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G = C20.11Q16order 320 = 26·5

11st non-split extension by C20 of Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.11Q16, C20.20SD16, C42.226D10, C4⋊Q8.12D5, C4⋊C4.87D10, (C2×C20).162D4, C4.5(D4.D5), C10.45(C2×Q16), C4.5(C5⋊Q16), C54(C4.SD16), C20.87(C4○D4), C202Q8.23C2, C10.61(C2×SD16), (C2×C20).411C23, (C4×C20).140C22, C4.18(Q82D5), C10.Q16.15C2, C10.60(C4.4D4), C2.13(C20.23D4), (C2×Dic10).120C22, (C5×C4⋊Q8).12C2, (C4×C52C8).15C2, C2.15(C2×D4.D5), C2.16(C2×C5⋊Q16), (C2×C10).542(C2×D4), (C2×C4).139(C5⋊D4), (C5×C4⋊C4).134C22, (C2×C4).508(C22×D5), C22.214(C2×C5⋊D4), (C2×C52C8).273C22, SmallGroup(320,720)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.11Q16
C1C5C10C20C2×C20C2×Dic10C202Q8 — C20.11Q16
C5C10C2×C20 — C20.11Q16
C1C22C42C4⋊Q8

Generators and relations for C20.11Q16
 G = < a,b,c | a20=b8=1, c2=a10b4, bab-1=a9, cac-1=a11, cbc-1=a10b-1 >

Subgroups: 318 in 98 conjugacy classes, 47 normal (23 characteristic)
C1, C2 [×3], C4 [×6], C4 [×4], C22, C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×6], C10 [×3], C42, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×Q8 [×3], Dic5 [×2], C20 [×6], C20 [×2], C2×C10, C4×C8, Q8⋊C4 [×4], C4⋊Q8, C4⋊Q8, C52C8 [×2], Dic10 [×4], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C4.SD16, C2×C52C8 [×2], C4⋊Dic5 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], Q8×C10, C4×C52C8, C10.Q16 [×4], C202Q8, C5×C4⋊Q8, C20.11Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], Q16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C2×SD16, C2×Q16, C5⋊D4 [×2], C22×D5, C4.SD16, D4.D5 [×2], C5⋊Q16 [×2], Q82D5 [×2], C2×C5⋊D4, C2×D4.D5, C2×C5⋊Q16, C20.23D4, C20.11Q16

Smallest permutation representation of C20.11Q16
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)
(1 132 183 161 279 41 120 314)(2 121 184 170 280 50 101 303)(3 130 185 179 261 59 102 312)(4 139 186 168 262 48 103 301)(5 128 187 177 263 57 104 310)(6 137 188 166 264 46 105 319)(7 126 189 175 265 55 106 308)(8 135 190 164 266 44 107 317)(9 124 191 173 267 53 108 306)(10 133 192 162 268 42 109 315)(11 122 193 171 269 51 110 304)(12 131 194 180 270 60 111 313)(13 140 195 169 271 49 112 302)(14 129 196 178 272 58 113 311)(15 138 197 167 273 47 114 320)(16 127 198 176 274 56 115 309)(17 136 199 165 275 45 116 318)(18 125 200 174 276 54 117 307)(19 134 181 163 277 43 118 316)(20 123 182 172 278 52 119 305)(21 160 253 72 290 92 204 233)(22 149 254 61 291 81 205 222)(23 158 255 70 292 90 206 231)(24 147 256 79 293 99 207 240)(25 156 257 68 294 88 208 229)(26 145 258 77 295 97 209 238)(27 154 259 66 296 86 210 227)(28 143 260 75 297 95 211 236)(29 152 241 64 298 84 212 225)(30 141 242 73 299 93 213 234)(31 150 243 62 300 82 214 223)(32 159 244 71 281 91 215 232)(33 148 245 80 282 100 216 221)(34 157 246 69 283 89 217 230)(35 146 247 78 284 98 218 239)(36 155 248 67 285 87 219 228)(37 144 249 76 286 96 220 237)(38 153 250 65 287 85 201 226)(39 142 251 74 288 94 202 235)(40 151 252 63 289 83 203 224)
(1 287 269 28)(2 298 270 39)(3 289 271 30)(4 300 272 21)(5 291 273 32)(6 282 274 23)(7 293 275 34)(8 284 276 25)(9 295 277 36)(10 286 278 27)(11 297 279 38)(12 288 280 29)(13 299 261 40)(14 290 262 31)(15 281 263 22)(16 292 264 33)(17 283 265 24)(18 294 266 35)(19 285 267 26)(20 296 268 37)(41 236 122 65)(42 227 123 76)(43 238 124 67)(44 229 125 78)(45 240 126 69)(46 231 127 80)(47 222 128 71)(48 233 129 62)(49 224 130 73)(50 235 131 64)(51 226 132 75)(52 237 133 66)(53 228 134 77)(54 239 135 68)(55 230 136 79)(56 221 137 70)(57 232 138 61)(58 223 139 72)(59 234 140 63)(60 225 121 74)(81 177 159 320)(82 168 160 311)(83 179 141 302)(84 170 142 313)(85 161 143 304)(86 172 144 315)(87 163 145 306)(88 174 146 317)(89 165 147 308)(90 176 148 319)(91 167 149 310)(92 178 150 301)(93 169 151 312)(94 180 152 303)(95 171 153 314)(96 162 154 305)(97 173 155 316)(98 164 156 307)(99 175 157 318)(100 166 158 309)(101 212 194 251)(102 203 195 242)(103 214 196 253)(104 205 197 244)(105 216 198 255)(106 207 199 246)(107 218 200 257)(108 209 181 248)(109 220 182 259)(110 211 183 250)(111 202 184 241)(112 213 185 252)(113 204 186 243)(114 215 187 254)(115 206 188 245)(116 217 189 256)(117 208 190 247)(118 219 191 258)(119 210 192 249)(120 201 193 260)

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

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

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

50 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J5A5B8A···8H10A···10F20A···20L20M···20T
order12224···44444558···810···1020···2020···20
size11112···28840402210···102···24···48···8

50 irreducible representations

dim1111122222222444
type+++++++-++--+
imageC1C2C2C2C2D4D5SD16Q16C4○D4D10D10C5⋊D4D4.D5C5⋊Q16Q82D5
kernelC20.11Q16C4×C52C8C10.Q16C202Q8C5×C4⋊Q8C2×C20C4⋊Q8C20C20C20C42C4⋊C4C2×C4C4C4C4
# reps1141122444248444

Matrix representation of C20.11Q16 in GL6(𝔽41)

3410000
3310000
001000
000100
00003231
000009
,
1120000
22300000
0038000
00102700
00001414
0000038
,
100000
010000
003900
0083800
00001324
00003428

G:=sub<GL(6,GF(41))| [34,33,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,31,9],[11,22,0,0,0,0,2,30,0,0,0,0,0,0,38,10,0,0,0,0,0,27,0,0,0,0,0,0,14,0,0,0,0,0,14,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,8,0,0,0,0,9,38,0,0,0,0,0,0,13,34,0,0,0,0,24,28] >;

C20.11Q16 in GAP, Magma, Sage, TeX

C_{20}._{11}Q_{16}
% in TeX

G:=Group("C20.11Q16");
// GroupNames label

G:=SmallGroup(320,720);
// by ID

G=gap.SmallGroup(320,720);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,219,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^10*b^4,b*a*b^-1=a^9,c*a*c^-1=a^11,c*b*c^-1=a^10*b^-1>;
// generators/relations

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