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## G = C7×Dic11order 308 = 22·7·11

### Direct product of C7 and Dic11

Aliases: C7×Dic11, C11⋊C28, C772C4, C22.C14, C154.2C2, C14.2D11, C2.(C7×D11), SmallGroup(308,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C7×Dic11
 Chief series C1 — C11 — C22 — C154 — C7×Dic11
 Lower central C11 — C7×Dic11
 Upper central C1 — C14

Generators and relations for C7×Dic11
G = < a,b,c | a7=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×Dic11
Regular action on 308 points
Generators in S308
(1 133 119 102 74 48 26)(2 134 120 103 75 49 27)(3 135 121 104 76 50 28)(4 136 122 105 77 51 29)(5 137 123 106 78 52 30)(6 138 124 107 79 53 31)(7 139 125 108 80 54 32)(8 140 126 109 81 55 33)(9 141 127 110 82 56 34)(10 142 128 89 83 57 35)(11 143 129 90 84 58 36)(12 144 130 91 85 59 37)(13 145 131 92 86 60 38)(14 146 132 93 87 61 39)(15 147 111 94 88 62 40)(16 148 112 95 67 63 41)(17 149 113 96 68 64 42)(18 150 114 97 69 65 43)(19 151 115 98 70 66 44)(20 152 116 99 71 45 23)(21 153 117 100 72 46 24)(22 154 118 101 73 47 25)(155 298 276 243 221 210 177)(156 299 277 244 222 211 178)(157 300 278 245 223 212 179)(158 301 279 246 224 213 180)(159 302 280 247 225 214 181)(160 303 281 248 226 215 182)(161 304 282 249 227 216 183)(162 305 283 250 228 217 184)(163 306 284 251 229 218 185)(164 307 285 252 230 219 186)(165 308 286 253 231 220 187)(166 287 265 254 232 199 188)(167 288 266 255 233 200 189)(168 289 267 256 234 201 190)(169 290 268 257 235 202 191)(170 291 269 258 236 203 192)(171 292 270 259 237 204 193)(172 293 271 260 238 205 194)(173 294 272 261 239 206 195)(174 295 273 262 240 207 196)(175 296 274 263 241 208 197)(176 297 275 264 242 209 198)
(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)(265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286)(287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308)
(1 155 12 166)(2 176 13 165)(3 175 14 164)(4 174 15 163)(5 173 16 162)(6 172 17 161)(7 171 18 160)(8 170 19 159)(9 169 20 158)(10 168 21 157)(11 167 22 156)(23 180 34 191)(24 179 35 190)(25 178 36 189)(26 177 37 188)(27 198 38 187)(28 197 39 186)(29 196 40 185)(30 195 41 184)(31 194 42 183)(32 193 43 182)(33 192 44 181)(45 213 56 202)(46 212 57 201)(47 211 58 200)(48 210 59 199)(49 209 60 220)(50 208 61 219)(51 207 62 218)(52 206 63 217)(53 205 64 216)(54 204 65 215)(55 203 66 214)(67 228 78 239)(68 227 79 238)(69 226 80 237)(70 225 81 236)(71 224 82 235)(72 223 83 234)(73 222 84 233)(74 221 85 232)(75 242 86 231)(76 241 87 230)(77 240 88 229)(89 256 100 245)(90 255 101 244)(91 254 102 243)(92 253 103 264)(93 252 104 263)(94 251 105 262)(95 250 106 261)(96 249 107 260)(97 248 108 259)(98 247 109 258)(99 246 110 257)(111 284 122 273)(112 283 123 272)(113 282 124 271)(114 281 125 270)(115 280 126 269)(116 279 127 268)(117 278 128 267)(118 277 129 266)(119 276 130 265)(120 275 131 286)(121 274 132 285)(133 298 144 287)(134 297 145 308)(135 296 146 307)(136 295 147 306)(137 294 148 305)(138 293 149 304)(139 292 150 303)(140 291 151 302)(141 290 152 301)(142 289 153 300)(143 288 154 299)

