direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×D17, C17⋊C18, C51.C6, C153⋊2C2, C3.(C3×D17), (C3×D17).C3, SmallGroup(306,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C9×D17 |
Generators and relations for C9×D17
G = < a,b,c | a9=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D17
►On 153 points
Generators in S153
(1 146 86 37 124 69 20 118 58)(2 147 87 38 125 70 21 119 59)(3 148 88 39 126 71 22 103 60)(4 149 89 40 127 72 23 104 61)(5 150 90 41 128 73 24 105 62)(6 151 91 42 129 74 25 106 63)(7 152 92 43 130 75 26 107 64)(8 153 93 44 131 76 27 108 65)(9 137 94 45 132 77 28 109 66)(10 138 95 46 133 78 29 110 67)(11 139 96 47 134 79 30 111 68)(12 140 97 48 135 80 31 112 52)(13 141 98 49 136 81 32 113 53)(14 142 99 50 120 82 33 114 54)(15 143 100 51 121 83 34 115 55)(16 144 101 35 122 84 18 116 56)(17 145 102 36 123 85 19 117 57)
(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)(137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 21)(19 20)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(35 38)(36 37)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(64 68)(65 67)(69 85)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 78)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(116 119)(117 118)(120 127)(121 126)(122 125)(123 124)(128 136)(129 135)(130 134)(131 133)(138 153)(139 152)(140 151)(141 150)(142 149)(143 148)(144 147)(145 146)
G:=sub<Sym(153)| (1,146,86,37,124,69,20,118,58)(2,147,87,38,125,70,21,119,59)(3,148,88,39,126,71,22,103,60)(4,149,89,40,127,72,23,104,61)(5,150,90,41,128,73,24,105,62)(6,151,91,42,129,74,25,106,63)(7,152,92,43,130,75,26,107,64)(8,153,93,44,131,76,27,108,65)(9,137,94,45,132,77,28,109,66)(10,138,95,46,133,78,29,110,67)(11,139,96,47,134,79,30,111,68)(12,140,97,48,135,80,31,112,52)(13,141,98,49,136,81,32,113,53)(14,142,99,50,120,82,33,114,54)(15,143,100,51,121,83,34,115,55)(16,144,101,35,122,84,18,116,56)(17,145,102,36,123,85,19,117,57), (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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,38)(36,37)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(64,68)(65,67)(69,85)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(116,119)(117,118)(120,127)(121,126)(122,125)(123,124)(128,136)(129,135)(130,134)(131,133)(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)(145,146)>;
G:=Group( (1,146,86,37,124,69,20,118,58)(2,147,87,38,125,70,21,119,59)(3,148,88,39,126,71,22,103,60)(4,149,89,40,127,72,23,104,61)(5,150,90,41,128,73,24,105,62)(6,151,91,42,129,74,25,106,63)(7,152,92,43,130,75,26,107,64)(8,153,93,44,131,76,27,108,65)(9,137,94,45,132,77,28,109,66)(10,138,95,46,133,78,29,110,67)(11,139,96,47,134,79,30,111,68)(12,140,97,48,135,80,31,112,52)(13,141,98,49,136,81,32,113,53)(14,142,99,50,120,82,33,114,54)(15,143,100,51,121,83,34,115,55)(16,144,101,35,122,84,18,116,56)(17,145,102,36,123,85,19,117,57), (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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,38)(36,37)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(64,68)(65,67)(69,85)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(116,119)(117,118)(120,127)(121,126)(122,125)(123,124)(128,136)(129,135)(130,134)(131,133)(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)(145,146) );
G=PermutationGroup([[(1,146,86,37,124,69,20,118,58),(2,147,87,38,125,70,21,119,59),(3,148,88,39,126,71,22,103,60),(4,149,89,40,127,72,23,104,61),(5,150,90,41,128,73,24,105,62),(6,151,91,42,129,74,25,106,63),(7,152,92,43,130,75,26,107,64),(8,153,93,44,131,76,27,108,65),(9,137,94,45,132,77,28,109,66),(10,138,95,46,133,78,29,110,67),(11,139,96,47,134,79,30,111,68),(12,140,97,48,135,80,31,112,52),(13,141,98,49,136,81,32,113,53),(14,142,99,50,120,82,33,114,54),(15,143,100,51,121,83,34,115,55),(16,144,101,35,122,84,18,116,56),(17,145,102,36,123,85,19,117,57)], [(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),(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,21),(19,20),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(35,38),(36,37),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(64,68),(65,67),(69,85),(70,84),(71,83),(72,82),(73,81),(74,80),(75,79),(76,78),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(116,119),(117,118),(120,127),(121,126),(122,125),(123,124),(128,136),(129,135),(130,134),(131,133),(138,153),(139,152),(140,151),(141,150),(142,149),(143,148),(144,147),(145,146)]])
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 17A | ··· | 17H | 18A | ··· | 18F | 51A | ··· | 51P | 153A | ··· | 153AV |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 17 | ··· | 17 | 18 | ··· | 18 | 51 | ··· | 51 | 153 | ··· | 153 |
| size | 1 | 17 | 1 | 1 | 17 | 17 | 1 | ··· | 1 | 2 | ··· | 2 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D17 | C3×D17 | C9×D17 |
| kernel | C9×D17 | C153 | C3×D17 | C51 | D17 | C17 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 8 | 16 | 48 |
Matrix representation of C9×D17 ►in GL2(𝔽307) generated by
| 93 | 0 |
| 0 | 93 |
| 269 | 1 |
| 119 | 13 |
| 301 | 149 |
| 76 | 6 |
G:=sub<GL(2,GF(307))| [93,0,0,93],[269,119,1,13],[301,76,149,6] >;
C9×D17 in GAP, Magma, Sage, TeX
C_9\times D_{17} % in TeX
G:=Group("C9xD17"); // GroupNames label
G:=SmallGroup(306,2);
// by ID
G=gap.SmallGroup(306,2);
# by ID
G:=PCGroup([4,-2,-3,-3,-17,29,4611]);
// Polycyclic
G:=Group<a,b,c|a^9=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export