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G = C20.2D8order 320 = 26·5

2nd non-split extension by C20 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.2D8, C4.10D40, C20.22Q16, C20.1SD16, C42.6D10, C4⋊C8.6D5, C4⋊Dic5.3C4, (C2×C20).466D4, (C2×C4).124D20, C4.7(C40⋊C2), C53(C4.10D8), C203C8.10C2, C4.10(D4.D5), (C4×C20).44C22, C202Q8.10C2, C4.10(C5⋊Q16), C2.5(D205C4), C2.5(C10.Q16), C10.28(D4⋊C4), C10.10(Q8⋊C4), C2.5(C4.12D20), C10.9(C4.10D4), C22.63(D10⋊C4), (C5×C4⋊C8).6C2, (C2×C4).17(C4×D5), (C2×C20).202(C2×C4), (C2×C4).230(C5⋊D4), (C2×C10).112(C22⋊C4), SmallGroup(320,44)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.2D8
C1C5C10C2×C10C2×C20C4×C20C202Q8 — C20.2D8
C5C2×C10C2×C20 — C20.2D8
C1C22C42C4⋊C8

Generators and relations for C20.2D8
 G = < a,b,c | a20=b8=1, c2=a15, bab-1=a11, cac-1=a9, cbc-1=a5b-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 [×3], C2×C8 [×2], C2×Q8, Dic5 [×2], C20 [×4], C20, C2×C10, C4⋊C8, C4⋊C8, C4⋊Q8, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×C20 [×3], C4.10D8, C2×C52C8, C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C203C8, C5×C4⋊C8, C202Q8, C20.2D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16 [×2], Q16, D10, C4.10D4, D4⋊C4, Q8⋊C4, C4×D5, D20, C5⋊D4, C4.10D8, C40⋊C2, D40, D10⋊C4, D4.D5, C5⋊Q16, C10.Q16, D205C4, C4.12D20, C20.2D8

Smallest permutation representation of C20.2D8
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)
(1 135 160 101 197 80 96 50)(2 126 141 112 198 71 97 41)(3 137 142 103 199 62 98 52)(4 128 143 114 200 73 99 43)(5 139 144 105 181 64 100 54)(6 130 145 116 182 75 81 45)(7 121 146 107 183 66 82 56)(8 132 147 118 184 77 83 47)(9 123 148 109 185 68 84 58)(10 134 149 120 186 79 85 49)(11 125 150 111 187 70 86 60)(12 136 151 102 188 61 87 51)(13 127 152 113 189 72 88 42)(14 138 153 104 190 63 89 53)(15 129 154 115 191 74 90 44)(16 140 155 106 192 65 91 55)(17 131 156 117 193 76 92 46)(18 122 157 108 194 67 93 57)(19 133 158 119 195 78 94 48)(20 124 159 110 196 69 95 59)(21 166 314 259 287 219 227 277)(22 177 315 250 288 210 228 268)(23 168 316 241 289 201 229 279)(24 179 317 252 290 212 230 270)(25 170 318 243 291 203 231 261)(26 161 319 254 292 214 232 272)(27 172 320 245 293 205 233 263)(28 163 301 256 294 216 234 274)(29 174 302 247 295 207 235 265)(30 165 303 258 296 218 236 276)(31 176 304 249 297 209 237 267)(32 167 305 260 298 220 238 278)(33 178 306 251 299 211 239 269)(34 169 307 242 300 202 240 280)(35 180 308 253 281 213 221 271)(36 171 309 244 282 204 222 262)(37 162 310 255 283 215 223 273)(38 173 311 246 284 206 224 264)(39 164 312 257 285 217 225 275)(40 175 313 248 286 208 226 266)
(1 288 16 283 11 298 6 293)(2 297 17 292 12 287 7 282)(3 286 18 281 13 296 8 291)(4 295 19 290 14 285 9 300)(5 284 20 299 15 294 10 289)(21 183 36 198 31 193 26 188)(22 192 37 187 32 182 27 197)(23 181 38 196 33 191 28 186)(24 190 39 185 34 200 29 195)(25 199 40 194 35 189 30 184)(41 214 56 209 51 204 46 219)(42 203 57 218 52 213 47 208)(43 212 58 207 53 202 48 217)(44 201 59 216 54 211 49 206)(45 210 60 205 55 220 50 215)(61 262 76 277 71 272 66 267)(62 271 77 266 72 261 67 276)(63 280 78 275 73 270 68 265)(64 269 79 264 74 279 69 274)(65 278 80 273 75 268 70 263)(81 233 96 228 91 223 86 238)(82 222 97 237 92 232 87 227)(83 231 98 226 93 221 88 236)(84 240 99 235 94 230 89 225)(85 229 100 224 95 239 90 234)(101 162 116 177 111 172 106 167)(102 171 117 166 112 161 107 176)(103 180 118 175 113 170 108 165)(104 169 119 164 114 179 109 174)(105 178 120 173 115 168 110 163)(121 249 136 244 131 259 126 254)(122 258 137 253 132 248 127 243)(123 247 138 242 133 257 128 252)(124 256 139 251 134 246 129 241)(125 245 140 260 135 255 130 250)(141 304 156 319 151 314 146 309)(142 313 157 308 152 303 147 318)(143 302 158 317 153 312 148 307)(144 311 159 306 154 301 149 316)(145 320 160 315 155 310 150 305)

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

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

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

59 conjugacy classes

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

59 irreducible representations

dim11111222222222224444
type+++++++-+++----
imageC1C2C2C2C4D4D5D8SD16Q16D10C4×D5D20C5⋊D4C40⋊C2D40C4.10D4D4.D5C5⋊Q16C4.12D20
kernelC20.2D8C203C8C5×C4⋊C8C202Q8C4⋊Dic5C2×C20C4⋊C8C20C20C20C42C2×C4C2×C4C2×C4C4C4C10C4C4C2
# reps11114222422444881224

Matrix representation of C20.2D8 in GL4(𝔽41) generated by

35100
40000
00139
00140
,
43100
102600
002538
00316
,
133200
282800
001130
00260
G:=sub<GL(4,GF(41))| [35,40,0,0,1,0,0,0,0,0,1,1,0,0,39,40],[4,10,0,0,31,26,0,0,0,0,25,3,0,0,38,16],[13,28,0,0,32,28,0,0,0,0,11,26,0,0,30,0] >;

C20.2D8 in GAP, Magma, Sage, TeX

C_{20}._2D_8
% in TeX

G:=Group("C20.2D8");
// GroupNames label

G:=SmallGroup(320,44);
// by ID

G=gap.SmallGroup(320,44);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,141,36,1094,268,1123,346,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^15,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=a^5*b^-1>;
// generators/relations

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