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G = C20.23Q16order 320 = 26·5

2nd non-split extension by C20 of Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q84Dic10, C20.23Q16, C42.54D10, (C5×Q8)⋊4Q8, (C4×Q8).3D5, C54(C4.Q16), (C2×C20).64D4, (Q8×C20).3C2, C20.29(C2×Q8), C4⋊C4.249D10, C10.34(C2×Q16), C203C8.15C2, C20.56(C4○D4), C4.63(C4○D20), (C4×C20).92C22, (C2×Q8).156D10, C202Q8.15C2, C4.13(C2×Dic10), Q8⋊Dic5.8C2, C4.11(C5⋊Q16), C2.9(D4⋊D10), (C2×C20).343C23, C10.D8.10C2, C10.65(C22⋊Q8), C10.110(C8⋊C22), C4⋊Dic5.140C22, (Q8×C10).191C22, C2.16(C20.48D4), C2.6(C2×C5⋊Q16), (C2×C10).474(C2×D4), (C2×C4).248(C5⋊D4), (C5×C4⋊C4).280C22, (C2×C52C8).98C22, (C2×C4).443(C22×D5), C22.153(C2×C5⋊D4), SmallGroup(320,648)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.23Q16
C1C5C10C20C2×C20C4⋊Dic5C202Q8 — C20.23Q16
C5C10C2×C20 — C20.23Q16
C1C22C42C4×Q8

Generators and relations for C20.23Q16
 G = < a,b,c | a20=b8=1, c2=a10b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 310 in 96 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×6], C22, C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×3], C10 [×3], C42, C42, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×Q8, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], C20 [×4], C2×C10, Q8⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×Q8, C4⋊Q8, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C5×Q8, C4.Q16, C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, Q8×C10, C203C8, C10.D8 [×2], Q8⋊Dic5 [×2], C202Q8, Q8×C20, C20.23Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, Q16 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×Q16, C8⋊C22, Dic10 [×2], C5⋊D4 [×2], C22×D5, C4.Q16, C5⋊Q16 [×2], C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, C2×C5⋊Q16, D4⋊D10, C20.23Q16

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

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

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

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

59 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4I4J4K5A5B8A8B8C8D10A···10F20A···20H20I···20AF
order122244444···44455888810···1020···2020···20
size111122224···4404022202020202···22···24···4

59 irreducible representations

dim11111122222222222444
type+++++++-+-+++-+-+
imageC1C2C2C2C2C2D4Q8D5Q16C4○D4D10D10D10C5⋊D4Dic10C4○D20C8⋊C22C5⋊Q16D4⋊D10
kernelC20.23Q16C203C8C10.D8Q8⋊Dic5C202Q8Q8×C20C2×C20C5×Q8C4×Q8C20C20C42C4⋊C4C2×Q8C2×C4Q8C4C10C4C2
# reps11221122242222888144

Matrix representation of C20.23Q16 in GL6(𝔽41)

3510000
4000000
000100
0040000
0000400
0000040
,
8290000
36330000
0003200
0032000
00002424
0000290
,
4000000
0400000
000100
0040000
0000374
000064

G:=sub<GL(6,GF(41))| [35,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[8,36,0,0,0,0,29,33,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,24,29,0,0,0,0,24,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,37,6,0,0,0,0,4,4] >;

C20.23Q16 in GAP, Magma, Sage, TeX

C_{20}._{23}Q_{16}
% in TeX

G:=Group("C20.23Q16");
// GroupNames label

G:=SmallGroup(320,648);
// by ID

G=gap.SmallGroup(320,648);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,336,253,120,254,268,1123,297,136,12550]);
// Polycyclic

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

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