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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, C5⋊2(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×C5⋊2C8).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

Derived series
C1 — C10 — C20.31M4(2)
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C4×C5⋊C8 — C20.31M4(2)
Lower central
C5 — C10 — C20.31M4(2)
Upper central
C1 — C2×C4 — C2×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, C4, C4, C22, C5, C8, C2×C4, C2×C4, C10, C42, C2×C8, C2×C8, Dic5, C20, C2×C10, C4×C8, C5⋊2C8, C40, C5⋊C8, C5⋊C8, C2×Dic5, C2×C20, C8⋊C8, C2×C5⋊2C8, C4×Dic5, C2×C40, C2×C5⋊C8, C8×Dic5, C4×C5⋊C8, C20.31M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C42, C2×C8, M4(2), F5, C4×C8, C8⋊C4, 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)

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

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

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×C5⋊2C8C2×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
 

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