metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.17D8, C20.8Q16, C42.220D10, C5⋊2C8⋊7Q8, C4⋊Q8.4D5, C5⋊3(C8⋊2Q8), C4.34(Q8×D5), C4⋊C4.79D10, C4.6(D4⋊D5), C10.59(C2×D8), C20.35(C2×Q8), (C2×C20).151D4, C10.30(C4⋊Q8), C10.40(C2×Q16), C4.3(C5⋊Q16), C20⋊2Q8.19C2, (C2×C20).396C23, (C4×C20).125C22, C10.D8.15C2, C4⋊Dic5.156C22, C2.10(Dic5⋊Q8), (C5×C4⋊Q8).4C2, C2.14(C2×D4⋊D5), (C4×C5⋊2C8).11C2, C2.11(C2×C5⋊Q16), (C2×C10).527(C2×D4), (C2×C4).133(C5⋊D4), (C5×C4⋊C4).126C22, (C2×C4).493(C22×D5), C22.199(C2×C5⋊D4), (C2×C5⋊2C8).267C22, SmallGroup(320,705)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.17D8
G = < a,b,c | a20=b8=1, c2=a10, bab-1=a9, cac-1=a-1, cbc-1=b-1 >
Subgroups: 318 in 98 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C4×C8, C2.D8, C4⋊Q8, C4⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C8⋊2Q8, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, Q8×C10, C4×C5⋊2C8, C10.D8, C20⋊2Q8, C5×C4⋊Q8, C20.17D8
Quotients: C1, C2, C22, D4, Q8, C23, D5, D8, Q16, C2×D4, C2×Q8, D10, C4⋊Q8, C2×D8, C2×Q16, C5⋊D4, C22×D5, C8⋊2Q8, D4⋊D5, C5⋊Q16, Q8×D5, C2×C5⋊D4, C2×D4⋊D5, C2×C5⋊Q16, Dic5⋊Q8, C20.17D8
►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 37 202 141 136 222 316 243)(2 26 203 150 137 231 317 252)(3 35 204 159 138 240 318 241)(4 24 205 148 139 229 319 250)(5 33 206 157 140 238 320 259)(6 22 207 146 121 227 301 248)(7 31 208 155 122 236 302 257)(8 40 209 144 123 225 303 246)(9 29 210 153 124 234 304 255)(10 38 211 142 125 223 305 244)(11 27 212 151 126 232 306 253)(12 36 213 160 127 221 307 242)(13 25 214 149 128 230 308 251)(14 34 215 158 129 239 309 260)(15 23 216 147 130 228 310 249)(16 32 217 156 131 237 311 258)(17 21 218 145 132 226 312 247)(18 30 219 154 133 235 313 256)(19 39 220 143 134 224 314 245)(20 28 201 152 135 233 315 254)(41 102 290 268 193 174 65 97)(42 111 291 277 194 163 66 86)(43 120 292 266 195 172 67 95)(44 109 293 275 196 161 68 84)(45 118 294 264 197 170 69 93)(46 107 295 273 198 179 70 82)(47 116 296 262 199 168 71 91)(48 105 297 271 200 177 72 100)(49 114 298 280 181 166 73 89)(50 103 299 269 182 175 74 98)(51 112 300 278 183 164 75 87)(52 101 281 267 184 173 76 96)(53 110 282 276 185 162 77 85)(54 119 283 265 186 171 78 94)(55 108 284 274 187 180 79 83)(56 117 285 263 188 169 80 92)(57 106 286 272 189 178 61 81)(58 115 287 261 190 167 62 90)(59 104 288 270 191 176 63 99)(60 113 289 279 192 165 64 88)
(1 276 11 266)(2 275 12 265)(3 274 13 264)(4 273 14 263)(5 272 15 262)(6 271 16 261)(7 270 17 280)(8 269 18 279)(9 268 19 278)(10 267 20 277)(21 298 31 288)(22 297 32 287)(23 296 33 286)(24 295 34 285)(25 294 35 284)(26 293 36 283)(27 292 37 282)(28 291 38 281)(29 290 39 300)(30 289 40 299)(41 143 51 153)(42 142 52 152)(43 141 53 151)(44 160 54 150)(45 159 55 149)(46 158 56 148)(47 157 57 147)(48 156 58 146)(49 155 59 145)(50 154 60 144)(61 228 71 238)(62 227 72 237)(63 226 73 236)(64 225 74 235)(65 224 75 234)(66 223 76 233)(67 222 77 232)(68 221 78 231)(69 240 79 230)(70 239 80 229)(81 130 91 140)(82 129 92 139)(83 128 93 138)(84 127 94 137)(85 126 95 136)(86 125 96 135)(87 124 97 134)(88 123 98 133)(89 122 99 132)(90 