direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×Dic7, C63⋊3C4, C7⋊3C36, C18.2D7, C14.3C18, C126.3C2, C21.3C12, C42.10C6, C2.(C9×D7), C6.2(C3×D7), C3.(C3×Dic7), (C3×Dic7).2C3, SmallGroup(252,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C9×Dic7 |
Generators and relations for C9×Dic7
G = < a,b,c | a9=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×Dic7
►Regular action on 252 points
Generators in S252
(1 202 152 20 217 140 73 101 52)(2 203 153 21 218 127 74 102 53)(3 204 154 22 219 128 75 103 54)(4 205 141 23 220 129 76 104 55)(5 206 142 24 221 130 77 105 56)(6 207 143 25 222 131 78 106 43)(7 208 144 26 223 132 79 107 44)(8 209 145 27 224 133 80 108 45)(9 210 146 28 211 134 81 109 46)(10 197 147 15 212 135 82 110 47)(11 198 148 16 213 136 83 111 48)(12 199 149 17 214 137 84 112 49)(13 200 150 18 215 138 71 99 50)(14 201 151 19 216 139 72 100 51)(29 117 167 231 175 239 185 60 91)(30 118 168 232 176 240 186 61 92)(31 119 155 233 177 241 187 62 93)(32 120 156 234 178 242 188 63 94)(33 121 157 235 179 243 189 64 95)(34 122 158 236 180 244 190 65 96)(35 123 159 237 181 245 191 66 97)(36 124 160 238 182 246 192 67 98)(37 125 161 225 169 247 193 68 85)(38 126 162 226 170 248 194 69 86)(39 113 163 227 171 249 195 70 87)(40 114 164 228 172 250 196 57 88)(41 115 165 229 173 251 183 58 89)(42 116 166 230 174 252 184 59 90)
(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)(225 226 227 228 229 230 231 232 233 234 235 236 237 238)(239 240 241 242 243 244 245 246 247 248 249 250 251 252)
(1 180 8 173)(2 179 9 172)(3 178 10 171)(4 177 11 170)(5 176 12 169)(6 175 13 182)(7 174 14 181)(15 70 22 63)(16 69 23 62)(17 68 24 61)(18 67 25 60)(19 66 26 59)(20 65 27 58)(21 64 28 57)(29 138 36 131)(30 137 37 130)(31 136 38 129)(32 135 39 128)(33 134 40 127)(34 133 41 140)(35 132 42 139)(43 231 50 238)(44 230 51 237)(45 229 52 236)(46 228 53 235)(47 227 54 234)(48 226 55 233)(49 225 56 232)(71 124 78 117)(72 123 79 116)(73 122 80 115)(74 121 81 114)(75 120 82 113)(76 119 83 126)(77 118 84 125)(85 221 92 214)(86 220 93 213)(87 219 94 212)(88 218 95 211)(89 217 96 224)(90 216 97 223)(91 215 98 222)(99 160 106 167)(100 159 107 166)(101 158 108 165)(102 157 109 164)(103 156 110 163)(104 155 111 162)(105 168 112 161)(141 187 148 194)(142 186 149 193)(143 185 150 192)(144 184 151 191)(145 183 152 190)(146 196 153 189)(147 195 154 188)(197 249 204 242)(198 248 205 241)(199 247 206 240)(200 246 207 239)(201 245 208 252)(202 244 209 251)(203 243 210 250)
G:=sub<Sym(252)| (1,202,152,20,217,140,73,101,52)(2,203,153,21,218,127,74,102,53)(3,204,154,22,219,128,75,103,54)(4,205,141,23,220,129,76,104,55)(5,206,142,24,221,130,77,105,56)(6,207,143,25,222,131,78,106,43)(7,208,144,26,223,132,79,107,44)(8,209,145,27,224,133,80,108,45)(9,210,146,28,211,134,81,109,46)(10,197,147,15,212,135,82,110,47)(11,198,148,16,213,136,83,111,48)(12,199,149,17,214,137,84,112,49)(13,200,150,18,215,138,71,99,50)(14,201,151,19,216,139,72,100,51)(29,117,167,231,175,239,185,60,91)(30,118,168,232,176,240,186,61,92)(31,119,155,233,177,241,187,62,93)(32,120,156,234,178,242,188,63,94)(33,121,157,235,179,243,189,64,95)(34,122,158,236,180,244,190,65,96)(35,123,159,237,181,245,191,66,97)(36,124,160,238,182,246,192,67,98)(37,125,161,225,169,247,193,68,85)(38,126,162,226,170,248,194,69,86)(39,113,163,227,171,249,195,70,87)(40,114,164,228,172,250,196,57,88)(41,115,165,229,173,251,183,58,89)(42,116,