direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic34, C34⋊Q8, C4.11D34, C34.1C23, C22.8D34, C68.11C22, Dic17.1C22, C17⋊1(C2×Q8), (C2×C68).4C2, (C2×C4).4D17, (C2×C34).8C22, C2.3(C22×D17), (C2×Dic17).3C2, SmallGroup(272,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic34
G = < a,b,c | a2=b68=1, c2=b34, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic34
►Regular action on 272 points
Generators in S272
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(137 256)(138 257)(139 258)(140 259)(141 260)(142 261)(143 262)(144 263)(145 264)(146 265)(147 266)(148 267)(149 268)(150 269)(151 270)(152 271)(153 272)(154 205)(155 206)(156 207)(157 208)(158 209)(159 210)(160 211)(161 212)(162 213)(163 214)(164 215)(165 216)(166 217)(167 218)(168 219)(169 220)(170 221)(171 222)(172 223)(173 224)(174 225)(175 226)(176 227)(177 228)(178 229)(179 230)(180 231)(181 232)(182 233)(183 234)(184 235)(185 236)(186 237)(187 238)(188 239)(189 240)(190 241)(191 242)(192 243)(193 244)(194 245)(195 246)(196 247)(197 248)(198 249)(199 250)(200 251)(201 252)(202 253)(203 254)(204 255)
(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)
(1 171 35 137)(2 170 36 204)(3 169 37 203)(4 168 38 202)(5 167 39 201)(6 166 40 200)(7 165 41 199)(8 164 42 198)(9 163 43 197)(10 162 44 196)(11 161 45 195)(12 160 46 194)(13 159 47 193)(14 158 48 192)(15 157 49 191)(16 156 50 190)(17 155 51 189)(18 154 52 188)(19 153 53 187)(20 152 54 186)(21 151 55 185)(22 150 56 184)(23 149 57 183)(24 148 58 182)(25 147 59 181)(26 146 60 180)(27 145 61 179)(28 144 62 178)(29 143 63 177)(30 142 64 176)(31 141 65 175)(32 140 66 174)(33 139 67 173)(34 138 68 172)(69 238 103 272)(70 237 104 271)(71 236 105 270)(72 235 106 269)(73 234 107 268)(74 233 108 267)(75 232 109 266)(76 231 110 265)(77 230 111 264)(78 229 112 263)(79 228 113 262)(80 227 114 261)(81 226 115 260)(82 225 116 259)(83 224 117 258)(84 223 118 257)(85 222 119 256)(86 221 120 255)(87 220 121 254)(88 219 122 253)(89 218 123 252)(90 217 124 251)(91 216 125 250)(92 215 126 249)(93 214 127 248)(94 213 128 247)(95 212 129 246)(96 211 130 245)(97 210 131 244)(98 209 132 243)(99 208 133 242)(100 207 134 241)(101 206 135 240)(102 205 136 239)
G:=sub<Sym(272)| (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(137,256)(138,257)(139,258)(140,259)(141,260)(142,261)(143,262)(144,263)(145,264)(146,265)(147,266)(148,267)(149,268)(150,269)(151,270)(152,271)(153,272)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219)(169,220)(170,221)(171,222)(172,223)(173,224)(174,225)(175,226)(176,227)(177,228)(178,229)(179,230)(180,231)(181,232)(182,233)(183,234)(184,235)(185,236)(186,237)(187,238)(188,239)(189,240)(190,241)(191,242)(192,243)(193,244)(194,245)(195,246)(196,247)(197,248)(198,249)(199,250)(200,251)(201,252)(202,253)(203,254)(204,255), (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), (1,171,35,137)(2,170,36,204)(3,169,37,203)(4,168,38,202)(5,167,39,201)(6,166,40,200)(7,165,41,199)(8,164,42,198)(9,163,43,197)(10,162,44,196)(11,161,45,195)(12,160,46,194)(13,159,47,193)(14,158,48,192)(15,157,49,191)(16,156,50,190)(17,155,51,189)(18,154,52,188)(19,153,53,187)(20,152,54,186)(21,151,55,185)(22,150,56,184)(23,149,57,183)(24,148,58,182)(25,147,59,181)(26,146,60,180)(27,145,61,179)(28,144,62,178)(29,143,63,177)(30,142,64,176)(31,141,65,175)(32,140,66,174)(33,139,67,173)(34,138,68,172)(69,238,103,272)(70,237,104,271)(71,236,105,270)(72,235,106,269)(73,234,107,268)(74,233,108,267)(75,232,109,266)(76,231,110,265)(77,230,111,264)(78,229,112,263)(79,228,113,262)(80,227,114,261)(81,226,115,260)(82,225,116,259)(83,224,117,258)(84,223,118,257)(85,222,119,256)(86,221,120,255)(87,220,121,254)(88,219,122,253)(89,218,123,252)(90,217,124,251)(91,216,125,250)(92,215,126,249)(93,214,127,248)(94,213,128,247)(95,212,129,246)(96,211,130,245)(97,210,131,244)(98,209,132,243)(99,208,133,242)(100,207,134,241)(101,206,135,240)(102,205,136,239)>;