G:=sub<Sym(308)| (1,133,119,102,74,48,26)(2,134,120,103,75,49,27)(3,135,121,104,76,50,28)(4,136,122,105,77,51,29)(5,137,123,106,78,52,30)(6,138,124,107,79,53,31)(7,139,125,108,80,54,32)(8,140,126,109,81,55,33)(9,141,127,110,82,56,34)(10,142,128,89,83,57,35)(11,143,129,90,84,58,36)(12,144,130,91,85,59,37)(13,145,131,92,86,60,38)(14,146,132,93,87,61,39)(15,147,111,94,88,62,40)(16,148,112,95,67,63,41)(17,149,113,96,68,64,42)(18,150,114,97,69,65,43)(19,151,115,98,70,66,44)(20,152,116,99,71,45,23)(21,153,117,100,72,46,24)(22,154,118,101,73,47,25)(155,298,276,243,221,210,177)(156,299,277,244,222,211,178)(157,300,278,245,223,212,179)(158,301,279,246,224,213,180)(159,302,280,247,225,214,181)(160,303,281,248,226,215,182)(161,304,282,249,227,216,183)(162,305,283,250,228,217,184)(163,306,284,251,229,218,185)(164,307,285,252,230,219,186)(165,308,286,253,231,220,187)(166,287,265,254,232,199,188)(167,288,266,255,233,200,189)(168,289,267,256,234,201,190)(169,290,268,257,235,202,191)(170,291,269,258,236,203,192)(171,292,270,259,237,204,193)(172,293,271,260,238,205,194)(173,294,272,261,239,206,195)(174,295,273,262,240,207,196)(175,296,274,263,241,208,197)(176,297,275,264,242,209,198), (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)(265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286)(287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308), (1,155,12,166)(2,176,13,165)(3,175,14,164)(4,174,15,163)(5,173,16,162)(6,172,17,161)(7,171,18,160)(8,170,19,159)(9,169,20,158)(10,168,21,157)(11,167,22,156)(23,180,34,191)(24,179,35,190)(25,178,36,189)(26,177,37,188)(27,198,38,187)(28,197,39,186)(29,196,40,185)(30,195,41,184)(31,194,42,183)(32,193,43,182)(33,192,44,181)(45,213,56,202)(46,212,57,201)(47,211,58,200)(48,210,59,199)(49,209,60,220)(50,208,61,219)(51,207,62,218)(52,206,63,217)(53,205,64,216)(54,204,65,215)(55,203,66,214)(67,228,78,239)(68,227,79,238)(69,226,80,237)(70,225,81,236)(71,224,82,235)(72,223,83,234)(73,222,84,233)(74,221,85,232)(75,242,86,231)(76,241,87,230)(77,240,88,229)(89,256,100,245)(90,255,101,244)(91,254,102,243)(92,253,103,264)(93,252,104,263)(94,251,105,262)(95,250,106,261)(96,249,107,260)(97,248,108,259)(98,247,109,258)(99,246,110,257)(111,284,122,273)(112,283,123,272)(113,282,124,271)(114,281,125,270)(115,280,126,269)(116,279,127,268)(117,278,128,267)(118,277,129,266)(119,276,130,265)(120,275,131,286)(121,274,132,285)(133,298,144,287)(134,297,145,308)(135,296,146,307)(136,295,147,306)(137,294,148,305)(138,293,149,304)(139,292,150,303)(140,291,151,302)(141,290,152,301)(142,289,153,300)(143,288,154,299)>;

