direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D9×C26, C18⋊C26, C234⋊3C2, C78.6S3, C39.3D6, C117⋊4C22, C9⋊(C2×C26), C3.(S3×C26), C6.2(S3×C13), SmallGroup(468,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C26 |
Generators and relations for D9×C26
G = < a,b,c | a26=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D9×C26
►On 234 points
Generators in S234
(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)(183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)
(1 91 149 201 161 108 224 58 43)(2 92 150 202 162 109 225 59 44)(3 93 151 203 163 110 226 60 45)(4 94 152 204 164 111 227 61 46)(5 95 153 205 165 112 228 62 47)(6 96 154 206 166 113 229 63 48)(7 97 155 207 167 114 230 64 49)(8 98 156 208 168 115 231 65 50)(9 99 131 183 169 116 232 66 51)(10 100 132 184 170 117 233 67 52)(11 101 133 185 171 118 234 68 27)(12 102 134 186 172 119 209 69 28)(13 103 135 187 173 120 210 70 29)(14 104 136 188 174 121 211 71 30)(15 79 137 189 175 122 212 72 31)(16 80 138 190 176 123 213 73 32)(17 81 139 191 177 124 214 74 33)(18 82 140 192 178 125 215 75 34)(19 83 141 193 179 126 216 76 35)(20 84 142 194 180 127 217 77 36)(21 85 143 195 181 128 218 78 37)(22 86 144 196 182 129 219 53 38)(23 87 145 197 157 130 220 54 39)(24 88 146 198 158 105 221 55 40)(25 89 147 199 159 106 222 56 41)(26 90 148 200 160 107 223 57 42)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(53 86)(54 87)(55 88)(56 89)(57 90)(58 91)(59 92)(60 93)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(78 85)(105 198)(106 199)(107 200)(108 201)(109 202)(110 203)(111 204)(112 205)(113 206)(114 207)(115 208)(116 183)(117 184)(118 185)(119 186)(120 187)(121 188)(122 189)(123 190)(124 191)(125 192)(126 193)(127 194)(128 195)(129 196)(130 197)(131 232)(132 233)(133 234)(134 209)(135 210)(136 211)(137 212)(138 213)(139 214)(140 215)(141 216)(142 217)(143 218)(144 219)(145 220)(146 221)(147 222)(148 223)(149 224)(150 225)(151 226)(152 227)(153 228)(154 229)(155 230)(156 231)
G:=sub<Sym(234)| (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)(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234), (1,91,149,201,161,108,224,58,43)(2,92,150,202,162,109,225,59,44)(3,93,151,203,163,110,226,60,45)(4,94,152,204,164,111,227,61,46)(5,95,153,205,165,112,228,62,47)(6,96,154,206,166,113,229,63,48)(7,97,155,207,167,114,230,64,49)(8,98,156,208,168,115,231,65,50)(9,99,131,183,169,116,232,66,51)(10,100,132,184,170,117,233,67,52)(11,101,133,185,171,118,234,68,27)(12,102,134,186,172,119,209,69,28)(13,103,135,187,173,120,210,70,29)(14,104,136,188,174,121,211,71,30)(15,79,137,189,175,122,212,72,31)(16,80,138,190,176,123,213,73,32)(17,81,139,191,177,124,214,74,33)(18,82,140,192,178,125,215,75,34)(19,83,141,193,179,126,216,76,35)(20,84,142,194,180,127,217,77,36)(21,85,143,195,181,128,218,78,37)(22,86,144,196,182,129,219,53,38)(23,87,145,197,157,130,220,54,39)(24,88,146,198,158,105,221,55,40)(25,89,147,199,159,106,222,56,41)(26,90,148,200,160,107,223,57,42), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(53,86)(54,87)(55,88)(56,89)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(78,85)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,206)(114,207)(115,208)(116,183)(117,184)(118,185)(119,186)(120,187)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,196)(130,197)(131,232)(132,233)(133,234)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231)>;
