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G = C412C8order 328 = 23·41

The semidirect product of C41 and C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C412C8, C82.C4, Dic41.2C2, C2.(C41⋊C4), SmallGroup(328,3)

Series: Derived Chief Lower central Upper central

C1C41 — C412C8
C1C41C82Dic41 — C412C8
C41 — C412C8
C1C2

Generators and relations for C412C8
 G = < a,b | a41=b8=1, bab-1=a9 >

41C4
41C8

Character table of C412C8

 class 124A4B8A8B8C8D41A41B41C41D41E41F41G41H41I41J82A82B82C82D82E82F82G82H82I82J
 size 1141414141414144444444444444444444
ρ11111111111111111111111111111    trivial
ρ21111-1-1-1-111111111111111111111    linear of order 2
ρ311-1-1-ii-ii11111111111111111111    linear of order 4
ρ411-1-1i-ii-i11111111111111111111    linear of order 4
ρ51-1i-iζ87ζ85ζ83ζ81111111111-1-1-1-1-1-1-1-1-1-1    linear of order 8
ρ61-1-iiζ8ζ83ζ85ζ871111111111-1-1-1-1-1-1-1-1-1-1    linear of order 8
ρ71-1-iiζ85ζ87ζ8ζ831111111111-1-1-1-1-1-1-1-1-1-1    linear of order 8
ρ81-1i-iζ83ζ8ζ87ζ851111111111-1-1-1-1-1-1-1-1-1-1    linear of order 8
ρ944000000ζ413441224119417ζ4130412441174111ζ4140413241941ζ413941234118412ζ41374136415414ζ413541284113416ζ413341314110418ζ4129412641154112ζ413841274114413ζ4125412141204116ζ4125412141204116ζ413441224119417ζ4130412441174111ζ4140413241941ζ413941234118412ζ41374136415414ζ413541284113416ζ413341314110418ζ4129412641154112ζ413841274114413    orthogonal lifted from C41⋊C4
ρ1044000000ζ4140413241941ζ4125412141204116ζ413541284113416ζ4129412641154112ζ4130412441174111ζ41374136415414ζ413441224119417ζ413341314110418ζ413941234118412ζ413841274114413ζ413841274114413ζ4140413241941ζ4125412141204116ζ413541284113416ζ4129412641154112ζ4130412441174111ζ41374136415414ζ413441224119417ζ413341314110418ζ413941234118412    orthogonal lifted from C41⋊C4
ρ1144000000ζ4129412641154112ζ413541284113416ζ413341314110418ζ4125412141204116ζ4140413241941ζ413441224119417ζ413941234118412ζ413841274114413ζ4130412441174111ζ41374136415414ζ41374136415414ζ4129412641154112ζ413541284113416ζ413341314110418ζ4125412141204116ζ4140413241941ζ413441224119417ζ413941234118412ζ413841274114413ζ4130412441174111    orthogonal lifted from C41⋊C4
ρ1244000000ζ4125412141204116ζ413341314110418ζ413841274114413ζ413541284113416ζ4129412641154112ζ413941234118412ζ4130412441174111ζ41374136415414ζ4140413241941ζ413441224119417ζ413441224119417ζ4125412141204116ζ413341314110418ζ413841274114413ζ413541284113416ζ4129412641154112ζ413941234118412ζ4130412441174111ζ41374136415414ζ4140413241941    orthogonal lifted from C41⋊C4
ρ1344000000ζ413941234118412ζ4140413241941ζ4129412641154112ζ4130412441174111ζ413441224119417ζ413341314110418ζ413841274114413ζ4125412141204116ζ41374136415414ζ413541284113416ζ413541284113416ζ413941234118412ζ4140413241941ζ4129412641154112ζ4130412441174111ζ413441224119417ζ413341314110418ζ413841274114413ζ4125412141204116ζ41374136415414    orthogonal lifted from C41⋊C4
ρ1444000000ζ4130412441174111ζ4129412641154112ζ4125412141204116ζ4140413241941ζ413941234118412ζ413841274114413ζ41374136415414ζ413541284113416ζ413441224119417ζ413341314110418ζ413341314110418ζ4130412441174111ζ4129412641154112ζ4125412141204116ζ4140413241941ζ413941234118412ζ413841274114413ζ41374136415414ζ413541284113416ζ413441224119417    orthogonal lifted from C41⋊C4
ρ1544000000ζ413841274114413ζ413441224119417ζ413941234118412ζ41374136415414ζ413341314110418ζ4129412641154112ζ4125412141204116ζ4130412441174111ζ413541284113416ζ4140413241941ζ4140413241941ζ413841274114413ζ413441224119417ζ413941234118412ζ41374136415414ζ413341314110418ζ4129412641154112ζ4125412141204116ζ4130412441174111ζ413541284113416    orthogonal lifted from C41⋊C4