121 100 131)(101 201 111 211)(102 220 112 210)(103 219 113 209)(104 218 114 208)(105 217 115 207)(106 216 116 206)(107 215 117 205)(108 214 118 204)(109 213 119 203)(110 212 120 202)(161 307 171 317)(162 306 172 316)(163 305 173 315)(164 304 174 314)(165 303 175 313)(166 302 176 312)(167 301 177 311)(168 320 178 310)(169 319 179 309)(170 318 180 308)(181 257 191 247)(182 256 192 246)(183 255 193 245)(184 254 194 244)(185 253 195 243)(186 252 196 242)(187 251 197 241)(188 250 198 260)(189 249 199 259)(190 248 200 258)
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), (1,37,202,141,136,222,316,243)(2,26,203,150,137,231,317,252)(3,35,204,159,138,240,318,241)(4,24,205,148,139,229,319,250)(5,33,206,157,140,238,320,259)(6,22,207,146,121,227,301,248)(7,31,208,155,122,236,302,257)(8,40,209,144,123,225,303,246)(9,29,210,153,124,234,304,255)(10,38,211,142,125,223,305,244)(11,27,212,151,126,232,306,253)(12,36,213,160,127,221,307,242)(13,25,214,149,128,230,308,251)(14,34,215,158,129,239,309,260)(15,23,216,147,130,228,310,249)(16,32,217,156,131,237,311,258)(17,21,218,145,132,226,312,247)(18,30,219,154,133,235,313,256)(19,39,220,143,134,224,314,245)(20,28,201,152,135,233,315,254)(41,102,290,268,193,174,65,97)(42,111,291,277,194,163,66,86)(43,120,292,266,195,172,67,95)(44,109,293,275,196,161,68,84)(45,118,294,264,197,170,69,93)(46,107,295,273,198,179,70,82)(47,116,296,262,199,168,71,91)(48,105,297,271,200,177,72,100)(49,114,298,280,181,166,73,89)(50,103,299,269,182,175,74,98)(51,112,300,278,183,164,75,87)(52,101,281,267,184,173,76,96)(53,110,282,276,185,162,77,85)(54,119,283,265,186,171,78,94)(55,108,284,274,187,180,79,83)(56,117,285,263,188,169,80,92)(57,106,286,272,189,178,61,81)(58,115,287,261,190,167,62,90)(59,104,288,270,191,176,63,99)(60,113,289,279,192,165,64,88), (1,276,11,266)(2,275,12,265)(3,274,13,264)(4,273,14,263)(5,272,15,262)(6,271,16,261)(7,270,17,280)(8,269,18,279)(9,268,19,278)(10,267,20,277)(21,298,31,288)(22,297,32,287)(23,296,33,286)(24,295,34,285)(25,294,35,284)(26,293,36,283)(27,292,37,282)(28,291,38,281)(29,290,39,300)(30,289,40,299)(41,143,51,153)(42,142,52,152)(43,141,53,151)(44,160,54,150)(45,159,55,149)(46,158,56,148)(47,157,57,147)(48,156,58,146)(49,155,59,145)(50,154,60,144)(61,228,71,238)(62,227,72,237)(63,226,73,236)(64,225,74,235)(65,224,75,234)(66,223,76,233)(67,222,77,232)(68,221,78,231)(69,240,79,230)(70,239,80,229)(81,130,91,140)(82,129,92,139)(83,128,93,138)(84,127,94,137)(85,126,95,136)(86,125,96,135)(87,124,97,134)(88,123,98,133)(89,122,99,132)(90,121,100,131)(101,201,111,211)(102,220,112,210)(103,219,113,209)(104,218,114,208)(105,217,115,207)(106,216,116,206)(107,215,117,205)(108,214,118,204)(109,213,119,203)(110,212,120,202)(161,307,171,317)(162,306,172,316)(163,305,173,315)(164,304,174,314)(165,303,175,313)(166,302,176,312)(167,301,177,311)(168,320,178,310)(169,319,179,309)(170,318,180,308)(181,257,191,247)(182,256,192,246)(183,255,193,245)(184,254,194,244)(185,253,195,243)(186,252,196,242)(187,251,197,241)(188,250,198,260)(189,249,199,259)(190,248,200,258)>;
