Copied to
clipboard

## G = C40.13Q8order 320 = 26·5

### 3rd non-split extension by C40 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C40.13Q8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C20.6Q8 — C40.13Q8
 Lower central C5 — C10 — C2×C20 — C40.13Q8
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C40.13Q8
G = < a,b,c | a40=b4=1, c2=a20b2, ab=ba, cac-1=a-1, cbc-1=a20b-1 >

Subgroups: 302 in 86 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, C10, C10, C42, C4⋊C4, C2×C8, Dic5, C20, C20, C2×C10, C4×C8, C4.Q8, C2.D8, C42.C2, C40, C2×Dic5, C2×C20, C2×C20, C8.5Q8, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C406C4, C405C4, C4×C40, C20.6Q8, C40.13Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, C4○D8, Dic10, D20, C22×D5, C8.5Q8, C2×Dic10, C2×D20, C202Q8, D407C2, C40.13Q8

Smallest permutation representation of C40.13Q8
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)
(1 119 138 190)(2 120 139 191)(3 81 140 192)(4 82 141 193)(5 83 142 194)(6 84 143 195)(7 85 144 196)(8 86 145 197)(9 87 146 198)(10 88 147 199)(11 89 148 200)(12 90 149 161)(13 91 150 162)(14 92 151 163)(15 93 152 164)(16 94 153 165)(17 95 154 166)(18 96 155 167)(19 97 156 168)(20 98 157 169)(21 99 158 170)(22 100 159 171)(23 101 160 172)(24 102 121 173)(25 103 122 174)(26 104 123 175)(27 105 124 176)(28 106 125 177)(29 107 126 178)(30 108 127 179)(31 109 128 180)(32 110 129 181)(33 111 130 182)(34 112 131 183)(35 113 132 184)(36 114 133 185)(37 115 134 186)(38 116 135 187)(39 117 136 188)(40 118 137 189)(41 287 202 245)(42 288 203 246)(43 289 204 247)(44 290 205 248)(45 291 206 249)(46 292 207 250)(47 293 208 251)(48 294 209 252)(49 295 210 253)(50 296 211 254)(51 297 212 255)(52 298 213 256)(53 299 214 257)(54 300 215 258)(55 301 216 259)(56 302 217 260)(57 303 218 261)(58 304 219 262)(59 305 220 263)(60 306 221 264)(61 307 222 265)(62 308 223 266)(63 309 224 267)(64 310 225 268)(65 311 226 269)(66 312 227 270)(67 313 228 271)(68 314 229 272)(69 315 230 273)(70 316 231 274)(71 317 232 275)(72 318 233 276)(73 319 234 277)(74 320 235 278)(75 281 236 279)(76 282 237 280)(77 283 238 241)(78 284 239 242)(79 285 240 243)(80 286 201 244)
(1 284 158 262)(2 283 159 261)(3 282 160 260)(4 281 121 259)(5 320 122 258)(6 319 123 257)(7 318 124 256)(8 317 125 255)(9 316 126 254)(10 315 127 253)(11 314 128 252)(12 313 129 251)(13 312 130 250)(14 311 131 249)(15 310 132 248)(16 309 133 247)(17 308 134 246)(18 307 135 245)(19 306 136 244)(20 305 137 243)(21 304 138 242)(22 303 139 241)(23 302 140 280)(24 301 141 279)(25 300 142 278)(26 299 143 277)(27 298 144 276)(28 297 145 275)(29 296 146 274)(30 295 147 273)(31 294 148 272)(32 293 149 271)(33 292 150 270)(34 291 151 269)(35 290 152 268)(36 289 153 267)(37 288 154 266)(38 287 155 265)(39 286 156 264)(40 285 157 263)(41 187 222 96)(42 186 223 95)(43 185 224 94)(44 184 225 93)(45 183 226 92)(46 182 227 91)(47 181 228 90)(48 180 229 89)(49 179 230 88)(50 178 231 87)(51 177 232 86)(52 176 233 85)(53 175 234 84)(54 174 235 83)(55 173 236 82)(56 172 237 81)(57 171 238 120)(58 170 239 119)(59 169 240 118)(60 168 201 117)(61 167 202 116)(62 166 203 115)(63 165 204 114)(64 164 205 113)(65 163 206 112)(66 162 207 111)(67 161 208 110)(68 200 209 109)(69 199 210 108)(70 198 211 107)(71 197 212 106)(72 196 213 105)(73 195 214 104)(74 194 215 103)(75 193 216 102)(76 192 217 101)(77 191 218 100)(78 190 219 99)(79 189 220 98)(80 188 221 97)```

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

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

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

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 1 2 2 2 2 2 2 2 2 2 type + + + + + - + + + + - + image C1 C2 C2 C2 C2 Q8 D4 D5 D10 D10 C4○D8 Dic10 D20 D40⋊7C2 kernel C40.13Q8 C40⋊6C4 C40⋊5C4 C4×C40 C20.6Q8 C40 C2×C20 C4×C8 C42 C2×C8 C10 C8 C2×C4 C2 # reps 1 2 2 1 2 4 2 2 2 4 8 16 8 32

Matrix representation of C40.13Q8 in GL4(𝔽41) generated by

 40 6 0 0 35 35 0 0 0 0 12 18 0 0 23 38
,
 2 13 0 0 28 39 0 0 0 0 23 6 0 0 35 18
,
 26 39 0 0 31 15 0 0 0 0 9 28 0 0 0 32
`G:=sub<GL(4,GF(41))| [40,35,0,0,6,35,0,0,0,0,12,23,0,0,18,38],[2,28,0,0,13,39,0,0,0,0,23,35,0,0,6,18],[26,31,0,0,39,15,0,0,0,0,9,0,0,0,28,32] >;`

C40.13Q8 in GAP, Magma, Sage, TeX

`C_{40}._{13}Q_8`
`% in TeX`

`G:=Group("C40.13Q8");`
`// GroupNames label`

`G:=SmallGroup(320,310);`
`// by ID`

`G=gap.SmallGroup(320,310);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,58,1123,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^4=1,c^2=a^20*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^20*b^-1>;`
`// generators/relations`

׿
×
𝔽