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
(1 121 51 129 85 32 79 103 34)(2 35 104 80 17 86 130 52 122)(3 123 53 131 87 18 65 105 36)(4 37 106 66 19 88 132 54 124)(5 125 55 133 89 20 67 107 38)(6 39 108 68 21 90 134 56 126)(7 127 57 135 91 22 69 109 40)(8 41 110 70 23 92 136 58 128)(9 113 59 137 93 24 71 111 42)(10 43 112 72 25 94 138 60 114)(11 115 61 139 95 26 73 97 44)(12 45 98 74 27 96 140 62 116)(13 117 63 141 81 28 75 99 46)(14 47 100 76 29 82 142 64 118)(15 119 49 143 83 30 77 101 48)(16 33 102 78 31 84 144 50 120)
(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 92)(18 83)(19 90)(20 81)(21 88)(22 95)(23 86)(24 93)(25 84)(26 91)(27 82)(28 89)(29 96)(30 87)(31 94)(32 85)(33 114)(34 121)(35 128)(36 119)(37 126)(38 117)(39 124)(40 115)(41 122)(42 113)(43 120)(44 127)(45 118)(46 125)(47 116)(48 123)(49 105)(50 112)(51 103)(52 110)(53 101)(54 108)(55 99)(56 106)(57 97)(58 104)(59 111)(60 102)(61 109)(62 100)(63 107)(64 98)(65 143)(66 134)(67 141)(68 132)(69 139)(70 130)(71 137)(72 144)(73 135)(74 142)(75 133)(76 140)(77 131)(78 138)(79 129)(80 136)
G:=sub<Sym(144)| (1,121,51,129,85,32,79,103,34)(2,35,104,80,17,86,130,52,122)(3,123,53,131,87,18,65,105,36)(4,37,106,66,19,88,132,54,124)(5,125,55,133,89,20,67,107,38)(6,39,108,68,21,90,134,56,126)(7,127,57,135,91,22,69,109,40)(8,41,110,70,23,92,136,58,128)(9,113,59,137,93,24,71,111,42)(10,43,112,72,25,94,138,60,114)(11,115,61,139,95,26,73,97,44)(12,45,98,74,27,96,140,62,116)(13,117,63,141,81,28,75,99,46)(14,47,100,76,29,82,142,64,118)(15,119,49,143,83,30,77,101,48)(16,33,102,78,31,84,144,50,120), (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)(17,92)(18,83)(19,90)(20,81)(21,88)(22,95)(23,86)(24,93)(25,84)(26,91)(27,82)(28,89)(29,96)(30,87)(31,94)(32,85)(33,114)(34,121)(35,128)(36,119)(37,126)(38,117)(39,124)(40,115)(41,122)(42,113)(43,120)(44,127)(45,118)(46,125)(47,116)(48,123)(49,105)(50,112)(51,103)(52,110)(53,101)(54,108)(55,99)(56,106)(57,97)(58,104)(59,111)(60,102)(61,109)(62,100)(63,107)(64,98)(65,143)(66,134)(67,141)(68,132)(69,139)(70,130)(71,137)(72,144)(73,135)(74,142)(75,133)(76,140)(77,131)(78,138)(79,129)(80,136)>;
G:=Group( (1,121,51,129,85,32,79,103,34)(2,35,104,80,17,86,130,52,122)(3,123,53,131,87,18,65,105,36)(4,37,106,66,19,88,132,54,124)(5,125,55,133,89,20,67,107,38)(6,39,108,68,21,90,134,56,126)(7,127,57,135,91,22,69,109,40)(8,41,110,70,23,92,136,58,128)(9,113,59,137,93,24,71,111,42)(10,43,112,72,25,94,138,60,114)(11,115,61,139,95,26,73,97,44)(12,45,98,74,27,96,140,62,116)(13,117,63,141,81,28,75,99,46)(14,47,100,76,29,82,142,64,118)(15,119,49,143,83,30,77,101,48)(16,33,102,78,31,84,144,50,120), (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)(17,92)(18,83)(19,90)(20,81)(21,88)(22,95)(23,86)(24,93)(25,84)(26,91)(27,82)(28,89)(29,96)(30,87)(31,94)(32,85)(33,114)(34,121)(35,128)(36,119)(37,126)(38,117)(39,124)(40,115)(41,122)(42,113)(43,120)(44,127)(45,118)(46,125)(47,116)(48,123)(49,105)(50,112)(51,103)(52,110)(53,101)(54,108)(55,99)(56,106)(57,97)(58,104)(59,111)(60,102)(61,109)(62,100)(63,107)(64,98)(65,143)(66,134)(67,141)(68,132)(69,139)(70,130)(71,137)(72,144)(73,135)(74,142)(75,133)(76,140)(77,131)(78,138)(79,129)(80,136) );
G=PermutationGroup([[(1,121,51,129,85,32,79,103,34),(2,35,104,80,17,86,130,52,122),(3,123,53,131,87,18,65,105,36),(4,37,106,66,19,88,132,54,124),(5,125,55,133,89,20,67,107,38),(6,39,108,68,21,90,134,56,126),(7,127,57,135,91,22,69,109,40),(8,41,110,70,23,92,136,58,128),(9,113,59,137,93,24,71,111,42),(10,43,112,72,25,94,138,60,114),(11,115,61,139,95,26,73,97,44),(12,45,98,74,27,96,140,62,116),(13,117,63,141,81,28,75,99,46),(14,47,100,76,29,82,142,64,118),(15,119,49,143,83,30,77,101,48),(16,33,102,78,31,84,144,50,120)], [(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),(17,92),(18,83),(19,90),(20,81),(21,88),(22,95),(23,86),(24,93),(25,84),(26,91),(27,82),(28,89),(29,96),(30,87),(31,94),(32,85),(33,114),(34,121),(35,128),(36,119),(37,126),(38,117),(39,124),(40,115),(41,122),(42,113),(43,120),(44,127),(45,118),(46,125),(47,116),(48,123),(49,105),(50,112),(51,103),(52,110),(53,101),(54,108),(55,99),(56,106),(57,97),(58,104),(59,111),(60,102),(61,109),(62,100),(63,107),(64,98),(65,143),(66,134),(67,141),(68,132),(69,139),(70,130),(71,137),(72,144),(73,135),(74,142),(75,133),(76,140),(77,131),(78,138),(79,129),(80,136)]])
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