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), (1,37,202,141,136,222,316,243)(2,26,203,150,137,231,317,252)(3,35,204,159,138,240,318,241)(4,24,205,148,139,229,319,250)(5,33,206,157,140,238,320,259)(6,22,207,146,121,227,301,248)(7,31,208,155,122,236,302,257)(8,40,209,144,123,225,303,246)(9,29,210,153,124,234,304,255)(10,38,211,142,125,223,305,244)(11,27,212,151,126,232,306,253)(12,36,213,160,127,221,307,242)(13,25,214,149,128,230,308,251)(14,34,215,158,129,239,309,260)(15,23,216,147,130,228,310,249)(16,32,217,156,131,237,311,258)(17,21,218,145,132,226,312,247)(18,30,219,154,133,235,313,256)(19,39,220,143,134,224,314,245)(20,28,201,152,135,233,315,254)(41,102,290,268,193,174,65,97)(42,111,291,277,194,163,66,86)(43,120,292,266,195,172,67,95)(44,109,293,275,196,161,68,84)(45,118,294,264,197,170,69,93)(46,107,295,273,198,179,70,82)(47,116,296,262,199,168,71,91)(48,105,297,271,200,177,72,100)(49,114,298,280,181,166,73,89)(50,103,299,269,182,175,74,98)(51,112,300,278,183,164,75,87)(52,101,281,267,184,173,76,96)(53,110,282,276,185,162,77,85)(54,119,283,265,186,171,78,94)(55,108,284,274,187,180,79,83)(56,117,285,263,188,169,80,92)(57,106,286,272,189,178,61,81)(58,115,287,261,190,167,62,90)(59,104,288,270,191,176,63,99)(60,113,289,279,192,165,64,88), (1,276,11,266)(2,275,12,265)(3,274,13,264)(4,273,14,263)(5,272,15,262)(6,271,16,261)(7,270,17,280)(8,269,18,279)(9,268,19,278)(10,267,20,277)(21,298,31,288)(22,297,32,287)(23,296,33,286)(24,295,34,285)(25,294,35,284)(26,293,36,283)(27,292,37,282)(28,291,38,281)(29,290,39,300)(30,289,40,299)(41,143,51,153)(42,142,52,152)(43,141,53,151)(44,160,54,150)(45,159,55,149)(46,158,56,148)(47,157,57,147)(48,156,58,146)(49,155,59,145)(50,154,60,144)(61,228,71,238)(62,227,72,237)(63,226,73,236)(64,225,74,235)(65,224,75,234)(66,223,76,233)(67,222,77,232)(68,221,78,231)(69,240,79,230)(70,239,80,229)(81,130,91,140)(82,129,92,139)(83,128,93,138)(84,127,94,137)(85,126,95,136)(86,125,96,135)(87,124,97,134)(88,123,98,133)(89,122,99,132)(90,121,100,131)(101,201,111,211)(102,220,112,210)(103,219,113,209)(104,218,114,208)(105,217,115,207)(106,216,116,206)(107,215,117,205)(108,214,118,204)(109,213,119,203)(110,212,120,202)(161,307,171,317)(162,306,172,316)(163,305,173,315)(164,304,174,314)(165,303,175,313)(166,302,176,312)(167,301,177,311)(168,320,178,310)(169,319,179,309)(170,318,180,308)(181,257,191,247)(182,256,192,246)(183,255,193,245)(184,254,194,244)(185,253,195,243)(186,252,196,242)(187,251,197,241)(188,250,198,260)(189,249,199,259)(190,248,200,258) );
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)], [(1,37,202,141,136,222,316,243),(2,26,203,150,137,231,317,252),(3,35,204,159,138,240,318,241),(4,24,205,148,139,229,319,250),(5,33,206,157,140,238,320,259),(6,22,207,146,121,227,301,248),(7,31,208,155,122,236,302,257),(8,40,209,144,123,225,303,246),(9,29,210,153,124,234,304,255),(10,38,211,142,125,223,305,244),(11,27,212,151,126,232,306,253),(12,36,213,160,127,221,307,242),(13,25,214,149,128,230,308,251),(14,34,215,158,129,239,309,260),(15,23,216,147,130,228,310,249),(16,32,217,156,131,237,311,258),(17,21,218,145,132,226,312,247),(18,30,219,154,133,235,313,256),(19,39,220,143,134,224,314,245),(20,28,201,152,135,233,