166,230,174,252,184,59,90), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238)(239,240,241,242,243,244,245,246,247,248,249,250,251,252), (1,180,8,173)(2,179,9,172)(3,178,10,171)(4,177,11,170)(5,176,12,169)(6,175,13,182)(7,174,14,181)(15,70,22,63)(16,69,23,62)(17,68,24,61)(18,67,25,60)(19,66,26,59)(20,65,27,58)(21,64,28,57)(29,138,36,131)(30,137,37,130)(31,136,38,129)(32,135,39,128)(33,134,40,127)(34,133,41,140)(35,132,42,139)(43,231,50,238)(44,230,51,237)(45,229,52,236)(46,228,53,235)(47,227,54,234)(48,226,55,233)(49,225,56,232)(71,124,78,117)(72,123,79,116)(73,122,80,115)(74,121,81,114)(75,120,82,113)(76,119,83,126)(77,118,84,125)(85,221,92,214)(86,220,93,213)(87,219,94,212)(88,218,95,211)(89,217,96,224)(90,216,97,223)(91,215,98,222)(99,160,106,167)(100,159,107,166)(101,158,108,165)(102,157,109,164)(103,156,110,163)(104,155,111,162)(105,168,112,161)(141,187,148,194)(142,186,149,193)(143,185,150,192)(144,184,151,191)(145,183,152,190)(146,196,153,189)(147,195,154,188)(197,249,204,242)(198,248,205,241)(199,247,206,240)(200,246,207,239)(201,245,208,252)(202,244,209,251)(203,243,210,250)>;
G:=Group( (1,202,152,20,217,140,73,101,52)(2,203,153,21,218,127,74,102,53)(3,204,154,22,219,128,75,103,54)(4,205,141,23,220,129,76,104,55)(5,206,142,24,221,130,77,105,56)(6,207,143,25,222,131,78,106,43)(7,208,144,26,223,132,79,107,44)(8,209,145,27,224,133,80,108,45)(9,210,146,28,211,134,81,109,46)(10,197,147,15,212,135,82,110,47)(11,198,148,16,213,136,83,111,48)(12,199,149,17,214,137,84,112,49)(13,200,150,18,215,138,71,99,50)(14,201,151,19,216,139,72,100,51)(29,117,167,231,175,239,185,60,91)(30,118,168,232,176,240,186,61,92)(31,119,155,233,177,241,187,62,93)(32,120,156,234,178,242,188,63,94)(33,121,157,235,179,243,189,64,95)(34,122,158,236,180,244,190,65,96)(35,123,159,237,181,245,191,66,97)(36,124,160,238,182,246,192,67,98)(37,125,161,225,169,247,193,68,85)(38,126,162,226,170,248,194,69,86)(39,113,163,227,171,249,195,70,87)(40,114,164,228,172,250,196,57,88)(41,115,165,229,173,251,183,58,89)(42,116,166,230,174,252,184,59,90), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238)(239,240,241,242,243,244,245,246,247,248,249,250,251,252), (1,180,8,173)(2,179,9,172)(3,178,10,171)(4,177,11,170)(5,176,12,169)(6,175,13,182)(7,174,14,181)(15,70,22,63)(16,69,23,62)(17,68,24,61)(18,67,25,60)(19,66,26,59)(20,65,27,58)(21,64,28,57)(29,138,36,131)(30,137,37,130)(31,136,38,129)(32,135,39,128)(33,134,40,127)(34,133,41,140)(35,132,42,139)(43,231,50,238)(44,230,51,237)(45,229,52,236)(46,228,53,235)(47,227,54,234)(48,226,55,233)(49,225,56,232)(71,124,78,117)(72,123,79,116)(73,122,80,115)(74,121,81,114)(75,120,82,113)(76,119,83,126)(77,118,84,125)(85,221,92,214)(86,220,93,213)(87,219,94,212)(88,218,95,211)(89,217,96,224)(90,216,97,223)(91,215,98,222)(99,160,106,167)(100,159,107,166)(101,158,108,165)(102,157,109,164)(103,156,110,163)(104,155,111,162)(105,168,112,161)(141,187,148,194)(142,186,149,193)(143,185,150,192)(144,184,151,191)(145,183,152,190)(146,196,153,189)(147,195,154,188)(197,249,204,242)(198,248,205,241)(199,247,206,240)(200,246,207,239)(201,245,208,252)(202,244,209,251)(203,243,210,250) );
