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G = C20.14Q16order 320 = 26·5

3rd non-split extension by C20 of Q16 acting via Q16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C20.14Q16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C20⋊2Q8 — C20.14Q16
 Lower central C5 — C10 — C2×C20 — C20.14Q16
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C20.14Q16
G = < a,b,c | a20=b8=1, c2=a10b4, ab=ba, cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 398 in 98 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C2×C10, C4×C8, Q8⋊C4, C4⋊Q8, C40, Dic10, C2×Dic5, C2×C20, C4.SD16, C4⋊Dic5, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C2×Dic10, C20.44D4, C4×C40, C202Q8, C20.14Q16
Quotients: C1, C2, C22, D4, C23, D5, SD16, Q16, C2×D4, C4○D4, D10, C4.4D4, C2×SD16, C2×Q16, D20, C22×D5, C4.SD16, C40⋊C2, Dic20, C2×D20, C4○D20, C4.D20, C2×C40⋊C2, C2×Dic20, C20.14Q16

Smallest permutation representation of C20.14Q16
Regular action on 320 points
Generators in S320
```(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)(301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)
(1 221 133 165 45 106 98 66)(2 222 134 166 46 107 99 67)(3 223 135 167 47 108 100 68)(4 224 136 168 48 109 81 69)(5 225 137 169 49 110 82 70)(6 226 138 170 50 111 83 71)(7 227 139 171 51 112 84 72)(8 228 140 172 52 113 85 73)(9 229 121 173 53 114 86 74)(10 230 122 174 54 115 87 75)(11 231 123 175 55 116 88 76)(12 232 124 176 56 117 89 77)(13 233 125 177 57 118 90 78)(14 234 126 178 58 119 91 79)(15 235 127 179 59 120 92 80)(16 236 128 180 60 101 93 61)(17 237 129 161 41 102 94 62)(18 238 130 162 42 103 95 63)(19 239 131 163 43 104 96 64)(20 240 132 164 44 105 97 65)(21 272 292 192 217 301 242 141)(22 273 293 193 218 302 243 142)(23 274 294 194 219 303 244 143)(24 275 295 195 220 304 245 144)(25 276 296 196 201 305 246 145)(26 277 297 197 202 306 247 146)(27 278 298 198 203 307 248 147)(28 279 299 199 204 308 249 148)(29 280 300 200 205 309 250 149)(30 261 281 181 206 310 251 150)(31 262 282 182 207 311 252 151)(32 263 283 183 208 312 253 152)(33 264 284 184 209 313 254 153)(34 265 285 185 210 314 255 154)(35 266 286 186 211 315 256 155)(36 267 287 187 212 316 257 156)(37 268 288 188 213 317 258 157)(38 269 289 189 214 318 259 158)(39 270 290 190 215 319 260 159)(40 271 291 191 216 320 241 160)
(1 306 55 267)(2 305 56 266)(3 304 57 265)(4 303 58 264)(5 302 59 263)(6 301 60 262)(7 320 41 261)(8 319 42 280)(9 318 43 279)(10 317 44 278)(11 316 45 277)(12 315 46 276)(13 314 47 275)(14 313 48 274)(15 312 49 273)(16 311 50 272)(17 310 51 271)(18 309 52 270)(19 308 53 269)(20 307 54 268)(21 226 207 101)(22 225 208 120)(23 224 209 119)(24 223 210 118)(25 222 211 117)(26 221 212 116)(27 240 213 115)(28 239 214 114)(29 238 215 113)(30 237 216 112)(31 236 217 111)(32 235 218 110)(33 234 219 109)(34 233 220 108)(35 232 201 107)(36 231 202 106)(37 230 203 105)(38 229 204 104)(39 228 205 103)(40 227 206 102)(61 242 170 282)(62 241 171 281)(63 260 172 300)(64 259 173 299)(65 258 174 298)(66 257 175 297)(67 256 176 296)(68 255 177 295)(69 254 178 294)(70 253 179 293)(71 252 180 292)(72 251 161 291)(73 250 162 290)(74 249 163 289)(75 248 164 288)(76 247 165 287)(77 246 166 286)(78 245 167 285)(79 244 168 284)(80 243 169 283)(81 143 126 184)(82 142 127 183)(83 141 128 182)(84 160 129 181)(85 159 130 200)(86 158 131 199)(87 157 132 198)(88 156 133 197)(89 155 134 196)(90 154 135 195)(91 153 136 194)(92 152 137 193)(93 151 138 192)(94 150 139 191)(95 149 140 190)(96 148 121 189)(97 147 122 188)(98 146 123 187)(99 145 124 186)(100 144 125 185)```