G:=Group( (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(137,256)(138,257)(139,258)(140,259)(141,260)(142,261)(143,262)(144,263)(145,264)(146,265)(147,266)(148,267)(149,268)(150,269)(151,270)(152,271)(153,272)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219)(169,220)(170,221)(171,222)(172,223)(173,224)(174,225)(175,226)(176,227)(177,228)(178,229)(179,230)(180,231)(181,232)(182,233)(183,234)(184,235)(185,236)(186,237)(187,238)(188,239)(189,240)(190,241)(191,242)(192,243)(193,244)(194,245)(195,246)(196,247)(197,248)(198,249)(199,250)(200,251)(201,252)(202,253)(203,254)(204,255), (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), (1,171,35,137)(2,170,36,204)(3,169,37,203)(4,168,38,202)(5,167,39,201)(6,166,40,200)(7,165,41,199)(8,164,42,198)(9,163,43,197)(10,162,44,196)(11,161,45,195)(12,160,46,194)(13,159,47,193)(14,158,48,192)(15,157,49,191)(16,156,50,190)(17,155,51,189)(18,154,52,188)(19,153,53,187)(20,152,54,186)(21,151,55,185)(22,150,56,184)(23,149,57,183)(24,148,58,182)(25,147,59,181)(26,146,60,180)(27,145,61,179)(28,144,62,178)(29,143,63,177)(30,142,64,176)(31,141,65,175)(32,140,66,174)(33,139,67,173)(34,138,68,172)(69,238,103,272)(70,237,104,271)(71,236,105,270)(72,235,106,269)(73,234,107,268)(74,233,108,267)(75,232,109,266)(76,231,110,265)(77,230,111,264)(78,229,112,263)(79,228,113,262)(80,227,114,261)(81,226,115,260)(82,225,116,259)(83,224,117,258)(84,223,118,257)(85,222,119,256)(86,221,120,255)(87,220,121,254)(88,219,122,253)(89,218,123,252)(90,217,124,251)(91,216,125,250)(92,215,126,249)(93,214,127,248)(94,213,128,247)(95,212,129,246)(96,211,130,245)(97,210,131,244)(98,209,132,243)(99,208,133,242)(100,207,134,241)(101,206,135,240)(102,205,136,239) );
G=PermutationGroup([[(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,113),(30,114),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,121),(38,122),(39,123),(40,124),(41,125),(42,126),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,69),(54,70),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,79),(64,80),(65,81),(66,82),(67,83),(68,84),(137,256),(138,257),(139,258),(140,259),(141,260),(142,261),(143,262),(144,263),(145,264),(146,265),(147,266),(148,267),(149,268),(150,269),(151,270),(152,271),(153,272),(154,205),(155,206),(156,207),(157,208),(158,209),(159,210),(160,211),(161,212),(162,213),(163,214),(164,215),(165,216),(166,217),(167,218),(168,219),(169,220),(170,221),(171,222),(172,223),(173,224),(174,225),(175,226),(176,227),(177,228),(178,229),(179,230),(180,231),(181,232),(182,233),(183,234),(184,235),(185,236),(186,237),(187,238),(188,239),(189,240),(190,241),(191,242),(192,243),(193,244),(194,245),(195,246),(196,247),(197,248),(198,249),(199,250),(200,251),(201,252),(202,253),(203,254),(204,255)], [(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)], [(1,171,35,137),(2,170,36,204),(3,169,37,203),(4,168,38,202),(5,167,39,201),(6,166,40,200),(7,165,41,199),(8,164,42,198),(9,163,43,197),(10,162,44,196),(11,161,45,195),(12,160,46,194),(13,159,47,193),(14,158,48,192),(15,157,49,191),(16,156,50,190),(17,155,51,189),(18,154,52,188),(19,153,53,187),(20,152,54,186),(21,151,55,185),(22,150,56,184),(23,149,57,183),(24,148,58,182),(25,147,59,181),(26,146,60,180),(27,145,61,179),(28,144,62,178),(29,143,63,177),(30,142,64,176),(31,141,65,175),(32,140,66,174),(33,139,67,173),(34,138,68,172),(69,238,103,272),(70,237,104,271),(71,236,105,270),(72,235,106,269),(73,234,107,268),(74,233,108,267),(75,232,109,266),(76,231,110,265),(77,230,111,264),(78,229,112,263),(79,228,113,262),(80,227,114,261),(81,226,115,260),(82,225,116,259),(83,224,117,258),(84,223,118,257),(85,222,119,256),(86,221,120,255),(87,220,121,254),(88,219,122,253),(89,218,123,252),(90,217,124,251),(91,216,125,250),(92,215,126,249),(93,214,127,248),(94,213,128,247),(95,212,129,246),(96,211,130,245),(97,210,131,244),(98,209,132,243),(99,208,133,242),(100,207,134,241),(101,206,135,240),(102,205,136,239)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 2 | 2 | 34 | 34 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D17 | D34 | D34 | Dic34 |
| kernel | C2×Dic34 | Dic34 | C2×Dic17 | C2×C68 | C34 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 16 | 8 | 32 |
Matrix representation of C2×Dic34 ►in GL3(𝔽137) generated by
| 136 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 136 | 0 | 0 |
| 0 | 38 | 115 |
| 0 | 22 | 63 |
| 136 | 0 | 0 |
| 0 | 128 | 121 |
| 0 | 125 | 9 |
G:=sub<GL(3,GF(137))| [136,0,0,0,1,0,0,0,1],[136,0,0,0,38,22,0,115,63],[136,0,0,0,128,125,0,121,9] >;
C2×Dic34 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{34} % in TeX
G:=Group("C2xDic34"); // GroupNames label
G:=SmallGroup(272,36);
// by ID
G=gap.SmallGroup(272,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,40,182,42,6404]);
// Polycyclic
G:=Group<a,b,c|a^2=b^68=1,c^2=b^34,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export