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G = C409Q8order 320 = 26·5

2nd semidirect product of C40 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C409Q8, C88Dic10, C20.24SD16, C42.252D10, C51(C83Q8), (C4×C8).13D5, (C4×C40).15C2, (C2×C4).76D20, C10.2(C4⋊Q8), C20.69(C2×Q8), C406C4.4C2, (C2×C8).315D10, (C2×C20).373D4, C4.3(C40⋊C2), C202Q8.2C2, C10.1(C2×SD16), C2.6(C202Q8), C4.35(C2×Dic10), C22.85(C2×D20), C4⋊Dic5.1C22, (C4×C20).302C22, (C2×C40).387C22, (C2×C20).716C23, C2.5(C2×C40⋊C2), (C2×C10).99(C2×D4), (C2×C4).659(C22×D5), SmallGroup(320,307)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C409Q8
C1C5C10C20C2×C20C4⋊Dic5C202Q8 — C409Q8
C5C10C2×C20 — C409Q8
C1C22C42C4×C8

Generators and relations for C409Q8
 G = < a,b,c | a40=b4=1, c2=b2, ab=ba, cac-1=a19, cbc-1=b-1 >

Subgroups: 398 in 98 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2 [×2], C4 [×6], C4 [×4], C22, C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10, C10 [×2], C42, C4⋊C4 [×6], C2×C8 [×2], C2×Q8 [×2], Dic5 [×4], C20 [×6], C2×C10, C4×C8, C4.Q8 [×4], C4⋊Q8 [×2], C40 [×4], Dic10 [×4], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C83Q8, C4⋊Dic5 [×4], C4⋊Dic5 [×2], C4×C20, C2×C40 [×2], C2×Dic10 [×2], C406C4 [×4], C4×C40, C202Q8 [×2], C409Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, SD16 [×4], C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, C2×SD16 [×2], Dic10 [×4], D20 [×2], C22×D5, C83Q8, C40⋊C2 [×4], C2×Dic10 [×2], C2×D20, C202Q8, C2×C40⋊C2 [×2], C409Q8

Smallest permutation representation of C409Q8
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)
(1 164 314 144)(2 165 315 145)(3 166 316 146)(4 167 317 147)(5 168 318 148)(6 169 319 149)(7 170 320 150)(8 171 281 151)(9 172 282 152)(10 173 283 153)(11 174 284 154)(12 175 285 155)(13 176 286 156)(14 177 287 157)(15 178 288 158)(16 179 289 159)(17 180 290 160)(18 181 291 121)(19 182 292 122)(20 183 293 123)(21 184 294 124)(22 185 295 125)(23 186 296 126)(24 187 297 127)(25 188 298 128)(26 189 299 129)(27 190 300 130)(28 191 301 131)(29 192 302 132)(30 193 303 133)(31 194 304 134)(32 195 305 135)(33 196 306 136)(34 197 307 137)(35 198 308 138)(36 199 309 139)(37 200 310 140)(38 161 311 141)(39 162 312 142)(40 163 313 143)(41 91 210 258)(42 92 211 259)(43 93 212 260)(44 94 213 261)(45 95 214 262)(46 96 215 263)(47 97 216 264)(48 98 217 265)(49 99 218 266)(50 100 219 267)(51 101 220 268)(52 102 221 269)(53 103 222 270)(54 104 223 271)(55 105 224 272)(56 106 225 273)(57 107 226 274)(58 108 227 275)(59 109 228 276)(60 110 229 277)(61 111 230 278)(62 112 231 279)(63 113 232 280)(64 114 233 241)(65 115 234 242)(66 116 235 243)(67 117 236 244)(68 118 237 245)(69 119 238 246)(70 120 239 247)(71 81 240 248)(72 82 201 249)(73 83 202 250)(74 84 203 251)(75 85 204 252)(76 86 205 253)(77 87 206 254)(78 88 207 255)(79 89 208 256)(80 90 209 257)
(1 235 314 66)(2 214 315 45)(3 233 316 64)(4 212 317 43)(5 231 318 62)(6 210 319 41)(7 229 320 60)(8 208 281 79)(9 227 282 58)(10 206 283 77)(11 225 284 56)(12 204 285 75)(13 223 286 54)(14 202 287 73)(15 221 288 52)(16 240 289 71)(17 219 290 50)(18 238 291 69)(19 217 292 48)(20 236 293 67)(21 215 294 46)(22 234 295 65)(23 213 296 44)(24 232 297 63)(25 211 298 42)(26 230 299 61)(27 209 300 80)(28 228 301 59)(29 207 302 78)(30 226 303 57)(31 205 304 76)(32 224 305 55)(33 203 306 74)(34 222 307 53)(35 201 308 72)(36 220 309 51)(37 239 310 70)(38 218 311 49)(39 237 312 68)(40 216 313 47)(81 159 248 179)(82 138 249 198)(83 157 250 177)(84 136 251 196)(85 155 252 175)(86 134 253 194)(87 153 254 173)(88 132 255 192)(89 151 256 171)(90 130 257 190)(91 149 258 169)(92 128 259 188)(93 147 260 167)(94 126 261 186)(95 145 262 165)(96 124 263 184)(97 143 264 163)(98 122 265 182)(99 141 266 161)(100 160 267 180)(101 139 268 199)(102 158 269 178)(103 137 270 197)(104 156 271 176)(105 135 272 195)(106 154 273 174)(107 133 274 193)(108 152 275 172)(109 131 276 191)(110 150 277 170)(111 129 278 189)(112 148 279 168)(113 127 280 187)(114 146 241 166)(115 125 242 185)(116 144 243 164)(117 123 244 183)(118 142 245 162)(119 121 246 181)(120 140 247 200)

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

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

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

86 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J5A5B8A···8H10A···10F20A···20X40A···40AF
order12224···44444558···810···1020···2040···40
size11112···240404040222···22···22···22···2

86 irreducible representations

dim1111222222222
type++++-++++-+
imageC1C2C2C2Q8D4D5SD16D10D10Dic10D20C40⋊C2
kernelC409Q8C406C4C4×C40C202Q8C40C2×C20C4×C8C20C42C2×C8C8C2×C4C4
# reps141242282416832

Matrix representation of C409Q8 in GL4(𝔽41) generated by

261500
262600
001614
002928
,
04000
1000
003932
00372
,
382100
21300
00213
00320
G:=sub<GL(4,GF(41))| [26,26,0,0,15,26,0,0,0,0,16,29,0,0,14,28],[0,1,0,0,40,0,0,0,0,0,39,37,0,0,32,2],[38,21,0,0,21,3,0,0,0,0,21,3,0,0,3,20] >;

C409Q8 in GAP, Magma, Sage, TeX

C_{40}\rtimes_9Q_8
% in TeX

G:=Group("C40:9Q8");
// GroupNames label

G:=SmallGroup(320,307);
// by ID

G=gap.SmallGroup(320,307);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^19,c*b*c^-1=b^-1>;
// generators/relations

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