`G:=sub<Sym(320)| (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)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,221,133,165,45,106,98,66)(2,222,134,166,46,107,99,67)(3,223,135,167,47,108,100,68)(4,224,136,168,48,109,81,69)(5,225,137,169,49,110,82,70)(6,226,138,170,50,111,83,71)(7,227,139,171,51,112,84,72)(8,228,140,172,52,113,85,73)(9,229,121,173,53,114,86,74)(10,230,122,174,54,115,87,75)(11,231,123,175,55,116,88,76)(12,232,124,176,56,117,89,77)(13,233,125,177,57,118,90,78)(14,234,126,178,58,119,91,79)(15,235,127,179,59,120,92,80)(16,236,128,180,60,101,93,61)(17,237,129,161,41,102,94,62)(18,238,130,162,42,103,95,63)(19,239,131,163,43,104,96,64)(20,240,132,164,44,105,97,65)(21,272,292,192,217,301,242,141)(22,273,293,193,218,302,243,142)(23,274,294,194,219,303,244,143)(24,275,295,195,220,304,245,144)(25,276,296,196,201,305,246,145)(26,277,297,197,202,306,247,146)(27,278,298,198,203,307,248,147)(28,279,299,199,204,308,249,148)(29,280,300,200,205,309,250,149)(30,261,281,181,206,310,251,150)(31,262,282,182,207,311,252,151)(32,263,283,183,208,312,253,152)(33,264,284,184,209,313,254,153)(34,265,285,185,210,314,255,154)(35,266,286,186,211,315,256,155)(36,267,287,187,212,316,257,156)(37,268,288,188,213,317,258,157)(38,269,289,189,214,318,259,158)(39,270,290,190,215,319,260,159)(40,271,291,191,216,320,241,160), (1,306,55,267)(2,305,56,266)(3,304,57,265)(4,303,58,264)(5,302,59,263)(6,301,60,262)(7,320,41,261)(8,319,42,280)(9,318,43,279)(10,317,44,278)(11,316,45,277)(12,315,46,276)(13,314,47,275)(14,313,48,274)(15,312,49,273)(16,311,50,272)(17,310,51,271)(18,309,52,270)(19,308,53,269)(20,307,54,268)(21,226,207,101)(22,225,208,120)(23,224,209,119)(24,223,210,118)(25,222,211,117)(26,221,212,116)(27,240,213,115)(28,239,214,114)(29,238,215,113)(30,237,216,112)(31,236,217,111)(32,235,218,110)(33,234,219,109)(34,233,220,108)(35,232,201,107)(36,231,202,106)(37,230,203,105)(38,229,204,104)(39,228,205,103)(40,227,206,102)(61,242,170,282)(62,241,171,281)(63,260,172,300)(64,259,173,299)(65,258,174,298)(66,257,175,297)(67,256,176,296)(68,255,177,295)(69,254,178,294)(70,253,179,293)(71,252,180,292)(72,251,161,291)(73,250,162,290)(74,249,163,289)(75,248,164,288)(76,247,165,287)(77,246,166,286)(78,245,167,285)(79,244,168,284)(80,243,169,283)(81,143,126,184)(82,142,127,183)(83,141,128,182)(84,160,129,181)(85,159,130,200)(86,158,131,199)(87,157,132,198)(88,156,133,197)(89,155,134,196)(90,154,135,195)(91,153,136,194)(92,152,137,193)(93,151,138,192)(94,150,139,191)(95,149,140,190)(96,148,121,189)(97,147,122,188)(98,146,123,187)(99,145,124,186)(100,144,125,185)>;`

