direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C17⋊3C8, C34⋊3C8, C68.6C4, C4.14D34, C4.3Dic17, C68.14C22, C22.2Dic17, C17⋊5(C2×C8), (C2×C68).6C2, (C2×C34).4C4, (C2×C4).5D17, C34.13(C2×C4), C2.1(C2×Dic17), SmallGroup(272,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C2×C17⋊3C8 |
Generators and relations for C2×C17⋊3C8
G = < a,b,c | a2=b17=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C17⋊3C8
►Regular action on 272 points
Generators in S272
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 129)(62 130)(63 131)(64 132)(65 133)(66 134)(67 135)(68 136)(137 214)(138 215)(139 216)(140 217)(141 218)(142 219)(143 220)(144 221)(145 205)(146 206)(147 207)(148 208)(149 209)(150 210)(151 211)(152 212)(153 213)(154 238)(155 222)(156 223)(157 224)(158 225)(159 226)(160 227)(161 228)(162 229)(163 230)(164 231)(165 232)(166 233)(167 234)(168 235)(169 236)(170 237)(171 244)(172 245)(173 246)(174 247)(175 248)(176 249)(177 250)(178 251)(179 252)(180 253)(181 254)(182 255)(183 239)(184 240)(185 241)(186 242)(187 243)(188 265)(189 266)(190 267)(191 268)(192 269)(193 270)(194 271)(195 272)(196 256)(197 257)(198 258)(199 259)(200 260)(201 261)(202 262)(203 263)(204 264)
(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)
(1 269 52 238 18 249 35 221)(2 268 53 237 19 248 36 220)(3 267 54 236 20 247 37 219)(4 266 55 235 21 246 38 218)(5 265 56 234 22 245 39 217)(6 264 57 233 23 244 40 216)(7 263 58 232 24 243 41 215)(8 262 59 231 25 242 42 214)(9 261 60 230 26 241 43 213)(10 260 61 229 27 240 44 212)(11 259 62 228 28 239 45 211)(12 258 63 227 29 255 46 210)(13 257 64 226 30 254 47 209)(14 256 65 225 31 253 48 208)(15 272 66 224 32 252 49 207)(16 271 67 223 33 251 50 206)(17 270 68 222 34 250 51 205)(69 192 120 154 86 176 103 144)(70 191 121 170 87 175 104 143)(71 190 122 169 88 174 105 142)(72 189 123 168 89 173 106 141)(73 188 124 167 90 172 107 140)(74 204 125 166 91 171 108 139)(75 203 126 165 92 187 109 138)(76 202 127 164 93 186 110 137)(77 201 128 163 94 185 111 153)(78 200 129 162 95 184 112 152)(79 199 130 161 96 183 113 151)(80 198 131 160 97 182 114 150)(81 197 132 159 98 181 115 149)(82 196 133 158 99 180 116 148)(83 195 134 157 100 179 117 147)(84 194 135 156 101 178 118 146)(85 193 136 155 102 177 119 145)
G:=sub<Sym(272)| (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,129)(62,130)(63,131)(64,132)(65,133)(66,134)(67,135)(68,136)(137,214)(138,215)(139,216)(140,217)(141,218)(142,219)(143,220)(144,221)(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)(152,212)(153,213)(154,238)(155,222)(156,223)(157,224)(158,225)(159,226)(160,227)(161,228)(162,229)(163,230)(164,231)(165,232)(166,233)(167,234)(168,235)(169,236)(170,237)(171,244)(172,245)(173,246)(174,247)(175,248)(176,249)(177,250)(178,251)(179,252)(180,253)(181,254)(182,255)(183,239)(184,240)(185,241)(186,242)(187,243)(188,265)(189,266)(190,267)(191,268)(192,269)(193,270)(194,271)(195,272)(196,256)(197,257)(198,258)(199,259)(200,260)(201,261)(202,262)(203,263)(204,264), (