direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic17, C68⋊4C4, C17⋊2C42, C22.3D34, (C2×C68).7C2, C2.2(C4×D17), (C2×C4).6D17, C34.10(C2×C4), (C2×C34).3C22, C2.2(C2×Dic17), (C2×Dic17).6C2, SmallGroup(272,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C4×Dic17 |
Generators and relations for C4×Dic17
G = < a,b,c | a4=b34=1, c2=b17, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×Dic17
►Regular action on 272 points
Generators in S272
(1 241 78 157)(2 242 79 158)(3 243 80 159)(4 244 81 160)(5 245 82 161)(6 246 83 162)(7 247 84 163)(8 248 85 164)(9 249 86 165)(10 250 87 166)(11 251 88 167)(12 252 89 168)(13 253 90 169)(14 254 91 170)(15 255 92 137)(16 256 93 138)(17 257 94 139)(18 258 95 140)(19 259 96 141)(20 260 97 142)(21 261 98 143)(22 262 99 144)(23 263 100 145)(24 264 101 146)(25 265 102 147)(26 266 69 148)(27 267 70 149)(28 268 71 150)(29 269 72 151)(30 270 73 152)(31 271 74 153)(32 272 75 154)(33 239 76 155)(34 240 77 156)(35 185 128 230)(36 186 129 231)(37 187 130 232)(38 188 131 233)(39 189 132 234)(40 190 133 235)(41 191 134 236)(42 192 135 237)(43 193 136 238)(44 194 103 205)(45 195 104 206)(46 196 105 207)(47 197 106 208)(48 198 107 209)(49 199 108 210)(50 200 109 211)(51 201 110 212)(52 202 111 213)(53 203 112 214)(54 204 113 215)(55 171 114 216)(56 172 115 217)(57 173 116 218)(58 174 117 219)(59 175 118 220)(60 176 119 221)(61 177 120 222)(62 178 121 223)(63 179 122 224)(64 180 123 225)(65 181 124 226)(66 182 125 227)(67 183 126 228)(68 184 127 229)
(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)(239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272)
(1 198 18 181)(2 197 19 180)(3 196 20 179)(4 195 21 178)(5 194 22 177)(6 193 23 176)(7 192 24 175)(8 191 25 174)(9 190 26 173)(10 189 27 172)(11 188 28 171)(12 187 29 204)(13 186 30 203)(14 185 31 202)(15 184 32 201)(16 183 33 200)(17 182 34 199)(35 153 52 170)(36 152 53 169)(37 151 54 168)(38 150 55 167)(39 149 56 166)(40 148 57 165)(41 147 58 164)(42 146 59 163)(43 145 60 162)(44 144 61 161)(45 143 62 160)(46 142 63 159)(47 141 64 158)(48 140 65 157)(49 139 66 156)(50 138 67 155)(51 137 68 154)(69 218 86 235)(70 217 87 234)(71 216 88 233)(72 215 89 232)(73 214 90 231)(74 213 91 230)(75 212 92 229)(76 211 93 228)(77 210 94 227)(78 209 95 226)(79 208 96 225)(80 207 97 224)(81 206 98 223)(82 205 99 222)(83 238 100 221)(84 237 101 220)(85 236 102 219)(103 262 120 245)(104 261 121 244)(105 260 122 243)(106 259 123 242)(107 258 124 241)(108 257 125 240)(109 256 126 239)(110 255 127 272)(111 254 128 271)(112 253 129 270)(113 252 130 269)(114 251 131 268)(115 250 132 267)(116 249 133 266)(117 248 134 265)(118 247 135 264)(119 246 136 263)
G:=sub<Sym(272)| (1,241,78,157)(2,242,79,158)(3,243,80,159)(4,244,81,160)(5,245,82,161)(6,246,83,162)(7,247,84,163)(8,248,85,164)(9,249,86,165)(10,250,87,166)(11,251,88,167)(12,252,89,168)(13,253,90,169)(14,254,91,170)(15,255,92,137)(16,256,93,138)(17,257,94,139)(18,258,95,140)(19,259,96,141)(20,260,97,142)(21,261,98,143)(22,262,99,144)(23,263,100,145)(24,264,101,146)(25,265,102,147)(26,266,69,148)(27,267,70,149)(28,268,71,150)(29,269,72,151)(30,270,73,152)(31,271,74,153)(32,272,75,154)(33,239,76,155)(34,240,77,156)(35,185,128,230)(36,186,129,231)(37,187,130,232)(38,188,131,233)(39,189,132,234)(40,190,133,235)(41,191,134,236)(42,192,135,237)(43,193,136,238)(44,194,103,205)(45,195,104,206)(46,196,105,207)(47,197,106,208)(48,198,107,209)(49,199,108,210)(50,200,109,211)(51,201,110,212)(52,202,111,213)(53,203,112,214)(54,204,113,215)(55,171,114,216)(56,172,115,217)(57,173,116,218)(58,174,117,219)(59,175,118,220)(60,176,119,221)(61,177,120,222)(62,178,121,223)(63,179,122,224)(64,180,123,225)(65,181,124,226)(66,182,125,227)(67,183,126,228)(68,184,127,229), (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)(239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272), (1,198,18,181)(2,197,19,180)(3,196,20,179)(4,195,21,178)(5,194,22,177)(6,193,23,176)(7,192,24,175)(8,191,25,174)(9,190,26,173)(10,189,27,172)(11,188,28,171)(12,187,29,204)(13,186,30,203)(14,185,31,202)(15,184,32,201)(16,183,33,200)(17,182,34,199)(35,153,52,170)(36,152,53,169)(37,151,54,168)(38,150,55,167)(39,149,56,166)(40,148,57,165)(41,147,58,164)(42,146,59,163)(43,145,60,162)(44,144,61,161)(45,143,62,160)(46,142,63,159)(47,141,64,158)(48,140,65,157)(49,139,66,156)(50,138,67,155)(51,137,68,154)(69,218,86,235)(70,217,87,234)(71,216,88,233)(72,215,89,232)(73,214,90,231)(74,213,91,230)(75,212,92,229)(76,211,93,228)(77,210,94,227)(78,209,95,226)(79,208,96,225)(80,207,97,224)(81,206,98,223)(82,205,99,222)(83,238,100,221)(84,237,101,220)(85,236,102,219)(103,262,120,245)(104,261,121,244)(105,260,122,243)(106,259,123,242)(107,258,124,241)(108,257,125,240)(109,256,126,239)(110,255,127,272)(111,254,128,271)(112,253,129,270)(113,252,130,269)(114,251,131,268)(115,250,132,267)(116,249,133,266)(117,248,134,265)(118,247,135,264)(119,246,136,263)>;
G:=Group( (1,241,78,157)(2,242,79,158)(3,243,80,159)(4,244,81,160)(5,245,82,161)(6,246,83,162)(7,247,84,163)(8,248,85,164)(9,249,86,165)(10,250,87,166)(11,251,88,167)(12,252,89,168)(13,253,90,169)(14,254,91,170)(15,255,92,137)(16,256,93,138)(17,257,94,139)(18,258,95,140)(19,259,96,141)(20,260,97,142)(21,261,98,143)(22,262,99,144)(23,263,100,145)(24,264,101,146)(25,265,102,147)(26,266,69,148)(27,267,70,149)(28,268,71,150)(29,269,72,151)(30,270,73,152)(31,271,74,153)(32,272,75,154)(33,239,76,155)(34,240,77,156)(35,185,128,230)(36,186,129,231)(37,187,130,232)(38,188,131,233)(39,189,132,234)(40,190,133,235)(41,191,134,236)(42,192,135,237)(43,193,136,238)(44,194,103,205)(45,195,104,206)(46,196,105,207)(47,197,106,208)(48,198,107,209)(49,199,108,210)(50,200,109,211)(51,201,110,212)(52,202,111,213)(53,203,112,214)(54,204,113,215)(55,171,114,216)(56,172,115,217)(57,173,116,218)(58,174,117,219)(59,175,118,220)(60,176,119,221)(61,177,120,222)(62,178,121,223)(63,179,122,224)(64,180,123,225)(65,181,124,226)(66,182,125,227)(67,183,126,228)(68,184,127,229), (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)(239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272), (1,198,18,181)(2,197,19,180)(3,196,20,179)(4,195,21,178)(5,194,22,177)(6,193,23,176)(7,192,24,175)(8,191,25,174)(9,190,26,173)(10,189,27,172)(11,188,28,171)(12,187,29,204)(13,186,30,203)(14,185,31,202)(15,184,32,201)(16,183,33,200)(17,182,34,199)(35,153,52,170)(36,152,53,169)(37,151,54,168)(38,150,55,167)(39,149,56,166)(40,148,57,165)(41,147,58,164)(42,146,59,163)(43,145,60,162)(44,144,61,161)(45,143,62,160)(46,142,63,159)(47,141,64,158)(48,140,65,157)(49,139,66,156)(50,138,67,155)(51,137,68,154)(69,218,86,235)(70,217,87,234)(71,216,88,233)(72,215,89,232)(73,214,90,231)(74,213,91,230)(75,212,92,229)(76,211,93,228)(77,210,94,227)(78,209,95,226)(79,208,96,225)(80,207,97,224)(81,206,98,223)(82,205,99,222)(83,238,100,221)(84,237,101,220)(85,236,102,219)(103,262,120,245)(104,261,121,244)(105,260,122,243)(106,259,123,242)(107,258,124,241)(108,257,125,240)(109,256,126,239)(110,255,127,272)(111,254,128,271)(112,253,129,270)(113,252,130,269)(114,251,131,268)(115,250,132,267)(116,249,133,266)(117,248,134,265)(118,247,135,264)(119,246,136,263) );