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)(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234), (1,91,149,201,161,108,224,58,43)(2,92,150,202,162,109,225,59,44)(3,93,151,203,163,110,226,60,45)(4,94,152,204,164,111,227,61,46)(5,95,153,205,165,112,228,62,47)(6,96,154,206,166,113,229,63,48)(7,97,155,207,167,114,230,64,49)(8,98,156,208,168,115,231,65,50)(9,99,131,183,169,116,232,66,51)(10,100,132,184,170,117,233,67,52)(11,101,133,185,171,118,234,68,27)(12,102,134,186,172,119,209,69,28)(13,103,135,187,173,120,210,70,29)(14,104,136,188,174,121,211,71,30)(15,79,137,189,175,122,212,72,31)(16,80,138,190,176,123,213,73,32)(17,81,139,191,177,124,214,74,33)(18,82,140,192,178,125,215,75,34)(19,83,141,193,179,126,216,76,35)(20,84,142,194,180,127,217,77,36)(21,85,143,195,181,128,218,78,37)(22,86,144,196,182,129,219,53,38)(23,87,145,197,157,130,220,54,39)(24,88,146,198,158,105,221,55,40)(25,89,147,199,159,106,222,56,41)(26,90,148,200,160,107,223,57,42), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(53,86)(54,87)(55,88)(56,89)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(78,85)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,206)(114,207)(115,208)(116,183)(117,184)(118,185)(119,186)(120,187)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,196)(130,197)(131,232)(132,233)(133,234)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227)(153,228)(154,229)(155,230)(156,231) );
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),(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)], [(1,91,149,201,161,108,224,58,43),(2,92,150,202,162,109,225,59,44),(3,93,151,203,163,110,226,60,45),(4,94,152,204,164,111,227,61,46),(5,95,153,205,165,112,228,62,47),(6,96,154,206,166,113,229,63,48),(7,97,155,207,167,114,230,64,49),(8,98,156,208,168,115,231,65,50),(9,99,131,183,169,116,232,66,51),(10,100,132,184,170,117,233,67,52),(11,101,133,185,171,118,234,68,27),(12,102,134,186,172,119,209,69,28),(13,103,135,187,173,120,210,70,29),(14,104,136,188,174,121,211,71,30),(15,79,137,189,175,122,212,72,31),(16,80,138,190,176,123,213,73,32),(17,81,139,191,177,124,214,74,33),(18,82,140,192,178,125,215,75,34),(19,83,141,193,179,126,216,76,35),(20,84,142,194,180,127,217,77,36),(21,85,143,195,181,128,218,78,37),(22,86,144,196,182,129,219,53,38),(23,87,145,197,157,130,220,54,39),(24,88,146,198,158,105,221,55,40),(25,89,147,199,159,106,222,56,41),(26,90,148,200,160,107,223,57,42)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(53,86),(54,87),(55,88),(56,89),(57,90),(58,91),(59,92),(60,93),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(78,85),(105,198),(106,199),(107,200),(108,201),(109,202),(110,203),(111,204),(112,205),(113,206),(114,207),(115,208),(116,183),(117,184),(118,185),(119,186),(120,187),(121,188),(122,189),(123,190),(124,191),(125,192),(126,193),(127,194),(128,195),(129,196),(130,197),(131,232),(132,233),(133,234),(134,209),(135,210),(136,211),(137,212),(138,213),(139,214),(140,215),(141,216),(142,217),(143,218),(144,219),(145,220),(146,221),(147,222),(148,223),(149,224),(150,225),(151,226),(152,227),(153,228),(154,229),(155,230),(156,231)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 9A | 9B | 9C | 13A | ··· | 13L | 18A | 18B | 18C | 26A | ··· | 26L | 26M | ··· | 26AJ | 39A | ··· | 39L | 78A | ··· | 78L | 117A | ··· | 117AJ | 234A | ··· | 234AJ |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 9 | 9 | 9 | 13 | ··· | 13 | 18 | 18 | 18 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 | 117 | ··· | 117 | 234 | ··· | 234 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | S3 | D6 | D9 | D18 | S3×C13 | S3×C26 | C13×D9 | D9×C26 |
| kernel | D9×C26 | C13×D9 | C234 | D18 | D9 | C18 | C78 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 12 | 24 | 12 | 1 | 1 | 3 | 3 | 12 | 12 | 36 | 36 |
Matrix representation of D9×C26 ►in GL2(𝔽937) generated by
| 26 | 0 |
| 0 | 26 |
| 262 | 472 |
| 465 | 734 |
| 465 | 734 |
| 262 | 472 |
G:=sub<GL(2,GF(937))| [26,0,0,26],[262,465,472,734],[465,262,734,472] >;
D9×C26 in GAP, Magma, Sage, TeX
D_9\times C_{26} % in TeX
G:=Group("D9xC26"); // GroupNames label
G:=SmallGroup(468,16);
// by ID
G=gap.SmallGroup(468,16);
# by ID
G:=PCGroup([5,-2,-2,-13,-3,-3,5203,138,7804]);
// Polycyclic
G:=Group<a,b,c|a^26=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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