`G:=Group( (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)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,221,133,165,45,106,98,66)(2,222,134,166,46,107,99,67)(3,223,135,167,47,108,100,68)(4,224,136,168,48,109,81,69)(5,225,137,169,49,110,82,70)(6,226,138,170,50,111,83,71)(7,227,139,171,51,112,84,72)(8,228,140,172,52,113,85,73)(9,229,121,173,53,114,86,74)(10,230,122,174,54,115,87,75)(11,231,123,175,55,116,88,76)(12,232,124,176,56,117,89,77)(13,233,125,177,57,118,90,78)(14,234,126,178,58,119,91,79)(15,235,127,179,59,120,92,80)(16,236,128,180,60,101,93,61)(17,237,129,161,41,102,94,62)(18,238,130,162,42,103,95,63)(19,239,131,163,43,104,96,64)(20,240,132,164,44,105,97,65)(21,272,292,192,217,301,242,141)(22,273,293,193,218,302,243,142)(23,274,294,194,219,303,244,143)(24,275,295,195,220,304,245,144)(25,276,296,196,201,305,246,145)(26,277,297,197,202,306,247,146)(27,278,298,198,203,307,248,147)(28,279,299,199,204,308,249,148)(29,280,300,200,205,309,250,149)(30,261,281,181,206,310,251,150)(31,262,282,182,207,311,252,151)(32,263,283,183,208,312,253,152)(33,264,284,184,209,313,254,153)(34,265,285,185,210,314,255,154)(35,266,286,186,211,315,256,155)(36,267,287,187,212,316,257,156)(37,268,288,188,213,317,258,157)(38,269,289,189,214,318,259,158)(39,270,290,190,215,319,260,159)(40,271,291,191,216,320,241,160), (1,306,55,267)(2,305,56,266)(3,304,57,265)(4,303,58,264)(5,302,59,263)(6,301,60,262)(7,320,41,261)(8,319,42,280)(9,318,43,279)(10,317,44,278)(11,316,45,277)(12,315,46,276)(13,314,47,275)(14,313,48,274)(15,312,49,273)(16,311,50,272)(17,310,51,271)(18,309,52,270)(19,308,53,269)(20,307,54,268)(21,226,207,101)(22,225,208,120)(23,224,209,119)(24,223,210,118)(25,222,211,117)(26,221,212,116)(27,240,213,115)(28,239,214,114)(29,238,215,113)(30,237,216,112)(31,236,217,111)(32,235,218,110)(33,234,219,109)(34,233,220,108)(35,232,201,107)(36,231,202,106)(37,230,203,105)(38,229,204,104)(39,228,205,103)(40,227,206,102)(61,242,170,282)(62,241,171,281)(63,260,172,300)(64,259,173,299)(65,258,174,298)(66,257,175,297)(67,256,176,296)(68,255,177,295)(69,254,178,294)(70,253,179,293)(71,252,180,292)(72,251,161,291)(73,250,162,290)(74,249,163,289)(75,248,164,288)(76,247,165,287)(77,246,166,286)(78,245,167,285)(79,244,168,284)(80,243,169,283)(81,143,126,184)(82,142,127,183)(83,141,128,182)(84,160,129,181)(85,159,130,200)(86,158,131,199)(87,157,132,198)(88,156,133,197)(89,155,134,196)(90,154,135,195)(91,153,136,194)(92,152,137,193)(93,151,138,192)(94,150,139,191)(95,149,140,190)(96,148,121,189)(97,147,122,188)(98,146,123,187)(99,145,124,186)(100,144,125,185) );`

`G=PermutationGroup([[(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),(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)], [(1,221,133,165,45,106,98,66),(2,222,134,166,46,107,99,67),(3,223,135,167,47,108,100,68),(4,224,136,168,48,109,81,69),(5,225,137,169,49,110,82,70),(6,226,138,170,50,111,83,71),(7,227,139,171,51,112,84,72),(8,228,140,172,52,113,85,73),(9,229,121,173,53,114,86,74),(10,230,122,174,54,115,87,75),(11,231,123,175,55,116,88,76),(12,232,124,176,56,117,89,77),(13,233,125,177,57,118,90,78),(14,234,126,178,58,119,91,79),(15,235,127,179,59,120,92,80),(16,236,128,180,60,101,93,61),(17,237,129,161,41,102,94,62),(18,238,130,162,42,103,95,63),(19,239,131,163,43,104,96,64),(20,240,132,164,44,105,97,65),(21,272,292,192,217,301,242,141),(22,273,293,193,218,302,243,142),(23,274,294,194,219,303,244,143),(24,275,295