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G = C4.Dic20order 320 = 26·5

1st non-split extension by C4 of Dic20 acting via Dic20/Dic10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.1Q16, C4.5Dic20, C42.3D10, C20.35SD16, C4⋊C8.3D5, C203C8.8C2, (C2×C4).121D20, (C2×C20).463D4, C4.9(C40⋊C2), C4.9(D4.D5), C202Q8.8C2, C4.9(C5⋊Q16), C52(C4.6Q16), (C4×C20).39C22, (C2×Dic10).2C4, C2.4(C10.Q16), C10.14(Q8⋊C4), C10.10(C4.D4), C2.4(C20.46D4), C2.4(C20.44D4), C22.60(D10⋊C4), (C5×C4⋊C8).3C2, (C2×C4).14(C4×D5), (C2×C20).199(C2×C4), (C2×C4).227(C5⋊D4), (C2×C10).107(C22⋊C4), SmallGroup(320,39)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4.Dic20
C1C5C10C2×C10C2×C20C4×C20C203C8 — C4.Dic20
C5C2×C10C2×C20 — C4.Dic20
C1C22C42C4⋊C8

Generators and relations for C4.Dic20
 G = < a,b,c | a4=b40=1, c2=ab20, bab-1=a-1, ac=ca, cbc-1=a-1b-1 >

Subgroups: 254 in 64 conjugacy classes, 33 normal (31 characteristic)
C1, C2 [×3], C4 [×4], C4 [×3], C22, C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], Q8 [×2], C10 [×3], C42, C4⋊C4 [×2], C2×C8 [×2], C2×Q8 [×2], Dic5 [×2], C20 [×4], C20, C2×C10, C4⋊C8, C4⋊C8, C4⋊Q8, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×C20 [×3], C4.6Q16, C2×C52C8, C4⋊Dic5 [×2], C4×C20, C2×C40, C2×Dic10 [×2], C203C8, C5×C4⋊C8, C202Q8, C4.Dic20
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, SD16 [×2], Q16 [×2], D10, C4.D4, Q8⋊C4 [×2], C4×D5, D20, C5⋊D4, C4.6Q16, C40⋊C2, Dic20, D10⋊C4, D4.D5, C5⋊Q16, C10.Q16, C20.44D4, C20.46D4, C4.Dic20

Smallest permutation representation of C4.Dic20
Regular action on 320 points
Generators in S320
(1 109 164 287)(2 288 165 110)(3 111 166 289)(4 290 167 112)(5 113 168 291)(6 292 169 114)(7 115 170 293)(8 294 171 116)(9 117 172 295)(10 296 173 118)(11 119 174 297)(12 298 175 120)(13 81 176 299)(14 300 177 82)(15 83 178 301)(16 302 179 84)(17 85 180 303)(18 304 181 86)(19 87 182 305)(20 306 183 88)(21 89 184 307)(22 308 185 90)(23 91 186 309)(24 310 187 92)(25 93 188 311)(26 312 189 94)(27 95 190 313)(28 314 191 96)(29 97 192 315)(30 316 193 98)(31 99 194 317)(32 318 195 100)(33 101 196 319)(34 320 197 102)(35 103 198 281)(36 282 199 104)(37 105 200 283)(38 284 161 106)(39 107 162 285)(40 286 163 108)(41 262 202 150)(42 151 203 263)(43 264 204 152)(44 153 205 265)(45 266 206 154)(46 155 207 267)(47 268 208 156)(48 157 209 269)(49 270 210 158)(50 159 211 271)(51 272 212 160)(52 121 213 273)(53 274 214 122)(54 123 215 275)(55 276 216 124)(56 125 217 277)(57 278 218 126)(58 127 219 279)(59 280 220 128)(60 129 221 241)(61 242 222 130)(62 131 223 243)(63 244 224 132)(64 133 225 245)(65 246 226 134)(66 135 227 247)(67 248 228 136)(68 137 229 249)(69 250 230 138)(70 139 231 251)(71 252 232 140)(72 141 233 253)(73 254 234 142)(74 143 235 255)(75 256 236 144)(76 145 237 257)(77 258 238 146)(78 147 239 259)(79 260 240 148)(80 149 201 261)
(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)
(1 266 89 226 164 154 307 65)(2 205 308 245 165 44 90 133)(3 264 91 224 166 152 309 63)(4 203 310 243 167 42 92 131)(5 262 93 222 168 150 311 61)(6 201 312 241 169 80 94 129)(7 260 95 220 170 148 313 59)(8 239 314 279 171 78 96 127)(9 258 97 218 172 146 315 57)(10 237 316 277 173 76 98 125)(11 256 99 216 174 144 317 55)(12 235 318 275 175 74 100 123)(13 254 101 214 176 142 319 53)(14 233 320 273 177 72 102 121)(15 252 103 212 178 140 281 51)(16 231 282 271 179 70 104 159)(17 250 105 210 180 138 283 49)(18 229 284 269 181 68 106 157)(19 248 107 208 182 136 285 47)(20 227 286 267 183 66 108 155)(21 246 109 206 184 134 287 45)(22 225 288 265 185 64 110 153)(23 244 111 204 186 132 289 43)(24 223 290 263 187 62 112 151)(25 242 113 202 188 130 291 41)(26 221 292 261 189 60 114 149)(27 280 115 240 190 128 293 79)(28 219 294 259 191 58 116 147)(29 278 117 238 192 126 295 77)(30 217 296 257 193 56 118 145)(31 276 119 236 194 124 297 75)(32 215 298 255 195 54 120 143)(33 274 81 234 196 122 299 73)(34 213 300 253 197 52 82 141)(35 272 83 232 198 160 301 71)(36 211 302 251 199 50 84 139)(37 270 85 230 200 158 303 69)(38 209 304 249 161 48 86 137)(39 268 87 228 162 156 305 67)(40 207 306 247 163 46 88 135)

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

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

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

59 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H10A···10F20A···20H20I···20P40A···40P
order12224444444558888888810···1020···2020···2040···40
size1111222244040224444202020202···22···24···44···4

59 irreducible representations

dim1111122222222224444
type++++++-++-+--+
imageC1C2C2C2C4D4D5SD16Q16D10C4×D5D20C5⋊D4C40⋊C2Dic20C4.D4D4.D5C5⋊Q16C20.46D4
kernelC4.Dic20C203C8C5×C4⋊C8C202Q8C2×Dic10C2×C20C4⋊C8C20C20C42C2×C4C2×C4C2×C4C4C4C10C4C4C2
# reps1111422442444881224

Matrix representation of C4.Dic20 in GL4(𝔽41) generated by

1000
0100
004039
0011
,
62900
142000
001222
004029
,
282600
251300
00024
002917
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,1,0,0,39,1],[6,14,0,0,29,20,0,0,0,0,12,40,0,0,22,29],[28,25,0,0,26,13,0,0,0,0,0,29,0,0,24,17] >;

C4.Dic20 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_{20}
% in TeX

G:=Group("C4.Dic20");
// GroupNames label

G:=SmallGroup(320,39);
// by ID

G=gap.SmallGroup(320,39);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,85,92,422,387,268,570,136,12550]);
// Polycyclic

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

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