Copied to
clipboard

G = C20.31M4(2)  order 320 = 26·5

6th non-split extension by C20 of M4(2) acting via M4(2)/C22=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.31M4(2), Dic5.7M4(2), C5⋊C8⋊C8, C52(C8⋊C8), (C2×C8).4F5, C2.4(C8×F5), C10.3(C4×C8), (C2×C40).1C4, C2.2(C8⋊F5), C10.6(C8⋊C4), C4.10(C4.F5), C22.15(C4×F5), (C2×C10).10C42, Dic5.10(C2×C8), (C8×Dic5).10C2, C4.9(C22.F5), C2.1(C10.C42), (C4×Dic5).352C22, (C4×C5⋊C8).7C2, (C2×C5⋊C8).2C4, (C2×C52C8).34C4, (C2×C4).152(C2×F5), (C2×C20).158(C2×C4), (C2×Dic5).121(C2×C4), SmallGroup(320,218)

Series: Derived Chief Lower central Upper central

C1C10 — C20.31M4(2)
C1C5C10C2×C10C2×Dic5C4×Dic5C4×C5⋊C8 — C20.31M4(2)
C5C10 — C20.31M4(2)
C1C2×C4C2×C8

Generators and relations for C20.31M4(2)
 G = < a,b,c | a20=b8=1, c2=a5, bab-1=a13, ac=ca, cbc-1=a10b5 >

Subgroups: 178 in 66 conjugacy classes, 38 normal (26 characteristic)
C1, C2 [×3], C4 [×2], C4 [×4], C22, C5, C8 [×8], C2×C4, C2×C4 [×2], C10 [×3], C42, C2×C8, C2×C8 [×5], Dic5 [×4], C20 [×2], C2×C10, C4×C8 [×3], C52C8, C40, C5⋊C8 [×4], C5⋊C8 [×2], C2×Dic5 [×2], C2×C20, C8⋊C8, C2×C52C8, C4×Dic5, C2×C40, C2×C5⋊C8 [×4], C8×Dic5, C4×C5⋊C8 [×2], C20.31M4(2)
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], C42, C2×C8 [×2], M4(2) [×4], F5, C4×C8, C8⋊C4 [×2], C2×F5, C8⋊C8, C4.F5, C4×F5, C22.F5, C8×F5, C8⋊F5, C10.C42, C20.31M4(2)

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

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4L 5 8A8B8C8D8E···8X10A10B10C20A20B20C20D40A···40H
order122244444···4588888···81010102020202040···40
size111111115···54222210···1044444444···4

56 irreducible representations

dim1111111224444444
type+++++-
imageC1C2C2C4C4C4C8M4(2)M4(2)F5C2×F5C4.F5C22.F5C4×F5C8×F5C8⋊F5
kernelC20.31M4(2)C8×Dic5C4×C5⋊C8C2×C52C8C2×C40C2×C5⋊C8C5⋊C8Dic5C20C2×C8C2×C4C4C4C22C2C2
# reps11222816441122244

Matrix representation of C20.31M4(2) in GL8(𝔽41)

10000000
01000000
00900000
00090000
000040404040
00001000
00000100
00000010
,
3233000000
99000000
00640000
0022350000
00002428736
00002081337
00005293312
00004241217
,
4039000000
01000000
002150000
0012200000
00001000
00000100
00000010
00000001

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[32,9,0,0,0,0,0,0,33,9,0,0,0,0,0,0,0,0,6,22,0,0,0,0,0,0,4,35,0,0,0,0,0,0,0,0,24,20,5,4,0,0,0,0,28,8,29,24,0,0,0,0,7,13,33,12,0,0,0,0,36,37,12,17],[40,0,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,21,12,0,0,0,0,0,0,5,20,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C20.31M4(2) in GAP, Magma, Sage, TeX

C_{20}._{31}M_4(2)
% in TeX

G:=Group("C20.31M4(2)");
// GroupNames label

G:=SmallGroup(320,218);
// by ID

G=gap.SmallGroup(320,218);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,64,184,80,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^5,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^10*b^5>;
// generators/relations

׿
×
𝔽