metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.9Q16, C20.17SD16, C42.222D10, C4⋊Q8.7D5, C4.4(Q8⋊D5), (C2×C20).153D4, (C2×Q8).42D10, C10.41(C2×Q16), C5⋊3(C4.SD16), C4.4(C5⋊Q16), C20.78(C4○D4), C20⋊2Q8.21C2, C10.74(C2×SD16), C4.24(D4⋊2D5), (C4×C20).128C22, (C2×C20).399C23, Q8⋊Dic5.11C2, (Q8×C10).60C22, C10.45(C4.4D4), C4⋊Dic5.159C22, C2.12(C20.17D4), (C5×C4⋊Q8).7C2, C2.12(C2×Q8⋊D5), (C4×C5⋊2C8).13C2, C2.12(C2×C5⋊Q16), (C2×C10).530(C2×D4), (C2×C4).135(C5⋊D4), (C2×C4).496(C22×D5), C22.202(C2×C5⋊D4), (C2×C5⋊2C8).269C22, SmallGroup(320,708)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.Q16
G = < a,b,c | a20=b8=1, c2=b4, bab-1=a9, cac-1=a11, cbc-1=a10b-1 >
Subgroups: 318 in 98 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C2×C8, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C4×C8, Q8⋊C4, C4⋊Q8, C4⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4.SD16, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C4×C20, C5×C4⋊C4, C2×Dic10, Q8×C10, C4×C5⋊2C8, Q8⋊Dic5, C20⋊2Q8, C5×C4⋊Q8, C20.Q16
Quotients: C1, C2, C22, D4, C23, D5, SD16, Q16, C2×D4, C4○D4, D10, C4.4D4, C2×SD16, C2×Q16, C5⋊D4, C22×D5, C4.SD16, Q8⋊D5, C5⋊Q16, D4⋊2D5, C2×C5⋊D4, C20.17D4, C2×Q8⋊D5, C2×C5⋊Q16, C20.Q16
►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 85 69 249 210 280 177 136)(2 94 70 258 211 269 178 125)(3 83 71 247 212 278 179 134)(4 92 72 256 213 267 180 123)(5 81 73 245 214 276 161 132)(6 90 74 254 215 265 162 121)(7 99 75 243 216 274 163 130)(8 88 76 252 217 263 164 139)(9 97 77 241 218 272 165 128)(10 86 78 250 219 261 166 137)(11 95 79 259 220 270 167 126)(12 84 80 248 201 279 168 135)(13 93 61 257 202 268 169 124)(14 82 62 246 203 277 170 133)(15 91 63 255 204 266 171 122)(16 100 64 244 205 275 172 131)(17 89 65 253 206 264 173 140)(18 98 66 242 207 273 174 129)(19 87 67 251 208 262 175 138)(20 96 68 260 209 271 176 127)(21 308 287 119 190 50 234 151)(22 317 288 108 191 59 235 160)(23 306 289 117 192 48 236 149)(24 315 290 106 193 57 237 158)(25 304 291 115 194 46 238 147)(26 313 292 104 195 55 239 156)(27 302 293 113 196 44 240 145)(28 311 294 102 197 53 221 154)(29 320 295 111 198 42 222 143)(30 309 296 120 199 51 223 152)(31 318 297 109 200 60 224 141)(32 307 298 118 181 49 225 150)(33 316 299 107 182 58 226 159)(34 305 300 116 183 47 227 148)(35 314 281 105 184 56 228 157)(36 303 282 114 185 45 229 146)(37 312 283 103 186 54 230 155)(38 301 284 112 187 43 231 144)(39 310 285 101 188 52 232 153)(40 319 286 110 189 41 233 142)
(1 293 210 240)(2 284 211 231)(3 295 212 222)(4 286 213 233)(5 297 214 224)(6 288 215 235)(7 299 216 226)(8 290 217 237)(9 281 218 228)(10 292 219 239)(11 283 220 230)(12 294 201 221)(13 285 202 232)(14 296 203 223)(15 287 204 234)(16 298 205 225)(17 289 206 236)(18 300 207 227)(19 291 208 238)(20 282 209 229)(21 171 190 63)(22 162 191 74)(23 173 192 65)(24 164 193 76)(25 175 194 67)(26 166 195 78)(27 177 196 69)(28 168 197 80)(29 179 198 71)(30 170 199 62)(31 161 200 73)(32 172 181 64)(33 163 182 75)(34 174 183 66)(35 165 184 77)(36 176 185 68)(37 167 186 79)(38 178 187 70)(39 169 188 61)(40 180 189 72)(41 82 319 277)(42 93 320 268)(43 84 301 279)(44 95 302 270)(45 86 303 261)(46 97 304 272)(47 88 305 263)(48 99 306 274)(49 