G=PermutationGroup([[(1,241,78,157),(2,242,79,158),(3,243,80,159),(4,244,81,160),(5,245,82,161),(6,246,83,162),(7,247,84,163),(8,248,85,164),(9,249,86,165),(10,250,87,166),(11,251,88,167),(12,252,89,168),(13,253,90,169),(14,254,91,170),(15,255,92,137),(16,256,93,138),(17,257,94,139),(18,258,95,140),(19,259,96,141),(20,260,97,142),(21,261,98,143),(22,262,99,144),(23,263,100,145),(24,264,101,146),(25,265,102,147),(26,266,69,148),(27,267,70,149),(28,268,71,150),(29,269,72,151),(30,270,73,152),(31,271,74,153),(32,272,75,154),(33,239,76,155),(34,240,77,156),(35,185,128,230),(36,186,129,231),(37,187,130,232),(38,188,131,233),(39,189,132,234),(40,190,133,235),(41,191,134,236),(42,192,135,237),(43,193,136,238),(44,194,103,205),(45,195,104,206),(46,196,105,207),(47,197,106,208),(48,198,107,209),(49,199,108,210),(50,200,109,211),(51,201,110,212),(52,202,111,213),(53,203,112,214),(54,204,113,215),(55,171,114,216),(56,172,115,217),(57,173,116,218),(58,174,117,219),(59,175,118,220),(60,176,119,221),(61,177,120,222),(62,178,121,223),(63,179,122,224),(64,180,123,225),(65,181,124,226),(66,182,125,227),(67,183,126,228),(68,184,127,229)], [(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),(239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272)], [(1,198,18,181),(2,197,19,180),(3,196,20,179),(4,195,21,178),(5,194,22,177),(6,193,23,176),(7,192,24,175),(8,191,25,174),(9,190,26,173),(10,189,27,172),(11,188,28,171),(12,187,29,204),(13,186,30,203),(14,185,31,202),(15,184,32,201),(16,183,33,200),(17,182,34,199),(35,153,52,170),(36,152,53,169),(37,151,54,168),(38,150,55,167),(39,149,56,166),(40,148,57,165),(41,147,58,164),(42,146,59,163),(43,145,60,162),(44,144,61,161),(45,143,62,160),(46,142,63,159),(47,141,64,158),(48,140,65,157),(49,139,66,156),(50,138,67,155),(51,137,68,154),(69,218,86,235),(70,217,87,234),(71,216,88,233),(72,215,89,232),(73,214,90,231),(74,213,91,230),(75,212,92,229),(76,211,93,228),(77,210,94,227),(78,209,95,226),(79,208,96,225),(80,207,97,224),(81,206,98,223),(82,205,99,222),(83,238,100,221),(84,237,101,220),(85,236,102,219),(103,262,120,245),(104,261,121,244),(105,260,122,243),(106,259,123,242),(107,258,124,241),(108,257,125,240),(109,256,126,239),(110,255,127,272),(111,254,128,271),(112,253,129,270),(113,252,130,269),(114,251,131,268),(115,250,132,267),(116,249,133,266),(117,248,134,265),(118,247,135,264),(119,246,136,263)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 17A | ··· | 17H | 34A | ··· | 34X | 68A | ··· | 68AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D17 | Dic17 | D34 | C4×D17 |
| kernel | C4×Dic17 | C2×Dic17 | C2×C68 | Dic17 | C68 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 8 | 16 | 8 | 32 |
Matrix representation of C4×Dic17 ►in GL4(𝔽137) generated by
| 136 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 136 | 0 |
| 0 | 0 | 0 | 136 |
| 136 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 136 | 1 |
| 0 | 0 | 4 | 132 |
| 37 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 69 | 124 |
| 0 | 0 | 50 | 68 |
G:=sub<GL(4,GF(137))| [136,0,0,0,0,37,0,0,0,0,136,0,0,0,0,136],[136,0,0,0,0,1,0,0,0,0,136,4,0,0,1,132],[37,0,0,0,0,1,0,0,0,0,69,50,0,0,124,68] >;
C4×Dic17 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{17} % in TeX
G:=Group("C4xDic17"); // GroupNames label
G:=SmallGroup(272,11);
// by ID
G=gap.SmallGroup(272,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,46,6404]);
// Polycyclic
G:=Group<a,b,c|a^4=b^34=1,c^2=b^17,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export