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G = C20.10D8order 320 = 26·5

10th non-split extension by C20 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.10D8, C20.6Q16, C20.10SD16, C42.11D10, C4⋊Q8.2D5, C4.9(Q8⋊D5), C4⋊C4.2Dic5, C4.13(D4⋊D5), (C2×C20).108D4, C54(C4.10D8), C4.7(D4.D5), C203C8.13C2, C4.7(C5⋊Q16), (C4×C20).49C22, C2.5(Q8⋊Dic5), C2.5(D4⋊Dic5), C10.40(D4⋊C4), C10.20(Q8⋊C4), C2.4(C20.10D4), C10.14(C4.10D4), C22.43(C23.D5), (C5×C4⋊Q8).2C2, (C5×C4⋊C4).15C4, (C2×C20).346(C2×C4), (C2×C4).13(C2×Dic5), (C2×C4).178(C5⋊D4), (C2×C10).169(C22⋊C4), SmallGroup(320,105)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.10D8
C1C5C10C2×C10C2×C20C4×C20C203C8 — C20.10D8
C5C2×C10C2×C20 — C20.10D8
C1C22C42C4⋊Q8

Generators and relations for C20.10D8
 G = < a,b,c | a20=b8=1, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a15b-1 >

Subgroups: 174 in 64 conjugacy classes, 35 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], C4⋊C4, C2×C8 [×2], C2×Q8, C20 [×4], C20 [×3], C2×C10, C4⋊C8 [×2], C4⋊Q8, C52C8 [×2], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C4.10D8, C2×C52C8 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, Q8×C10, C203C8 [×2], C5×C4⋊Q8, C20.10D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16 [×2], Q16, Dic5 [×2], D10, C4.10D4, D4⋊C4, Q8⋊C4, C2×Dic5, C5⋊D4 [×2], C4.10D8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C23.D5, D4⋊Dic5, Q8⋊Dic5, C20.10D4, C20.10D8

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

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G5A5B8A···8H10A···10F20A···20L20M···20T
order12224444444558···810···1020···2020···20
size111122224882220···202···24···48···8

47 irreducible representations

dim111122222222444444
type++++++-+--+-+-
imageC1C2C2C4D4D5D8SD16Q16D10Dic5C5⋊D4C4.10D4D4⋊D5D4.D5Q8⋊D5C5⋊Q16C20.10D4
kernelC20.10D8C203C8C5×C4⋊Q8C5×C4⋊C4C2×C20C4⋊Q8C20C20C20C42C4⋊C4C2×C4C10C4C4C4C4C2
# reps121422242248122224

Matrix representation of C20.10D8 in GL6(𝔽41)

1690000
17250000
00344000
001000
0000400
0000040
,
11170000
29300000
00232500
00281800
0000317
00002540
,
40260000
40250000
006300
0023500
00003029
00001711

G:=sub<GL(6,GF(41))| [16,17,0,0,0,0,9,25,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[11,29,0,0,0,0,17,30,0,0,0,0,0,0,23,28,0,0,0,0,25,18,0,0,0,0,0,0,31,25,0,0,0,0,7,40],[40,40,0,0,0,0,26,25,0,0,0,0,0,0,6,2,0,0,0,0,3,35,0,0,0,0,0,0,30,17,0,0,0,0,29,11] >;

C20.10D8 in GAP, Magma, Sage, TeX

C_{20}._{10}D_8
% in TeX

G:=Group("C20.10D8");
// GroupNames label

G:=SmallGroup(320,105);
// by ID

G=gap.SmallGroup(320,105);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,100,1571,570,136,12550]);
// Polycyclic

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

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