90 307 265)(50 81 308 276)(51 92 309 267)(52 83 310 278)(53 94 311 269)(54 85 312 280)(55 96 313 271)(56 87 314 262)(57 98 315 273)(58 89 316 264)(59 100 317 275)(60 91 318 266)(101 247 153 134)(102 258 154 125)(103 249 155 136)(104 260 156 127)(105 251 157 138)(106 242 158 129)(107 253 159 140)(108 244 160 131)(109 255 141 122)(110 246 142 133)(111 257 143 124)(112 248 144 135)(113 259 145 126)(114 250 146 137)(115 241 147 128)(116 252 148 139)(117 243 149 130)(118 254 150 121)(119 245 151 132)(120 256 152 123)
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), (1,85,69,249,210,280,177,136)(2,94,70,258,211,269,178,125)(3,83,71,247,212,278,179,134)(4,92,72,256,213,267,180,123)(5,81,73,245,214,276,161,132)(6,90,74,254,215,265,162,121)(7,99,75,243,216,274,163,130)(8,88,76,252,217,263,164,139)(9,97,77,241,218,272,165,128)(10,86,78,250,219,261,166,137)(11,95,79,259,220,270,167,126)(12,84,80,248,201,279,168,135)(13,93,61,257,202,268,169,124)(14,82,62,246,203,277,170,133)(15,91,63,255,204,266,171,122)(16,100,64,244,205,275,172,131)(17,89,65,253,206,264,173,140)(18,98,66,242,207,273,174,129)(19,87,67,251,208,262,175,138)(20,96,68,260,209,271,176,127)(21,308,287,119,190,50,234,151)(22,317,288,108,191,59,235,160)(23,306,289,117,192,48,236,149)(24,315,290,106,193,57,237,158)(25,304,291,115,194,46,238,147)(26,313,292,104,195,55,239,156)(27,302,293,113,196,44,240,145)(28,311,294,102,197,53,221,154)(29,320,295,111,198,42,222,143)(30,309,296,120,199,51,223,152)(31,318,297,109,200,60,224,141)(32,307,298,118,181,49,225,150)(33,316,299,107,182,58,226,159)(34,305,300,116,183,47,227,148)(35,314,281,105,184,56,228,157)(36,303,282,114,185,45,229,146)(37,312,283,103,186,54,230,155)(38,301,284,112,187,43,231,144)(39,310,285,101,188,52,232,153)(40,319,286,110,189,41,233,142), (1,293,210,240)(2,284,211,231)(3,295,212,222)(4,286,213,233)(5,297,214,224)(6,288,215,235)(7,299,216,226)(8,290,217,237)(9,281,218,228)(10,292,219,239)(11,283,220,230)(12,294,201,221)(13,285,202,232)(14,296,203,223)(15,287,204,234)(16,298,205,225)(17,289,206,236)(18,300,207,227)(19,291,208,238)(20,282,209,229)(21,171,190,63)(22,162,191,74)(23,173,192,65)(24,164,193,76)(25,175,194,67)(26,166,195,78)(27,177,196,69)(28,168,197,80)(29,179,198,71)(30,170,199,62)(31,161,200,73)(32,172,181,64)(33,163,182,75)(34,174,183,66)(35,165,184,77)(36,176,185,68)(37,167,186,79)(38,178,187,70)(39,169,188,61)(40,180,189,72)(41,82,319,277)(42,93,320,268)(43,84,301,279)(44,95,302,270)(45,86,303,261)(46,97,304,272)(47,88,305,263)(48,99,306,274)(49,90,307,265)(50,81,308,276)(51,92,309,267)(52,83,310,278)(53,94,311,269)(54,85,312,280)(55,96,313,271)(56,87,314,262)(57,98,315,273)(58,89,316,264)(59,100,317,275)(60,91,318,266)(101,247,153,134)(102,258,154,125)(103,249,155,136)(104,260,156,127)(105,251,157,138)(106,242,158,129)(107,253,159,140)(108,244,160,131)(109,255,141,122)(110,246,142,133)(111,257,143,124)(112,248,144,135)(113,259,145,126)(114,250,146,137)(115,241,147,128)(116,252,148,139)(117,243,149,130)(118,254,150,121)(119,245,151,132)(120,256,152,123)>;
