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G = C20.14Q16order 320 = 26·5

3rd non-split extension by C20 of Q16 acting via Q16/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.14Q16, C4.3Dic20, C20.25SD16, C42.253D10, (C4×C8).5D5, (C4×C40).5C2, (C2×C4).77D20, C10.1(C2×Q16), (C2×C20).374D4, (C2×C8).281D10, C4.4(C40⋊C2), C202Q8.3C2, C10.2(C2×SD16), C2.4(C2×Dic20), C51(C4.SD16), C4.99(C4○D20), C22.86(C2×D20), C4⋊Dic5.2C22, C20.215(C4○D4), (C4×C20).303C22, (C2×C40).340C22, (C2×C20).717C23, C20.44D4.1C2, C2.9(C4.D20), C10.4(C4.4D4), (C2×Dic10).3C22, C2.6(C2×C40⋊C2), (C2×C10).100(C2×D4), (C2×C4).660(C22×D5), SmallGroup(320,308)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.14Q16
C1C5C10C20C2×C20C4⋊Dic5C202Q8 — C20.14Q16
C5C10C2×C20 — C20.14Q16
C1C22C42C4×C8

Generators and relations for C20.14Q16
 G = < a,b,c | a20=b8=1, c2=a10b4, ab=ba, cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 398 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 [×5], C2×C8 [×2], C2×Q8 [×3], Dic5 [×4], C20 [×6], C2×C10, C4×C8, Q8⋊C4 [×4], C4⋊Q8 [×2], C40 [×2], Dic10 [×6], C2×Dic5 [×4], C2×C20 [×3], C4.SD16, C4⋊Dic5 [×2], C4⋊Dic5 [×3], C4×C20, C2×C40 [×2], C2×Dic10 [×2], C2×Dic10, C20.44D4 [×4], C4×C40, C202Q8 [×2], C20.14Q16
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, D20 [×2], C22×D5, C4.SD16, C40⋊C2 [×2], Dic20 [×2], C2×D20, C4○D20 [×2], C4.D20, C2×C40⋊C2, C2×Dic20, C20.14Q16

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

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

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

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

86 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J5A5B8A···8H10A···10F20A···20X40A···40AF
order12224···44444558···810···1020···2040···40
size11112···240404040222···22···22···22···2

86 irreducible representations

dim111122222222222
type++++++-+++-
imageC1C2C2C2D4D5SD16Q16C4○D4D10D10D20C40⋊C2Dic20C4○D20
kernelC20.14Q16C20.44D4C4×C40C202Q8C2×C20C4×C8C20C20C20C42C2×C8C2×C4C4C4C4
# reps141222444248161616

Matrix representation of C20.14Q16 in GL4(𝔽41) generated by

143000
11900
002711
003032
,
32000
03200
002712
002925
,
122700
252900
003032
002711
G:=sub<GL(4,GF(41))| [14,11,0,0,30,9,0,0,0,0,27,30,0,0,11,32],[32,0,0,0,0,32,0,0,0,0,27,29,0,0,12,25],[12,25,0,0,27,29,0,0,0,0,30,27,0,0,32,11] >;

C20.14Q16 in GAP, Magma, Sage, TeX

C_{20}._{14}Q_{16}
% in TeX

G:=Group("C20.14Q16");
// GroupNames label

G:=SmallGroup(320,308);
// by ID

G=gap.SmallGroup(320,308);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,142,1123,136,12550]);
// Polycyclic

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

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