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## G = C10.Q32order 320 = 26·5

### 2nd non-split extension by C10 of Q32 acting via Q32/Q16=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C10.Q32
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C2×Dic20 — C10.Q32
 Lower central C5 — C10 — C20 — C40 — C10.Q32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2.D8

Generators and relations for C10.Q32
G = < a,b,c | a10=b16=1, c2=a5b8, bab-1=a-1, ac=ca, cbc-1=a5b-1 >

Subgroups: 238 in 58 conjugacy classes, 29 normal (27 characteristic)
C1, C2 [×3], C4 [×2], C4 [×3], C22, C5, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], C16, C4⋊C4, C2×C8, Q16 [×3], C2×Q8, Dic5 [×2], C20 [×2], C20, C2×C10, C2.D8, C2×C16, C2×Q16, C40 [×2], Dic10 [×3], C2×Dic5, C2×C20, C2×C20, C2.Q32, C52C16, Dic20 [×2], Dic20, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C52C16, C5×C2.D8, C2×Dic20, C10.Q32
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16, D10, D4⋊C4, SD32, Q32, C4×D5, D20, C5⋊D4, C2.Q32, D10⋊C4, D4⋊D5, Q8⋊D5, D206C4, D8.D5, C5⋊Q32, C10.Q32

Smallest permutation representation of C10.Q32
Regular action on 320 points
Generators in S320
```(1 177 100 125 197 18 55 36 168 140)(2 141 169 37 56 19 198 126 101 178)(3 179 102 127 199 20 57 38 170 142)(4 143 171 39 58 21 200 128 103 180)(5 181 104 113 201 22 59 40 172 144)(6 129 173 41 60 23 202 114 105 182)(7 183 106 115 203 24 61 42 174 130)(8 131 175 43 62 25 204 116 107 184)(9 185 108 117 205 26 63 44 176 132)(10 133 161 45 64 27 206 118 109 186)(11 187 110 119 207 28 49 46 162 134)(12 135 163 47 50 29 208 120 111 188)(13 189 112 121 193 30 51 48 164 136)(14 137 165 33 52 31 194 122 97 190)(15 191 98 123 195 32 53 34 166 138)(16 139 167 35 54 17 196 124 99 192)(65 83 148 246 222 314 285 257 229 304)(66 289 230 258 286 315 223 247 149 84)(67 85 150 248 224 316 287 259 231 290)(68 291 232 260 288 317 209 249 151 86)(69 87 152 250 210 318 273 261 233 292)(70 293 234 262 274 319 211 251 153 88)(71 89 154 252 212 320 275 263 235 294)(72 295 236 264 276 305 213 253 155 90)(73 91 156 254 214 306 277 265 237 296)(74 297 238 266 278 307 215 255 157 92)(75 93 158 256 216 308 279 267 239 298)(76 299 240 268 280 309 217 241 159 94)(77 95 160 242 218 310 281 269 225 300)(78 301 226 270 282 311 219 243 145 96)(79 81 146 244 220 312 283 271 227 302)(80 303 228 272 284 313 221 245 147 82)
(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)
(1 296 26 222)(2 213 27 303)(3 294 28 220)(4 211 29 301)(5 292 30 218)(6 209 31 299)(7 290 32 216)(8 223 17 297)(9 304 18 214)(10 221 19 295)(11 302 20 212)(12 219 21 293)(13 300 22 210)(14 217 23 291)(15 298 24 224)(16 215 25 289)(33 94 105 288)(34 279 106 85)(35 92 107 286)(36 277 108 83)(37 90 109 284)(38 275 110 81)(39 88 111 282)(40 273 112 95)(41 86 97 280)(42 287 98 93)(43 84 99 278)(44 285 100 91)(45 82 101 276)(46 283 102 89)(47 96 103 274)(48 281 104 87)(49 312 179 71)(50 78 180 319)(51 310 181 69)(52 76 182 317)(53 308 183 67)(54 74 184 315)(55 306 185 65)(56 72 186 313)(57 320 187 79)(58 70 188 311)(59 318 189 77)(60 68 190 309)(61 316 191 75)(62 66 192 307)(63 314 177 73)(64 80 178 305)(113 152 164 269)(114 260 165 159)(115 150 166 267)(116 258 167 157)(117 148 168 265)(118 272 169 155)(119 146 170 263)(120 270 171 153)(121 160 172 261)(122 268 173 151)(123 158 174 259)(124 266 175 149)(125 156 176 257)(126 264 161 147)(127 154 162 271)(128 262 163 145)(129 249 194 240)(130 231 195 256)(131 247 196 238)(132 229 197 254)(133 245 198 236)(134 227 199 252)(135 243 200 234)(136 225 201 250)(137 241 202 232)(138 239 203 248)(139 255 204 230)(140 237 205 246)(141 253 206 228)(142 235 207 244)(143 251 208 226)(144 233 193 242)```

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

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

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

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 2 4 4 4 4 type + + + + + + + + + - + + + - - image C1 C2 C2 C2 C4 D4 D4 D5 SD16 D8 D10 SD32 Q32 C4×D5 D20 C5⋊D4 Q8⋊D5 D4⋊D5 D8.D5 C5⋊Q32 kernel C10.Q32 C2×C5⋊2C16 C5×C2.D8 C2×Dic20 Dic20 C40 C2×C20 C2.D8 C20 C2×C10 C2×C8 C10 C10 C8 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 4 4 2 2 4 4

Matrix representation of C10.Q32 in GL4(𝔽241) generated by

 51 51 0 0 190 1 0 0 0 0 1 0 0 0 0 1
,
 166 60 0 0 91 75 0 0 0 0 27 85 0 0 156 27
,
 64 0 0 0 0 64 0 0 0 0 145 69 0 0 69 96
`G:=sub<GL(4,GF(241))| [51,190,0,0,51,1,0,0,0,0,1,0,0,0,0,1],[166,91,0,0,60,75,0,0,0,0,27,156,0,0,85,27],[64,0,0,0,0,64,0,0,0,0,145,69,0,0,69,96] >;`

C10.Q32 in GAP, Magma, Sage, TeX

`C_{10}.Q_{32}`
`% in TeX`

`G:=Group("C10.Q32");`
`// GroupNames label`

`G:=SmallGroup(320,50);`
`// by ID`

`G=gap.SmallGroup(320,50);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,675,346,192,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=b^16=1,c^2=a^5*b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1>;`
`// generators/relations`

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