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), (1,85,69,249,210,280,177,136)(2,94,70,258,211,269,178,125)(3,83,71,247,212,278,179,134)(4,92,72,256,213,267,180,123)(5,81,73,245,214,276,161,132)(6,90,74,254,215,265,162,121)(7,99,75,243,216,274,163,130)(8,88,76,252,217,263,164,139)(9,97,77,241,218,272,165,128)(10,86,78,250,219,261,166,137)(11,95,79,259,220,270,167,126)(12,84,80,248,201,279,168,135)(13,93,61,257,202,268,169,124)(14,82,62,246,203,277,170,133)(15,91,63,255,204,266,171,122)(16,100,64,244,205,275,172,131)(17,89,65,253,206,264,173,140)(18,98,66,242,207,273,174,129)(19,87,67,251,208,262,175,138)(20,96,68,260,209,271,176,127)(21,308,287,119,190,50,234,151)(22,317,288,108,191,59,235,160)(23,306,289,117,192,48,236,149)(24,315,290,106,193,57,237,158)(25,304,291,115,194,46,238,147)(26,313,292,104,195,55,239,156)(27,302,293,113,196,44,240,145)(28,311,294,102,197,53,221,154)(29,320,295,111,198,42,222,143)(30,309,296,120,199,51,223,152)(31,318,297,109,200,60,224,141)(32,307,298,118,181,49,225,150)(33,316,299,107,182,58,226,159)(34,305,300,116,183,47,227,148)(35,314,281,105,184,56,228,157)(36,303,282,114,185,45,229,146)(37,312,283,103,186,54,230,155)(38,301,284,112,187,43,231,144)(39,310,285,101,188,52,232,153)(40,319,286,110,189,41,233,142), (1,293,210,240)(2,284,211,231)(3,295,212,222)(4,286,213,233)(5,297,214,224)(6,288,215,235)(7,299,216,226)(8,290,217,237)(9,281,218,228)(10,292,219,239)(11,283,220,230)(12,294,201,221)(13,285,202,232)(14,296,203,223)(15,287,204,234)(16,298,205,225)(17,289,206,236)(18,300,207,227)(19,291,208,238)(20,282,209,229)(21,171,190,63)(22,162,191,74)(23,173,192,65)(24,164,193,76)(25,175,194,67)(26,166,195,78)(27,177,196,69)(28,168,197,80)(29,179,198,71)(30,170,199,62)(31,161,200,73)(32,172,181,64)(33,163,182,75)(34,174,183,66)(35,165,184,77)(36,176,185,68)(37,167,186,79)(38,178,187,70)(39,169,188,61)(40,180,189,72)(41,82,319,277)(42,93,320,268)(43,84,301,279)(44,95,302,270)(45,86,303,261)(46,97,304,272)(47,88,305,263)(48,99,306,274)(49,90,307,265)(50,81,308,276)(51,92,309,267)(52,83,310,278)(53,94,311,269)(54,85,312,280)(55,96,313,271)(56,87,314,262)(57,98,315,273)(58,89,316,264)(59,100,317,275)(60,91,318,266)(101,247,153,134)(102,258,154,125)(103,249,155,136)(104,260,156,127)(105,251,157,138)(106,242,158,129)(107,253,159,140)(108,244,160,131)(109,255,141,122)(110,246,142,133)(111,257,143,124)(112,248,144,135)(113,259,145,126)(114,250,146,137)(115,241,147,128)(116,252,148,139)(117,243,149,130)(118,254,150,121)(119,245,151,132)(120,256,152,123) );
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)], [(1,85,69,249,210,280,177,136),(2,94,70,258,211,269,178,125),(3,83,71,247,212,278,179,134),(4,92,72,256,213,267,180,123),(5,81,73,245,214,276,161,132),(6,90,74,254,215,265,162,121),(7,99,75,243,216,274,163,130),(8,88,76,252,217,263,164,139),(9,97,77,241,218,272,165,128),(10,86,78,250,219,261,166,137),(11,95,79,259,220,270,167,126),(12,84,80,248,201,279,168,135),(13,93,61,257,202,268,169,124),(14,82,62,246,203,277,170,133),(15,91,63,255,204,266,171,122),(16,100,64,244,205,275,172,131),(17,89,65,253,206,264,173,140),(18,98,66,242,207,273,174,129),(19,87,67,251,208,262,175,138),(20,96,68,260,209,271,176,127),(21,308,287,119,190,50,234,151),(22,317,288,108,191,59,235,160),(23,306,289,117,192,48,236,149),(24,315,290,106,193,57,237,158),(25,304,291,115,194,46,238,147),(26,313,292,104,195,55,239,156