direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D9×C23, C9⋊C46, C207⋊3C2, C69.2S3, C3.(S3×C23), SmallGroup(414,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C23 |
Generators and relations for D9×C23
G = < a,b,c | a23=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D9×C23
►On 207 points
Generators in S207
(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)(139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161)(162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207)
(1 159 56 112 117 30 188 84 175)(2 160 57 113 118 31 189 85 176)(3 161 58 114 119 32 190 86 177)(4 139 59 115 120 33 191 87 178)(5 140 60 93 121 34 192 88 179)(6 141 61 94 122 35 193 89 180)(7 142 62 95 123 36 194 90 181)(8 143 63 96 124 37 195 91 182)(9 144 64 97 125 38 196 92 183)(10 145 65 98 126 39 197 70 184)(11 146 66 99 127 40 198 71 162)(12 147 67 100 128 41 199 72 163)(13 148 68 101 129 42 200 73 164)(14 149 69 102 130 43 201 74 165)(15 150 47 103 131 44 202 75 166)(16 151 48 104 132 45 203 76 167)(17 152 49 105 133 46 204 77 168)(18 153 50 106 134 24 205 78 169)(19 154 51 107 135 25 206 79 170)(20 155 52 108 136 26 207 80 171)(21 156 53 109 137 27 185 81 172)(22 157 54 110 138 28 186 82 173)(23 158 55 111 116 29 187 83 174)
(1 175)(2 176)(3 177)(4 178)(5 179)(6 180)(7 181)(8 182)(9 183)(10 184)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 168)(18 169)(19 170)(20 171)(21 172)(22 173)(23 174)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 113)(32 114)(33 115)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 105)(47 202)(48 203)(49 204)(50 205)(51 206)(52 207)(53 185)(54 186)(55 187)(56 188)(57 189)(58 190)(59 191)(60 192)(61 193)(62 194)(63 195)(64 196)(65 197)(66 198)(67 199)(68 200)(69 201)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)(81 156)(82 157)(83 158)(84 159)(85 160)(86 161)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)
G:=sub<Sym(207)| (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161)(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207), (1,159,56,112,117,30,188,84,175)(2,160,57,113,118,31,189,85,176)(3,161,58,114,119,32,190,86,177)(4,139,59,115,120,33,191,87,178)(5,140,60,93,121,34,192,88,179)(6,141,61,94,122,35,193,89,180)(7,142,62,95,123,36,194,90,181)(8,143,63,96,124,37,195,91,182)(9,144,64,97,125,38,196,92,183)(10,145,65,98,126,39,197,70,184)(11,146,66,99,127,40,198,71,162)(12,147,67,100,128,41,199,72,163)(13,148,68,101,129,42,200,73,164)(14,149,69,102,130,43,201,74,165)(15,150,47,103,131,44,202,75,166)(16,151,48,104,132,45,203,76,167)(17,152,49,105,133,46,204,77,168)(18,153,50,106,134,24,205,78,169)(19,154,51,107,135,25,206,79,170)(20,155,52,108,136,26,207,80,171)(21,156,53,109,137,27,185,81,172)(22,157,54,110,138,28,186,82,173)(23,158,55,111,116,29,187,83,174), (1,175)(2,176)(3,177)(4,178)(5,179)(6,180)(7,181)(8,182)(9,183)(10,184)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,171)(21,172)(22,173)(23,174)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,202)(48,203)(49,204)(50,205)(51,206)(52,207)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(61,193)(62,194)(63,195)(64,196)(65,197)(66,198)(67,199)(68,200)(69,201)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)>;