G:=Group( (1,133,119,102,74,48,26)(2,134,120,103,75,49,27)(3,135,121,104,76,50,28)(4,136,122,105,77,51,29)(5,137,123,106,78,52,30)(6,138,124,107,79,53,31)(7,139,125,108,80,54,32)(8,140,126,109,81,55,33)(9,141,127,110,82,56,34)(10,142,128,89,83,57,35)(11,143,129,90,84,58,36)(12,144,130,91,85,59,37)(13,145,131,92,86,60,38)(14,146,132,93,87,61,39)(15,147,111,94,88,62,40)(16,148,112,95,67,63,41)(17,149,113,96,68,64,42)(18,150,114,97,69,65,43)(19,151,115,98,70,66,44)(20,152,116,99,71,45,23)(21,153,117,100,72,46,24)(22,154,118,101,73,47,25)(155,298,276,243,221,210,177)(156,299,277,244,222,211,178)(157,300,278,245,223,212,179)(158,301,279,246,224,213,180)(159,302,280,247,225,214,181)(160,303,281,248,226,215,182)(161,304,282,249,227,216,183)(162,305,283,250,228,217,184)(163,306,284,251,229,218,185)(164,307,285,252,230,219,186)(165,308,286,253,231,220,187)(166,287,265,254,232,199,188)(167,288,266,255,233,200,189)(168,289,267,256,234,201,190)(169,290,268,257,235,202,191)(170,291,269,258,236,203,192)(171,292,270,259,237,204,193)(172,293,271,260,238,205,194)(173,294,272,261,239,206,195)(174,295,273,262,240,207,196)(175,296,274,263,241,208,197)(176,297,275,264,242,209,198), (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)(265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286)(287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308), (1,155,12,166)(2,176,13,165)(3,175,14,164)(4,174,15,163)(5,173,16,162)(6,172,17,161)(7,171,18,160)(8,170,19,159)(9,169,20,158)(10,168,21,157)(11,167,22,156)(23,180,34,191)(24,179,35,190)(25,178,36,189)(26,177,37,188)(27,198,38,187)(28,197,39,186)(29,196,40,185)(30,195,41,184)(31,194,42,183)(32,193,43,182)(33,192,44,181)(45,213,56,202)(46,212,57,201)(47,211,58,200)(48,210,59,199)(49,209,60,220)(50,208,61,219)(51,207,62,218)(52,206,63,217)(53,205,64,216)(54,204,65,215)(55,203,66,214)(67,228,78,239)(68,227,79,238)(69,226,80,237)(70,225,81,236)(71,224,82,235)(72,223,83,234)(73,222,84,233)(74,221,85,232)(75,242,86,231)(76,241,87,230)(77,240,88,229)(89,256,100,245)(90,255,101,244)(91,254,102,243)(92,253,103,264)(93,252,104,263)(94,251,105,262)(95,250,106,261)(96,249,107,260)(97,248,108,259)(98,247,109,258)(99,246,110,257)(111,284,122,273)(112,283,123,272)(113,282,124,271)(114,281,125,270)(115,280,126,269)(116,279,127,268)(117,278,128,267)(118,277,129,266)(119,276,130,265)(120,275,131,286)(121,274,132,285)(133,298,144,287)(134,297,145,308)(135,296,146,307)(136,295,147,306)(137,294,148,305)(138,293,149,304)(139,292,150,303)(140,291,151,302)(141,290,152,301)(142,289,153,300)(143,288,154,299) );