G=PermutationGroup([[(1,202,152,20,217,140,73,101,52),(2,203,153,21,218,127,74,102,53),(3,204,154,22,219,128,75,103,54),(4,205,141,23,220,129,76,104,55),(5,206,142,24,221,130,77,105,56),(6,207,143,25,222,131,78,106,43),(7,208,144,26,223,132,79,107,44),(8,209,145,27,224,133,80,108,45),(9,210,146,28,211,134,81,109,46),(10,197,147,15,212,135,82,110,47),(11,198,148,16,213,136,83,111,48),(12,199,149,17,214,137,84,112,49),(13,200,150,18,215,138,71,99,50),(14,201,151,19,216,139,72,100,51),(29,117,167,231,175,239,185,60,91),(30,118,168,232,176,240,186,61,92),(31,119,155,233,177,241,187,62,93),(32,120,156,234,178,242,188,63,94),(33,121,157,235,179,243,189,64,95),(34,122,158,236,180,244,190,65,96),(35,123,159,237,181,245,191,66,97),(36,124,160,238,182,246,192,67,98),(37,125,161,225,169,247,193,68,85),(38,126,162,226,170,248,194,69,86),(39,113,163,227,171,249,195,70,87),(40,114,164,228,172,250,196,57,88),(41,115,165,229,173,251,183,58,89),(42,116,166,230,174,252,184,59,90)], [(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),(225,226,227,228,229,230,231,232,233,234,235,236,237,238),(239,240,241,242,243,244,245,246,247,248,249,250,251,252)], [(1,180,8,173),(2,179,9,172),(3,178,10,171),(4,177,11,170),(5,176,12,169),(6,175,13,182),(7,174,14,181),(15,70,22,63),(16,69,23,62),(17,68,24,61),(18,67,25,60),(19,66,26,59),(20,65,27,58),(21,64,28,57),(29,138,36,131),(30,137,37,130),(31,136,38,129),(32,135,39,128),(33,134,40,127),(34,133,41,140),(35,132,42,139),(43,231,50,238),(44,230,51,237),(45,229,52,236),(46,228,53,235),(47,227,54,234),(48,226,55,233),(49,225,56,232),(71,124,78,117),(72,123,79,116),(73,122,80,115),(74,121,81,114),(75,120,82,113),(76,119,83,126),(77,118,84,125),(85,221,92,214),(86,220,93,213),(87,219,94,212),(88,218,95,211),(89,217,96,224),(90,216,97,223),(91,215,98,222),(99,160,106,167),(100,159,107,166),(101,158,108,165),(102,157,109,164),(103,156,110,163),(104,155,111,162),(105,168,112,161),(141,187,148,194),(142,186,149,193),(143,185,150,192),(144,184,151,191),(145,183,152,190),(146,196,153,189),(147,195,154,188),(197,249,204,242),(198,248,205,241),(199,247,206,240),(200,246,207,239),(201,245,208,252),(202,244,209,251),(203,243,210,250)]])
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 7A | 7B | 7C | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 18A | ··· | 18F | 21A | ··· | 21F | 36A | ··· | 36L | 42A | ··· | 42F | 63A | ··· | 63R | 126A | ··· | 126R |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 7 | 7 | 7 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 18 | ··· | 18 | 21 | ··· | 21 | 36 | ··· | 36 | 42 | ··· | 42 | 63 | ··· | 63 | 126 | ··· | 126 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | D7 | Dic7 | C3×D7 | C3×Dic7 | C9×D7 | C9×Dic7 |
| kernel | C9×Dic7 | C126 | C3×Dic7 | C63 | C42 | Dic7 | C21 | C14 | C7 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 3 | 3 | 6 | 6 | 18 | 18 |
Matrix representation of C9×Dic7 ►in GL4(𝔽757) generated by
| 673 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 729 | 0 |
| 0 | 0 | 0 | 729 |
| 1 | 0 | 0 | 0 |
| 0 | 756 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 756 | 136 |
| 756 | 0 | 0 | 0 |
| 0 | 670 | 0 | 0 |
| 0 | 0 | 17 | 748 |
| 0 | 0 | 32 | 740 |
G:=sub<GL(4,GF(757))| [673,0,0,0,0,27,0,0,0,0,729,0,0,0,0,729],[1,0,0,0,0,756,0,0,0,0,0,756,0,0,1,136],[756,0,0,0,0,670,0,0,0,0,17,32,0,0,748,740] >;
C9×Dic7 in GAP, Magma, Sage, TeX
C_9\times {\rm Dic}_7 % in TeX
G:=Group("C9xDic7"); // GroupNames label
G:=SmallGroup(252,4);
// by ID
G=gap.SmallGroup(252,4);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-7,30,66,5404]);
// Polycyclic
G:=Group<a,b,c|a^9=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export