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

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

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

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

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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