G=PermutationGroup([(1,133,119,102,74,48,26),(2,134,120,103,75,49,27),(3,135,121,104,76,50,28),(4,136,122,105,77,51,29),(5,137,123,106,78,52,30),(6,138,124,107,79,53,31),(7,139,125,108,80,54,32),(8,140,126,109,81,55,33),(9,141,127,110,82,56,34),(10,142,128,89,83,57,35),(11,143,129,90,84,58,36),(12,144,130,91,85,59,37),(13,145,131,92,86,60,38),(14,146,132,93,87,61,39),(15,147,111,94,88,62,40),(16,148,112,95,67,63,41),(17,149,113,96,68,64,42),(18,150,114,97,69,65,43),(19,151,115,98,70,66,44),(20,152,116,99,71,45,23),(21,153,117,100,72,46,24),(22,154,118,101,73,47,25),(155,298,276,243,221,210,177),(156,299,277,244,222,211,178),(157,300,278,245,223,212,179),(158,301,279,246,224,213,180),(159,302,280,247,225,214,181),(160,303,281,248,226,215,182),(161,304,282,249,227,216,183),(162,305,283,250,228,217,184),(163,306,284,251,229,218,185),(164,307,285,252,230,219,186),(165,308,286,253,231,220,187),(166,287,265,254,232,199,188),(167,288,266,255,233,200,189),(168,289,267,256,234,201,190),(169,290,268,257,235,202,191),(170,291,269,258,236,203,192),(171,292,270,259,237,204,193),(172,293,271,260,238,205,194),(173,294,272,261,239,206,195),(174,295,273,262,240,207,196),(175,296,274,263,241,208,197),(176,297,275,264,242,209,198)], [(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),(265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286),(287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308)], [(1,155,12,166),(2,176,13,165),(3,175,14,164),(4,174,15,163),(5,173,16,162),(6,172,17,161),(7,171,18,160),(8,170,19,159),(9,169,20,158),(10,168,21,157),(11,167,22,156),(23,180,34,191),(24,179,35,190),(25,178,36,189),(26,177,37,188),(27,198,38,187),(28,197,39,186),(29,196,40,185),(30,195,41,184),(31,194,42,183),(32,193,43,182),(33,192,44,181),(45,213,56,202),(46,212,57,201),(47,211,58,200),(48,210,59,199),(49,209,60,220),(50,208,61,219),(51,207,62,218),(52,206,63,217),(53,205,64,216),(54,204,65,215),(55,203,66,214),(67,228,78,239),(68,227,79,238),(69,226,80,237),(70,225,81,236),(71,224,82,235),(72,223,83,234),(73,222,84,233),(74,221,85,232),(75,242,86,231),(76,241,87,230),(77,240,88,229),(89,256,100,245),(90,255,101,244),(91,254,102,243),(92,253,103,264),(93,252,104,263),(94,251,105,262),(95,250,106,261),(96,249,107,260),(97,248,108,259),(98,247,109,258),(99,246,110,257),(111,284,122,273),(112,283,123,272),(113,282,124,271),(114,281,125,270),(115,280,126,269),(116,279,127,268),(117,278,128,267),(118,277,129,266),(119,276,130,265),(120,275,131,286),(121,274,132,285),(133,298,144,287),(134,297,145,308),(135,296,146,307),(136,295,147,306),(137,294,148,305),(138,293,149,304),(139,292,150,303),(140,291,151,302),(141,290,152,301),(142,289,153,300),(143,288,154,299)])

98 conjugacy classes

 class 1 2 4A 4B 7A ··· 7F 11A ··· 11E 14A ··· 14F 22A ··· 22E 28A ··· 28L 77A ··· 77AD 154A ··· 154AD order 1 2 4 4 7 ··· 7 11 ··· 11 14 ··· 14 22 ··· 22 28 ··· 28 77 ··· 77 154 ··· 154 size 1 1 11 11 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 11 ··· 11 2 ··· 2 2 ··· 2

98 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C7 C14 C28 D11 Dic11 C7×D11 C7×Dic11 kernel C7×Dic11 C154 C77 Dic11 C22 C11 C14 C7 C2 C1 # reps 1 1 2 6 6 12 5 5 30 30

Matrix representation of C7×Dic11 in GL2(𝔽43) generated by

 16 0 0 16
,
 9 7 28 41
,
 0 22 41 0
G:=sub<GL(2,GF(43))| [16,0,0,16],[9,28,7,41],[0,41,22,0] >;

C7×Dic11 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_{11}
% in TeX

G:=Group("C7xDic11");
// GroupNames label

G:=SmallGroup(308,2);
// by ID

G=gap.SmallGroup(308,2);
# by ID

G:=PCGroup([4,-2,-7,-2,-11,56,4483]);
// Polycyclic

G:=Group<a,b,c|a^7=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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