315,254),(41,102,290,268,193,174,65,97),(42,111,291,277,194,163,66,86),(43,120,292,266,195,172,67,95),(44,109,293,275,196,161,68,84),(45,118,294,264,197,170,69,93),(46,107,295,273,198,179,70,82),(47,116,296,262,199,168,71,91),(48,105,297,271,200,177,72,100),(49,114,298,280,181,166,73,89),(50,103,299,269,182,175,74,98),(51,112,300,278,183,164,75,87),(52,101,281,267,184,173,76,96),(53,110,282,276,185,162,77,85),(54,119,283,265,186,171,78,94),(55,108,284,274,187,180,79,83),(56,117,285,263,188,169,80,92),(57,106,286,272,189,178,61,81),(58,115,287,261,190,167,62,90),(59,104,288,270,191,176,63,99),(60,113,289,279,192,165,64,88)], [(1,276,11,266),(2,275,12,265),(3,274,13,264),(4,273,14,263),(5,272,15,262),(6,271,16,261),(7,270,17,280),(8,269,18,279),(9,268,19,278),(10,267,20,277),(21,298,31,288),(22,297,32,287),(23,296,33,286),(24,295,34,285),(25,294,35,284),(26,293,36,283),(27,292,37,282),(28,291,38,281),(29,290,39,300),(30,289,40,299),(41,143,51,153),(42,142,52,152),(43,141,53,151),(44,160,54,150),(45,159,55,149),(46,158,56,148),(47,157,57,147),(48,156,58,146),(49,155,59,145),(50,154,60,144),(61,228,71,238),(62,227,72,237),(63,226,73,236),(64,225,74,235),(65,224,75,234),(66,223,76,233),(67,222,77,232),(68,221,78,231),(69,240,79,230),(70,239,80,229),(81,130,91,140),(82,129,92,139),(83,128,93,138),(84,127,94,137),(85,126,95,136),(86,125,96,135),(87,124,97,134),(88,123,98,133),(89,122,99,132),(90,121,100,131),(101,201,111,211),(102,220,112,210),(103,219,113,209),(104,218,114,208),(105,217,115,207),(106,216,116,206),(107,215,117,205),(108,214,118,204),(109,213,119,203),(110,212,120,202),(161,307,171,317),(162,306,172,316),(163,305,173,315),(164,304,174,314),(165,303,175,313),(166,302,176,312),(167,301,177,311),(168,320,178,310),(169,319,179,309),(170,318,180,308),(181,257,191,247),(182,256,192,246),(183,255,193,245),(184,254,194,244),(185,253,195,243),(186,252,196,242),(187,251,197,241),(188,250,198,260),(189,249,199,259),(190,248,200,258)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 40 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | D8 | Q16 | D10 | D10 | C5⋊D4 | D4⋊D5 | C5⋊Q16 | Q8×D5 |
| kernel | C20.17D8 | C4×C5⋊2C8 | C10.D8 | C20⋊2Q8 | C5×C4⋊Q8 | C5⋊2C8 | C2×C20 | C4⋊Q8 | C20 | C20 | C42 | C4⋊C4 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 4 |
Matrix representation of C20.17D8 ►in GL6(𝔽41)
| 37 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 30 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 29 |
| 0 | 0 | 0 | 0 | 12 | 29 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 23 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 32 | 0 |
G:=sub<GL(6,GF(41))| [37,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,24,15,0,0,0,0,30,0,0,0,0,0,0,0,29,12,0,0,0,0,29,29],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,23,40,0,0,0,0,0,0,0,32,0,0,0,0,32,0] >;
C20.17D8 in GAP, Magma, Sage, TeX
C_{20}._{17}D_8 % in TeX
G:=Group("C20.17D8"); // GroupNames label
G:=SmallGroup(320,705);
// by ID
G=gap.SmallGroup(320,705);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,64,422,135,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations