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G = C20.26M4(2)  order 320 = 26·5

1st non-split extension by C20 of M4(2) acting via M4(2)/C22=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.26M4(2), Dic5.12SD16, C52C84C8, C52(C82C8), (C2×C8).5F5, (C2×C40).2C4, C10.5(C4⋊C8), C20.12(C2×C8), C4.7(D5⋊C8), C20⋊C8.9C2, C2.2(C40⋊C4), C10.4(C4.Q8), (C8×Dic5).11C2, (C2×Dic5).29Q8, C22.17(C4⋊F5), C2.2(D10.Q8), C10.3(C8.C4), C4.4(C22.F5), (C2×Dic5).172D4, C2.4(Dic5⋊C8), (C4×Dic5).342C22, (C2×C52C8).21C4, (C2×C4).119(C2×F5), (C2×C10).10(C4⋊C4), (C2×C20).116(C2×C4), SmallGroup(320,221)

Series: Derived Chief Lower central Upper central

C1C20 — C20.26M4(2)
C1C5C10C2×C10C2×Dic5C4×Dic5C20⋊C8 — C20.26M4(2)
C5C10C20 — C20.26M4(2)
C1C22C2×C4C2×C8

Generators and relations for C20.26M4(2)
 G = < a,b,c | a20=b8=1, c2=a5, bab-1=a3, ac=ca, cbc-1=a15b5 >

5C4
5C4
10C4
2C8
5C8
5C2×C4
5C2×C4
5C8
20C8
20C8
2Dic5
5C2×C8
5C42
10C2×C8
10C2×C8
2C40
4C5⋊C8
4C5⋊C8
5C4⋊C8
5C4×C8
5C4⋊C8
2C2×C5⋊C8
2C2×C5⋊C8
5C82C8

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

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H 5 8A8B8C8D8E8F8G8H8I···8P10A10B10C20A20B20C20D40A···40H
order1222444444445888888888···81010102020202040···40
size11112255551010422221010101020···2044444444···4

44 irreducible representations

dim111111222224444444
type++++-++-
imageC1C2C2C4C4C8D4Q8SD16M4(2)C8.C4F5C2×F5D5⋊C8C22.F5C4⋊F5C40⋊C4D10.Q8
kernelC20.26M4(2)C8×Dic5C20⋊C8C2×C52C8C2×C40C52C8C2×Dic5C2×Dic5Dic5C20C10C2×C8C2×C4C4C4C22C2C2
# reps112228114241122244

Matrix representation of C20.26M4(2) in GL6(𝔽41)

35360000
3260000
00347734
00701414
002734270
000273427
,
060000
2200000
0016123534
002322294
00723191
0037191825
,
22220000
1520000
008055
00363360
00036336
005508

G:=sub<GL(6,GF(41))| [35,32,0,0,0,0,36,6,0,0,0,0,0,0,34,7,27,0,0,0,7,0,34,27,0,0,7,14,27,34,0,0,34,14,0,27],[0,22,0,0,0,0,6,0,0,0,0,0,0,0,16,23,7,37,0,0,12,22,23,19,0,0,35,29,19,18,0,0,34,4,1,25],[22,15,0,0,0,0,22,2,0,0,0,0,0,0,8,36,0,5,0,0,0,3,36,5,0,0,5,36,3,0,0,0,5,0,36,8] >;

C20.26M4(2) in GAP, Magma, Sage, TeX

C_{20}._{26}M_4(2)
% in TeX

G:=Group("C20.26M4(2)");
// GroupNames label

G:=SmallGroup(320,221);
// by ID

G=gap.SmallGroup(320,221);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,100,1123,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^5,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^15*b^5>;
// generators/relations

Export

Subgroup lattice of C20.26M4(2) in TeX

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