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G = C11×Dic7order 308 = 22·7·11

Direct product of C11 and Dic7

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C11×Dic7, C7⋊C44, C773C4, C14.C22, C22.2D7, C154.3C2, C2.(C11×D7), SmallGroup(308,1)

Series: Derived Chief Lower central Upper central

C1C7 — C11×Dic7
C1C7C14C154 — C11×Dic7
C7 — C11×Dic7
C1C22

Generators and relations for C11×Dic7
 G = < a,b,c | a11=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

7C4
7C44

Smallest permutation representation of C11×Dic7
Regular action on 308 points
Generators in S308
(1 144 131 125 103 98 74 57 46 41 15)(2 145 132 126 104 85 75 58 47 42 16)(3 146 133 113 105 86 76 59 48 29 17)(4 147 134 114 106 87 77 60 49 30 18)(5 148 135 115 107 88 78 61 50 31 19)(6 149 136 116 108 89 79 62 51 32 20)(7 150 137 117 109 90 80 63 52 33 21)(8 151 138 118 110 91 81 64 53 34 22)(9 152 139 119 111 92 82 65 54 35 23)(10 153 140 120 112 93 83 66 55 36 24)(11 154 127 121 99 94 84 67 56 37 25)(12 141 128 122 100 95 71 68 43 38 26)(13 142 129 123 101 96 72 69 44 39 27)(14 143 130 124 102 97 73 70 45 40 28)(155 295 288 274 260 246 232 218 204 183 169)(156 296 289 275 261 247 233 219 205 184 170)(157 297 290 276 262 248 234 220 206 185 171)(158 298 291 277 263 249 235 221 207 186 172)(159 299 292 278 264 250 236 222 208 187 173)(160 300 293 279 265 251 237 223 209 188 174)(161 301 294 280 266 252 238 224 210 189 175)(162 302 281 267 253 239 225 211 197 190 176)(163 303 282 268 254 240 226 212 198 191 177)(164 304 283 269 255 241 227 213 199 192 178)(165 305 284 270 256 242 228 214 200 193 179)(166 306 285 271 257 243 229 215 201 194 180)(167 307 286 272 258 244 230 216 202 195 181)(168 308 287 273 259 245 231 217 203 196 182)
(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)
(1 162 8 155)(2 161 9 168)(3 160 10 167)(4 159 11 166)(5 158 12 165)(6 157 13 164)(7 156 14 163)(15 176 22 169)(16 175 23 182)(17 174 24 181)(18 173 25 180)(19 172 26 179)(20 171 27 178)(21 170 28 177)(29 188 36 195)(30 187 37 194)(31 186 38 193)(32 185 39 192)(33 184 40 191)(34 183 41 190)(35 196 42 189)(43 200 50 207)(44 199 51 206)(45 198 52 205)(46 197 53 204)(47 210 54 203)(48 209 55 202)(49 208 56 201)(57 211 64 218)(58 224 65 217)(59 223 66 216)(60 222 67 215)(61 221 68 214)(62 220 69 213)(63 219 70 212)(71 228 78 235)(72 227 79 234)(73 226 80 233)(74 225 81 232)(75 238 82 231)(76 237 83 230)(77 236 84 229)(85 252 92 245)(86 251 93 244)(87 250 94 243)(88 249 95 242)(89 248 96 241)(90 247 97 240)(91 246 98 239)(99 257 106 264)(100 256 107 263)(101 255 108 262)(102 254 109 261)(103 253 110 260)(104 266 111 259)(105 265 112 258)(113 279 120 272)(114 278 121 271)(115 277 122 270)(116 276 123 269)(117 275 124 268)(118 274 125 267)(119 273 126 280)(127 285 134 292)(128 284 135 291)(129 283 136 290)(130 282 137 289)(131 281 138 288)(132 294 139 287)(133 293 140 286)(141 305 148 298)(142 304 149 297)(143 303 150 296)(144 302 151 295)(145 301 152 308)(146 300 153 307)(147 299 154 306)

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

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

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

110 conjugacy classes

class 1  2 4A4B7A7B7C11A···11J14A14B14C22A···22J44A···44T77A···77AD154A···154AD
order124477711···1114141422···2244···4477···77154···154
size11772221···12221···17···72···22···2

110 irreducible representations

dim1111112222
type+++-
imageC1C2C4C11C22C44D7Dic7C11×D7C11×Dic7
kernelC11×Dic7C154C77Dic7C14C7C22C11C2C1
# reps112101020333030

Matrix representation of C11×Dic7 in GL3(𝔽617) generated by

48900
010
001
,
100
00616
01271
,
100
0158508
0263459
G:=sub<GL(3,GF(617))| [489,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,616,271],[1,0,0,0,158,263,0,508,459] >;

C11×Dic7 in GAP, Magma, Sage, TeX

C_{11}\times {\rm Dic}_7
% in TeX

G:=Group("C11xDic7");
// GroupNames label

G:=SmallGroup(308,1);
// by ID

G=gap.SmallGroup(308,1);
# by ID

G:=PCGroup([4,-2,-11,-2,-7,88,4227]);
// Polycyclic

G:=Group<a,b,c|a^11=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C11×Dic7 in TeX

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