),(27,302,293,113,196,44,240,145),(28,311,294,102,197,53,221,154),(29,320,295,111,198,42,222,143),(30,309,296,120,199,51,223,152),(31,318,297,109,200,60,224,141),(32,307,298,118,181,49,225,150),(33,316,299,107,182,58,226,159),(34,305,300,116,183,47,227,148),(35,314,281,105,184,56,228,157),(36,303,282,114,185,45,229,146),(37,312,283,103,186,54,230,155),(38,301,284,112,187,43,231,144),(39,310,285,101,188,52,232,153),(40,319,286,110,189,41,233,142)], [(1,293,210,240),(2,284,211,231),(3,295,212,222),(4,286,213,233),(5,297,214,224),(6,288,215,235),(7,299,216,226),(8,290,217,237),(9,281,218,228),(10,292,219,239),(11,283,220,230),(12,294,201,221),(13,285,202,232),(14,296,203,223),(15,287,204,234),(16,298,205,225),(17,289,206,236),(18,300,207,227),(19,291,208,238),(20,282,209,229),(21,171,190,63),(22,162,191,74),(23,173,192,65),(24,164,193,76),(25,175,194,67),(26,166,195,78),(27,177,196,69),(28,168,197,80),(29,179,198,71),(30,170,199,62),(31,161,200,73),(32,172,181,64),(33,163,182,75),(34,174,183,66),(35,165,184,77),(36,176,185,68),(37,167,186,79),(38,178,187,70),(39,169,188,61),(40,180,189,72),(41,82,319,277),(42,93,320,268),(43,84,301,279),(44,95,302,270),(45,86,303,261),(46,97,304,272),(47,88,305,263),(48,99,306,274),(49,90,307,265),(50,81,308,276),(51,92,309,267),(52,83,310,278),(53,94,311,269),(54,85,312,280),(55,96,313,271),(56,87,314,262),(57,98,315,273),(58,89,316,264),(59,100,317,275),(60,91,318,266),(101,247,153,134),(102,258,154,125),(103,249,155,136),(104,260,156,127),(105,251,157,138),(106,242,158,129),(107,253,159,140),(108,244,160,131),(109,255,141,122),(110,246,142,133),(111,257,143,124),(112,248,144,135),(113,259,145,126),(114,250,146,137),(115,241,147,128),(116,252,148,139),(117,243,149,130),(118,254,150,121),(119,245,151,132),(120,256,152,123)]])
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 | D4 | D5 | SD16 | Q16 | C4○D4 | D10 | D10 | C5⋊D4 | Q8⋊D5 | C5⋊Q16 | D4⋊2D5 |
| kernel | C20.Q16 | C4×C5⋊2C8 | Q8⋊Dic5 | C20⋊2Q8 | C5×C4⋊Q8 | C2×C20 | C4⋊Q8 | C20 | C20 | C20 | C42 | C2×Q8 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 4 |
Matrix representation of C20.Q16 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 |
| 0 | 0 | 0 | 0 | 36 | 40 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 29 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 15 | 0 | 0 |
| 0 | 0 | 26 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 37 | 32 |
| 30 | 40 | 0 | 0 | 0 | 0 |
| 40 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 40 | 0 | 0 |
| 0 | 0 | 40 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 40 |
| 0 | 0 | 0 | 0 | 36 | 23 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,35,36,0,0,0,0,40,40],[12,29,0,0,0,0,12,12,0,0,0,0,0,0,26,26,0,0,0,0,15,26,0,0,0,0,0,0,9,37,0,0,0,0,0,32],[30,40,0,0,0,0,40,11,0,0,0,0,0,0,30,40,0,0,0,0,40,11,0,0,0,0,0,0,18,36,0,0,0,0,40,23] >;
C20.Q16 in GAP, Magma, Sage, TeX
C_{20}.Q_{16} % in TeX
G:=Group("C20.Q16"); // GroupNames label
G:=SmallGroup(320,708);
// by ID
G=gap.SmallGroup(320,708);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,64,590,135,184,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=1,c^2=b^4,b*a*b^-1=a^9,c*a*c^-1=a^11,c*b*c^-1=a^10*b^-1>;
// generators/relations