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## G = C10.SD32order 320 = 26·5

### 1st non-split extension by C10 of SD32 acting via SD32/Q16=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C10.SD32
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C2×C5⋊2C16 — C10.SD32
 Lower central C5 — C10 — C20 — C40 — C10.SD32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2.D8

Generators and relations for C10.SD32
G = < a,b,c | a10=b16=1, c2=a5, bab-1=cac-1=a-1, cbc-1=b7 >

Smallest permutation representation of C10.SD32
Regular action on 320 points
Generators in S320
(1 44 180 293 20 165 312 282 65 197)(2 198 66 283 313 166 21 294 181 45)(3 46 182 295 22 167 314 284 67 199)(4 200 68 285 315 168 23 296 183 47)(5 48 184 297 24 169 316 286 69 201)(6 202 70 287 317 170 25 298 185 33)(7 34 186 299 26 171 318 288 71 203)(8 204 72 273 319 172 27 300 187 35)(9 36 188 301 28 173 320 274 73 205)(10 206 74 275 305 174 29 302 189 37)(11 38 190 303 30 175 306 276 75 207)(12 208 76 277 307 176 31 304 191 39)(13 40 192 289 32 161 308 278 77 193)(14 194 78 279 309 162 17 290 177 41)(15 42 178 291 18 163 310 280 79 195)(16 196 80 281 311 164 19 292 179 43)(49 236 212 144 86 153 263 123 245 107)(50 108 246 124 264 154 87 129 213 237)(51 238 214 130 88 155 265 125 247 109)(52 110 248 126 266 156 89 131 215 239)(53 240 216 132 90 157 267 127 249 111)(54 112 250 128 268 158 91 133 217 225)(55 226 218 134 92 159 269 113 251 97)(56 98 252 114 270 160 93 135 219 227)(57 228 220 136 94 145 271 115 253 99)(58 100 254 116 272 146 95 137 221 229)(59 230 222 138 96 147 257 117 255 101)(60 102 256 118 258 148 81 139 223 231)(61 232 224 140 82 149 259 119 241 103)(62 104 242 120 260 150 83 141 209 233)(63 234 210 142 84 151 261 121 243 105)(64 106 244 122 262 152 85 143 211 235)
(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 53 165 157)(2 60 166 148)(3 51 167 155)(4 58 168 146)(5 49 169 153)(6 56 170 160)(7 63 171 151)(8 54 172 158)(9 61 173 149)(10 52 174 156)(11 59 175 147)(12 50 176 154)(13 57 161 145)(14 64 162 152)(15 55 163 159)(16 62 164 150)(17 262 194 235)(18 269 195 226)(19 260 196 233)(20 267 197 240)(21 258 198 231)(22 265 199 238)(23 272 200 229)(24 263 201 236)(25 270 202 227)(26 261 203 234)(27 268 204 225)(28 259 205 232)(29 266 206 239)(30 257 207 230)(31 264 208 237)(32 271 193 228)(33 98 317 93)(34 105 318 84)(35 112 319 91)(36 103 320 82)(37 110 305 89)(38 101 306 96)(39 108 307 87)(40 99 308 94)(41 106 309 85)(42 97 310 92)(43 104 311 83)(44 111 312 90)(45 102 313 81)(46 109 314 88)(47 100 315 95)(48 107 316 86)(65 216 293 127)(66 223 294 118)(67 214 295 125)(68 221 296 116)(69 212 297 123)(70 219 298 114)(71 210 299 121)(72 217 300 128)(73 224 301 119)(74 215 302 126)(75 222 303 117)(76 213 304 124)(77 220 289 115)(78 211 290 122)(79 218 291 113)(80 209 292 120)(129 191 246 277)(130 182 247 284)(131 189 248 275)(132 180 249 282)(133 187 250 273)(134 178 251 280)(135 185 252 287)(136 192 253 278)(137 183 254 285)(138 190 255 276)(139 181 256 283)(140 188 241 274)(141 179 242 281)(142 186 243 288)(143 177 244 279)(144 184 245 286)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 16A ··· 16H 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 16 ··· 16 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 8 8 40 40 2 2 2 2 2 2 2 ··· 2 10 ··· 10 4 4 4 4 8 ··· 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + + - + + - - + - + image C1 C2 C2 C2 C4 Q8 D4 D5 Q16 D8 D10 SD32 Dic10 C4×D5 C5⋊D4 C5⋊Q16 D4⋊D5 D8.D5 C5⋊SD32 kernel C10.SD32 C2×C5⋊2C16 C40⋊5C4 C5×C2.D8 C5⋊2C16 C40 C2×C20 C2.D8 C20 C2×C10 C2×C8 C10 C8 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 2 2 8 4 4 4 2 2 4 4

Matrix representation of C10.SD32 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 0 189 0 0 51 190
,
 138 200 0 0 41 138 0 0 0 0 110 195 0 0 174 131
,
 213 120 0 0 120 28 0 0 0 0 69 145 0 0 145 172
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,51,0,0,189,190],[138,41,0,0,200,138,0,0,0,0,110,174,0,0,195,131],[213,120,0,0,120,28,0,0,0,0,69,145,0,0,145,172] >;

C10.SD32 in GAP, Magma, Sage, TeX

C_{10}.{\rm SD}_{32}
% in TeX

G:=Group("C10.SD32");
// GroupNames label

G:=SmallGroup(320,48);
// by ID

G=gap.SmallGroup(320,48);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,589,36,346,192,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^10=b^16=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^7>;
// generators/relations

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