,195,220,304,245,144),(25,276,296,196,201,305,246,145),(26,277,297,197,202,306,247,146),(27,278,298,198,203,307,248,147),(28,279,299,199,204,308,249,148),(29,280,300,200,205,309,250,149),(30,261,281,181,206,310,251,150),(31,262,282,182,207,311,252,151),(32,263,283,183,208,312,253,152),(33,264,284,184,209,313,254,153),(34,265,285,185,210,314,255,154),(35,266,286,186,211,315,256,155),(36,267,287,187,212,316,257,156),(37,268,288,188,213,317,258,157),(38,269,289,189,214,318,259,158),(39,270,290,190,215,319,260,159),(40,271,291,191,216,320,241,160)], [(1,306,55,267),(2,305,56,266),(3,304,57,265),(4,303,58,264),(5,302,59,263),(6,301,60,262),(7,320,41,261),(8,319,42,280),(9,318,43,279),(10,317,44,278),(11,316,45,277),(12,315,46,276),(13,314,47,275),(14,313,48,274),(15,312,49,273),(16,311,50,272),(17,310,51,271),(18,309,52,270),(19,308,53,269),(20,307,54,268),(21,226,207,101),(22,225,208,120),(23,224,209,119),(24,223,210,118),(25,222,211,117),(26,221,212,116),(27,240,213,115),(28,239,214,114),(29,238,215,113),(30,237,216,112),(31,236,217,111),(32,235,218,110),(33,234,219,109),(34,233,220,108),(35,232,201,107),(36,231,202,106),(37,230,203,105),(38,229,204,104),(39,228,205,103),(40,227,206,102),(61,242,170,282),(62,241,171,281),(63,260,172,300),(64,259,173,299),(65,258,174,298),(66,257,175,297),(67,256,176,296),(68,255,177,295),(69,254,178,294),(70,253,179,293),(71,252,180,292),(72,251,161,291),(73,250,162,290),(74,249,163,289),(75,248,164,288),(76,247,165,287),(77,246,166,286),(78,245,167,285),(79,244,168,284),(80,243,169,283),(81,143,126,184),(82,142,127,183),(83,141,128,182),(84,160,129,181),(85,159,130,200),(86,158,131,199),(87,157,132,198),(88,156,133,197),(89,155,134,196),(90,154,135,195),(91,153,136,194),(92,152,137,193),(93,151,138,192),(94,150,139,191),(95,149,140,190),(96,148,121,189),(97,147,122,188),(98,146,123,187),(99,145,124,186),(100,144,125,185)]])`

86 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 5A 5B 8A ··· 8H 10A ··· 10F 20A ··· 20X 40A ··· 40AF order 1 2 2 2 4 ··· 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 ··· 2 40 40 40 40 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

86 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + + + - image C1 C2 C2 C2 D4 D5 SD16 Q16 C4○D4 D10 D10 D20 C40⋊C2 Dic20 C4○D20 kernel C20.14Q16 C20.44D4 C4×C40 C20⋊2Q8 C2×C20 C4×C8 C20 C20 C20 C42 C2×C8 C2×C4 C4 C4 C4 # reps 1 4 1 2 2 2 4 4 4 2 4 8 16 16 16

Matrix representation of C20.14Q16 in GL4(𝔽41) generated by

 14 30 0 0 11 9 0 0 0 0 27 11 0 0 30 32
,
 32 0 0 0 0 32 0 0 0 0 27 12 0 0 29 25
,
 12 27 0 0 25 29 0 0 0 0 30 32 0 0 27 11
`G:=sub<GL(4,GF(41))| [14,11,0,0,30,9,0,0,0,0,27,30,0,0,11,32],[32,0,0,0,0,32,0,0,0,0,27,29,0,0,12,25],[12,25,0,0,27,29,0,0,0,0,30,27,0,0,32,11] >;`

C20.14Q16 in GAP, Magma, Sage, TeX

`C_{20}._{14}Q_{16}`
`% in TeX`

`G:=Group("C20.14Q16");`
`// GroupNames label`

`G:=SmallGroup(320,308);`
`// by ID`

`G=gap.SmallGroup(320,308);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,142,1123,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^8=1,c^2=a^10*b^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;`
`// generators/relations`

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