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## G = C20.M4(2)  order 320 = 26·5

### 5th non-split extension by C20 of M4(2) acting via M4(2)/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — C20.M4(2)
 Lower central C5 — C2×C10 — C20.M4(2)
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C20.M4(2)
G = < a,b,c | a20=b8=1, c2=a10, bab-1=a7, cac-1=a-1, cbc-1=b5 >

Subgroups: 282 in 94 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×C8, C8⋊C4, C4⋊C8, C4×Q8, C5⋊C8, C5⋊C8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C84Q8, C4×Dic5, C4×Dic5, C10.D4, C5×C4⋊C4, C2×C5⋊C8, C2×C5⋊C8, C2×Dic10, C4×C5⋊C8, C20⋊C8, C20⋊C8, C10.C42, Dic53Q8, C20.M4(2)
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, M4(2), C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C2×M4(2), C8○D4, C2×F5, C84Q8, C4.F5, C22×F5, C2×C4.F5, D4.F5, Q8×F5, C20.M4(2)

Smallest permutation representation of C20.M4(2)
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 30 266 167 130 223 65 242)(2 33 275 174 131 226 74 249)(3 36 264 161 132 229 63 256)(4 39 273 168 133 232 72 243)(5 22 262 175 134 235 61 250)(6 25 271 162 135 238 70 257)(7 28 280 169 136 221 79 244)(8 31 269 176 137 224 68 251)(9 34 278 163 138 227 77 258)(10 37 267 170 139 230 66 245)(11 40 276 177 140 233 75 252)(12 23 265 164 121 236 64 259)(13 26 274 171 122 239 73 246)(14 29 263 178 123 222 62 253)(15 32 272 165 124 225 71 260)(16 35 261 172 125 228 80 247)(17 38 270 179 126 231 69 254)(18 21 279 166 127 234 78 241)(19 24 268 173 128 237 67 248)(20 27 277 180 129 240 76 255)(41 313 197 210 105 160 291 96)(42 316 186 217 106 143 300 83)(43 319 195 204 107 146 289 90)(44 302 184 211 108 149 298 97)(45 305 193 218 109 152 287 84)(46 308 182 205 110 155 296 91)(47 311 191 212 111 158 285 98)(48 314 200 219 112 141 294 85)(49 317 189 206 113 144 283 92)(50 320 198 213 114 147 292 99)(51 303 187 220 115 150 281 86)(52 306 196 207 116 153 290 93)(53 309 185 214 117 156 299 100)(54 312 194 201 118 159 288 87)(55 315 183 208 119 142 297 94)(56 318 192 215 120 145 286 81)(57 301 181 202 101 148 295 88)(58 304 190 209 102 151 284 95)(59 307 199 216 103 154 293 82)(60 310 188 203 104 157 282 89)
(1 183 11 193)(2 182 12 192)(3 181 13 191)(4 200 14 190)(5 199 15 189)(6 198 16 188)(7 197 17 187)(8 196 18 186)(9 195 19 185)(10 194 20 184)(21 83 31 93)(22 82 32 92)(23 81 33 91)(24 100 34 90)(25 99 35 89)(26 98 36 88)(27 97 37 87)(28 96 38 86)(29 95 39 85)(30 94 40 84)(41 69 51 79)(42 68 52 78)(43 67 53 77)(44 66 54 76)(45 65 55 75)(46 64 56 74)(47 63 57 73)(48 62 58 72)(49 61 59 71)(50 80 60 70)(101 274 111 264)(102 273 112 263)(103 272 113 262)(104 271 114 261)(105 270 115 280)(106 269 116 279)(107 268 117 278)(108 267 118 277)(109 266 119 276)(110 265 120 275)(121 286 131 296)(122 285 132 295)(123 284 133 294)(124 283 134 293)(125 282 135 292)(126 281 136 291)(127 300 137 290)(128 299 138 289)(129 298 139 288)(130 297 140 287)(141 253 151 243)(142 252 152 242)(143 251 153 241)(144 250 154 260)(145 249 155 259)(146 248 156 258)(147 247 157 257)(148 246 158 256)(149 245 159 255)(150 244 160 254)(161 301 171 311)(162 320 172 310)(163 319 173 309)(164 318 174 308)(165 317 175 307)(166 316 176 306)(167 315 177 305)(168 314 178 304)(169 313 179 303)(170 312 180 302)(201 240 211 230)(202 239 212 229)(203 238 213 228)(204 237 214 227)(205 236 215 226)(206 235 216 225)(207 234 217 224)(208 233 218 223)(209 232 219 222)(210 231 220 221)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5 8A ··· 8H 8I 8J 8K 8L 10A 10B 10C 20A ··· 20F order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 8 ··· 8 8 8 8 8 10 10 10 20 ··· 20 size 1 1 1 1 2 2 4 4 5 5 5 5 10 10 20 20 4 10 ··· 10 20 20 20 20 4 4 4 8 ··· 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 8 8 type + + + + + - + + - - image C1 C2 C2 C2 C2 C4 C4 C4 Q8 C4○D4 M4(2) C8○D4 F5 C2×F5 C4.F5 D4.F5 Q8×F5 kernel C20.M4(2) C4×C5⋊C8 C20⋊C8 C10.C42 Dic5⋊3Q8 C10.D4 C5×C4⋊C4 C2×Dic10 C5⋊C8 Dic5 C20 C10 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 1 3 2 1 4 2 2 2 2 4 4 1 3 4 1 1

Matrix representation of C20.M4(2) in GL6(𝔽41)

 26 26 0 0 0 0 26 15 0 0 0 0 0 0 37 10 16 0 0 0 38 10 16 0 0 0 37 11 16 0 0 0 14 8 21 18
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 31 34 5 2 0 0 4 18 24 22 0 0 32 1 38 17 0 0 23 5 14 36
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 18 14 3 11 0 0 5 4 15 4 0 0 23 14 20 1 0 0 40 39 19 40

`G:=sub<GL(6,GF(41))| [26,26,0,0,0,0,26,15,0,0,0,0,0,0,37,38,37,14,0,0,10,10,11,8,0,0,16,16,16,21,0,0,0,0,0,18],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,31,4,32,23,0,0,34,18,1,5,0,0,5,24,38,14,0,0,2,22,17,36],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,18,5,23,40,0,0,14,4,14,39,0,0,3,15,20,19,0,0,11,4,1,40] >;`

C20.M4(2) in GAP, Magma, Sage, TeX

`C_{20}.M_4(2)`
`% in TeX`

`G:=Group("C20.M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1047);`
`// by ID`

`G=gap.SmallGroup(320,1047);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,268,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

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