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), (1,269,52,238,18,249,35,221)(2,268,53,237,19,248,36,220)(3,267,54,236,20,247,37,219)(4,266,55,235,21,246,38,218)(5,265,56,234,22,245,39,217)(6,264,57,233,23,244,40,216)(7,263,58,232,24,243,41,215)(8,262,59,231,25,242,42,214)(9,261,60,230,26,241,43,213)(10,260,61,229,27,240,44,212)(11,259,62,228,28,239,45,211)(12,258,63,227,29,255,46,210)(13,257,64,226,30,254,47,209)(14,256,65,225,31,253,48,208)(15,272,66,224,32,252,49,207)(16,271,67,223,33,251,50,206)(17,270,68,222,34,250,51,205)(69,192,120,154,86,176,103,144)(70,191,121,170,87,175,104,143)(71,190,122,169,88,174,105,142)(72,189,123,168,89,173,106,141)(73,188,124,167,90,172,107,140)(74,204,125,166,91,171,108,139)(75,203,126,165,92,187,109,138)(76,202,127,164,93,186,110,137)(77,201,128,163,94,185,111,153)(78,200,129,162,95,184,112,152)(79,199,130,161,96,183,113,151)(80,198,131,160,97,182,114,150)(81,197,132,159,98,181,115,149)(82,196,133,158,99,180,116,148)(83,195,134,157,100,179,117,147)(84,194,135,156,101,178,118,146)(85,193,136,155,102,177,119,145)>;
G:=Group( (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,117)(50,118)(51,119)(52,120)(53,121)(54,122)(55,123)(56,124)(57,125)(58,126)(59,127)(60,128)(61,129)(62,130)(63,131)(64,132)(65,133)(66,134)(67,135)(68,136)(137,214)(138,215)(139,216)(140,217)(141,218)(142,219)(143,220)(144,221)(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)(152,212)(153,213)(154,238)(155,222)(156,223)(157,224)(158,225)(159,226)(160,227)(161,228)(162,229)(163,230)(164,231)(165,232)(166,233)(167,234)(168,235)(169,236)(170,237)(171,244)(172,245)(173,246)(174,247)(175,248)(176,249)(177,250)(178,251)(179,252)(180,253)(181,254)(182,255)(183,239)(184,240)(185,241)(186,242)(187,243)(188,265)(189,266)(190,267)(191,268)(192,269)(193,270)(194,271)(195,272)(196,256)(197,257)(198,258)(199,259)(200,260)(201,261)(202,262)(203,263)(204,264), (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), (1,269,52,238,18,249,35,221)(2,268,53,237,19,248,36,220)(3,267,54,236,20,247,37,219)(4,266,55,235,21,246,38,218)(5,265,56,234,22,245,39,217)(6,264,57,233,23,244,40,216)(7,263,58,232,24,243,41,215)(8,262,59,231,25,242,42,214)(9,261,60,230,26,241,43,213)(10,260,61,229,27,240,44,212)(11,259,62,228,28,239,45,211)(12,258,63,227,29,255,46,210)(13,257,64,226,30,254,47,209)(14,256,65,225,31,253,48,208)(15,272,66,224,32,252,49,207)(16,271,67,223,33,251,50,206)(17,270,68,222,34,250,51,205)(69,192,120,154,86,176,103,144)(70,191,121,170,87,175,104,143)(71,190,122,169,88,174,105,142)(72,189,123,168,89,173,106,141)(73,188,124,167,90,172,107,140)(74,204,125,166,91,171,108,139)(75,203,126,165,92,187,109,138)(76,202,127,164,93,186,110,137)(77,201,128,163,94,185,111,153)(78,200,129,162,95,184,112,152)(79,199,130,161,96,183,113,151)(80,198,131,160,97,182,114,150)(81,197,132,159,98,181,115,149)(82,196,133,158,99,180,116,148)(83,195,134,157,100,179,117,147)(84,194,135,156,101,178,118,146)(85,193,136,155,102,177,119,145) );