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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161)(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207), (1,159,56,112,117,30,188,84,175)(2,160,57,113,118,31,189,85,176)(3,161,58,114,119,32,190,86,177)(4,139,59,115,120,33,191,87,178)(5,140,60,93,121,34,192,88,179)(6,141,61,94,122,35,193,89,180)(7,142,62,95,123,36,194,90,181)(8,143,63,96,124,37,195,91,182)(9,144,64,97,125,38,196,92,183)(10,145,65,98,126,39,197,70,184)(11,146,66,99,127,40,198,71,162)(12,147,67,100,128,41,199,72,163)(13,148,68,101,129,42,200,73,164)(14,149,69,102,130,43,201,74,165)(15,150,47,103,131,44,202,75,166)(16,151,48,104,132,45,203,76,167)(17,152,49,105,133,46,204,77,168)(18,153,50,106,134,24,205,78,169)(19,154,51,107,135,25,206,79,170)(20,155,52,108,136,26,207,80,171)(21,156,53,109,137,27,185,81,172)(22,157,54,110,138,28,186,82,173)(23,158,55,111,116,29,187,83,174), (1,175)(2,176)(3,177)(4,178)(5,179)(6,180)(7,181)(8,182)(9,183)(10,184)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,168)(18,169)(19,170)(20,171)(21,172)(22,173)(23,174)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,202)(48,203)(49,204)(50,205)(51,206)(52,207)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(61,193)(62,194)(63,195)(64,196)(65,197)(66,198)(67,199)(68,200)(69,201)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144) );
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),(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161),(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184),(185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207)], [(1,159,56,112,117,30,188,84,175),(2,160,57,113,118,31,189,85,176),(3,161,58,114,119,32,190,86,177),(4,139,59,115,120,33,191,87,178),(5,140,60,93,121,34,192,88,179),(6,141,61,94,122,35,193,89,180),(7,142,62,95,123,36,194,90,181),(8,143,63,96,124,37,195,91,182),(9,144,64,97,125,38,196,92,183),(10,145,65,98,126,39,197,70,184),(11,146,66,99,127,40,198,71,162),(12,147,67,100,128,41,199,72,163),(13,148,68,101,129,42,200,73,164),(14,149,69,102,130,43,201,74,165),(15,150,47,103,131,44,202,75,166),(16,151,48,104,132,45,203,76,167),(17,152,49,105,133,46,204,77,168),(18,153,50,106,134,24,205,78,169),(19,154,51,107,135,25,206,79,170),(20,155,52,108,136,26,207,80,171),(21,156,53,109,137,27,185,81,172),(22,157,54,110,138,28,186,82,173),(23,158,55,111,116,29,187,83,174)], [(1,175),(2,176),(3,177),(4,178),(5,179),(6,180),(7,181),(8,182),(9,183),(10,184),(11,162),(12,163),(13,164),(14,165),(15,166),(16,167),(17,168),(18,169),(19,170),(20,171),(21,172),(22,173),(23,174),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,113),(32,114),(33,115),(34,93),(35,94),(36,95),(37,96),(38,97),(39,98),(40,99),(41,100),(42,101),(43,102),(44,103),(45,104),(46,105),(47,202),(48,203),(49,204),(50,205),(51,206),(52,207),(53,185),(54,186),(55,187),(56,188),(57,189),(58,190),(59,191),(60,192),(61,193),(62,194),(63,195),(64,196),(65,197),(66,198),(67,199),(68,200),(69,201),(70,145),(71,146),(72,147),(73,148),(74,149),(75,150),(76,151),(77,152),(78,153),(79,154),(80,155),(81,156),(82,157),(83,158),(84,159),(85,160),(86,161),(87,139),(88,140),(89,141),(90,142),(91,143),(92,144)]])
138 conjugacy classes
| class | 1 | 2 | 3 | 9A | 9B | 9C | 23A | ··· | 23V | 46A | ··· | 46V | 69A | ··· | 69V | 207A | ··· | 207BN |
| order | 1 | 2 | 3 | 9 | 9 | 9 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 | 207 | ··· | 207 |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
138 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C23 | C46 | S3 | D9 | S3×C23 | D9×C23 |
| kernel | D9×C23 | C207 | D9 | C9 | C69 | C23 | C3 | C1 |
| # reps | 1 | 1 | 22 | 22 | 1 | 3 | 22 | 66 |
Matrix representation of D9×C23 ►in GL2(𝔽829) generated by
| 616 | 0 |
| 0 | 616 |
| 465 | 70 |
| 759 | 535 |
| 759 | 535 |
| 465 | 70 |
G:=sub<GL(2,GF(829))| [616,0,0,616],[465,759,70,535],[759,465,535,70] >;
D9×C23 in GAP, Magma, Sage, TeX
D_9\times C_{23} % in TeX
G:=Group("D9xC23"); // GroupNames label
G:=SmallGroup(414,1);
// by ID
G=gap.SmallGroup(414,1);
# by ID
G:=PCGroup([4,-2,-23,-3,-3,2762,82,4419]);
// Polycyclic
G:=Group<a,b,c|a^23=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export