,137,64,98,17,71,44,89,123),(14,124,90,45,72,18,99,49,138),(15,139,50,100,19,73,46,91,125),(16,126,92,47,74,20,101,51,140)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,79),(18,70),(19,77),(20,68),(21,75),(22,66),(23,73),(24,80),(25,71),(26,78),(27,69),(28,76),(29,67),(30,74),(31,65),(32,72),(33,109),(34,100),(35,107),(36,98),(37,105),(38,112),(39,103),(40,110),(41,101),(42,108),(43,99),(44,106),(45,97),(46,104),(47,111),(48,102),(49,88),(50,95),(51,86),(52,93),(53,84),(54,91),(55,82),(56,89),(57,96),(58,87),(59,94),(60,85),(61,92),(62,83),(63,90),(64,81),(113,139),(114,130),(115,137),(116,144),(117,135),(118,142),(119,133),(120,140),(121,131),(122,138),(123,129),(124,136),(125,143),(126,134),(127,141),(128,132)])
39 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 18A | 18B | 18C | 24A | 24B | 36A | 36B | 36C | 36D | ··· | 36I | 72A | ··· | 72F |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 18 | 18 | 18 | 24 | 24 | 36 | 36 | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 72 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 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 | C8.6D6 | D4⋊D9 | C9⋊SD32 |
| kernel | C9⋊SD32 | C9⋊C16 | D72 | C9×Q16 | C3×Q16 | C36 | C24 | C18 | Q16 | 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 C9⋊SD32 ►in GL4(𝔽433) generated by
| 350 | 36 | 0 | 0 |
| 397 | 386 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 83 | 386 | 0 | 0 |
| 36 | 350 | 0 | 0 |
| 0 | 0 | 231 | 5 |
| 0 | 0 | 194 | 246 |
| 350 | 47 | 0 | 0 |
| 397 | 83 | 0 | 0 |
| 0 | 0 | 1 | 288 |
| 0 | 0 | 0 | 432 |
G:=sub<GL(4,GF(433))| [350,397,0,0,36,386,0,0,0,0,1,0,0,0,0,1],[83,36,0,0,386,350,0,0,0,0,231,194,0,0,5,246],[350,397,0,0,47,83,0,0,0,0,1,0,0,0,288,432] >;
C9⋊SD32 in GAP, Magma, Sage, TeX
C_9\rtimes {\rm SD}_{32} % in TeX
G:=Group("C9:SD32"); // GroupNames label
G:=SmallGroup(288,35);
// by ID
G=gap.SmallGroup(288,35);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,120,254,135,142,675,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^9=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^7>;
// generators/relations
Export