G=PermutationGroup([[(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,117),(50,118),(51,119),(52,120),(53,121),(54,122),(55,123),(56,124),(57,125),(58,126),(59,127),(60,128),(61,129),(62,130),(63,131),(64,132),(65,133),(66,134),(67,135),(68,136),(137,214),(138,215),(139,216),(140,217),(141,218),(142,219),(143,220),(144,221),(145,205),(146,206),(147,207),(148,208),(149,209),(150,210),(151,211),(152,212),(153,213),(154,238),(155,222),(156,223),(157,224),(158,225),(159,226),(160,227),(161,228),(162,229),(163,230),(164,231),(165,232),(166,233),(167,234),(168,235),(169,236),(170,237),(171,244),(172,245),(173,246),(174,247),(175,248),(176,249),(177,250),(178,251),(179,252),(180,253),(181,254),(182,255),(183,239),(184,240),(185,241),(186,242),(187,243),(188,265),(189,266),(190,267),(191,268),(192,269),(193,270),(194,271),(195,272),(196,256),(197,257),(198,258),(199,259),(200,260),(201,261),(202,262),(203,263),(204,264)], [(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)], [(1,269,52,238,18,249,35,221),(2,268,53,237,19,248,36,220),(3,267,54,236,20,247,37,219),(4,266,55,235,21,246,38,218),(5,265,56,234,22,245,39,217),(6,264,57,233,23,244,40,216),(7,263,58,232,24,243,41,215),(8,262,59,231,25,242,42,214),(9,261,60,230,26,241,43,213),(10,260,61,229,27,240,44,212),(11,259,62,228,28,239,45,211),(12,258,63,227,29,255,46,210),(13,257,64,226,30,254,47,209),(14,256,65,225,31,253,48,208),(15,272,66,224,32,252,49,207),(16,271,67,223,33,251,50,206),(17,270,68,222,34,250,51,205),(69,192,120,154,86,176,103,144),(70,191,121,170,87,175,104,143),(71,190,122,169,88,174,105,142),(72,189,123,168,89,173,106,141),(73,188,124,167,90,172,107,140),(74,204,125,166,91,171,108,139),(75,203,126,165,92,187,109,138),(76,202,127,164,93,186,110,137),(77,201,128,163,94,185,111,153),(78,200,129,162,95,184,112,152),(79,199,130,161,96,183,113,151),(80,198,131,160,97,182,114,150),(81,197,132,159,98,181,115,149),(82,196,133,158,99,180,116,148),(83,195,134,157,100,179,117,147),(84,194,135,156,101,178,118,146),(85,193,136,155,102,177,119,145)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 17A | ··· | 17H | 34A | ··· | 34X | 68A | ··· | 68AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 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 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D17 | Dic17 | D34 | Dic17 | C17⋊3C8 |
| kernel | C2×C17⋊3C8 | C17⋊3C8 | C2×C68 | C68 | C2×C34 | C34 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 32 |
Matrix representation of C2×C17⋊3C8 ►in GL3(𝔽137) generated by
| 1 | 0 | 0 |
| 0 | 136 | 0 |
| 0 | 0 | 136 |
| 1 | 0 | 0 |
| 0 | 74 | 136 |
| 0 | 1 | 0 |
| 41 | 0 | 0 |
| 0 | 45 | 70 |
| 0 | 112 | 92 |
G:=sub<GL(3,GF(137))| [1,0,0,0,136,0,0,0,136],[1,0,0,0,74,1,0,136,0],[41,0,0,0,45,112,0,70,92] >;
C2×C17⋊3C8 in GAP, Magma, Sage, TeX
C_2\times C_{17}\rtimes_3C_8 % in TeX
G:=Group("C2xC17:3C8"); // GroupNames label
G:=SmallGroup(272,9);
// by ID
G=gap.SmallGroup(272,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,42,6404]);
// Polycyclic
G:=Group<a,b,c|a^2=b^17=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export