ρ1644000000ζ413541284113416ζ413841274114413ζ41374136415414ζ413341314110418ζ4125412141204116ζ4130412441174111ζ4140413241941ζ413441224119417ζ4129412641154112ζ413941234118412ζ413941234118412ζ413541284113416ζ413841274114413ζ41374136415414ζ413341314110418ζ4125412141204116ζ4130412441174111ζ4140413241941ζ413441224119417ζ4129412641154112    orthogonal lifted from C41⋊C4
ρ1744000000ζ413341314110418ζ41374136415414ζ413441224119417ζ413841274114413ζ413541284113416ζ4140413241941ζ4129412641154112ζ413941234118412ζ4125412141204116ζ4130412441174111ζ4130412441174111ζ413341314110418ζ41374136415414ζ413441224119417ζ413841274114413ζ413541284113416ζ4140413241941ζ4129412641154112ζ413941234118412ζ4125412141204116    orthogonal lifted from C41⋊C4
ρ1844000000ζ41374136415414ζ413941234118412ζ4130412441174111ζ413441224119417ζ413841274114413ζ4125412141204116ζ413541284113416ζ4140413241941ζ413341314110418ζ4129412641154112ζ4129412641154112ζ41374136415414ζ413941234118412ζ4130412441174111ζ413441224119417ζ413841274114413ζ4125412141204116ζ413541284113416ζ4140413241941ζ413341314110418    orthogonal lifted from C41⋊C4
ρ194-4000000ζ413341314110418ζ41374136415414ζ413441224119417ζ413841274114413ζ413541284113416ζ4140413241941ζ4129412641154112ζ413941234118412ζ4125412141204116ζ4130412441174111413041244117411141334131411041841374136415414413441224119417413841274114413413541284113416414041324194141294126411541124139412341184124125412141204116    symplectic faithful, Schur index 2
ρ204-4000000ζ4130412441174111ζ4129412641154112ζ4125412141204116ζ4140413241941ζ413941234118412ζ413841274114413ζ41374136415414ζ413541284113416ζ413441224119417ζ413341314110418413341314110418413041244117411141294126411541124125412141204116414041324194141394123411841241384127411441341374136415414413541284113416413441224119417    symplectic faithful, Schur index 2
ρ214-4000000ζ413541284113416ζ413841274114413ζ41374136415414ζ413341314110418ζ4125412141204116ζ4130412441174111ζ4140413241941ζ413441224119417ζ4129412641154112ζ413941234118412413941234118412413541284113416413841274114413413741364154144133413141104184125412141204116413041244117411141404132419414134412241194174129412641154112    symplectic faithful, Schur index 2
ρ224-4000000ζ413941234118412ζ4140413241941ζ4129412641154112ζ4130412441174111ζ413441224119417ζ413341314110418ζ413841274114413ζ4125412141204116ζ41374136415414ζ413541284113416413541284113416413941234118412414041324194141294126411541124130412441174111413441224119417413341314110418413841274114413412541214120411641374136415414    symplectic faithful, Schur index 2
ρ234-4000000ζ41374136415414ζ413941234118412ζ4130412441174111ζ413441224119417ζ413841274114413ζ4125412141204116ζ413541284113416ζ4140413241941ζ413341314110418ζ4129412641154112412941264115411241374136415414413941234118412413041244117411141344122411941741384127411441341254121412041164135412841134164140413241941413341314110418    symplectic faithful, Schur index 2
ρ244-4000000ζ4129412641154112ζ413541284113416ζ413341314110418ζ4125412141204116ζ4140413241941ζ413441224119417ζ413941234118412ζ413841274114413ζ4130412441174111ζ41374136415414413741364154144129412641154112413541284113416413341314110418412541214120411641404132419414134412241194174139412341184124138412741144134130412441174111    symplectic faithful, Schur index 2
ρ254-4000000ζ413441224119417ζ4130412441174111ζ4140413241941ζ413941234118412ζ41374136415414ζ413541284113416ζ413341314110418ζ4129412641154112ζ413841274114413ζ4125412141204116412541214120411641344122411941741304124411741114140413241941413941234118412413741364154144135412841134164133413141104184129412641154112413841274114413    symplectic faithful, Schur index 2
ρ264-4000000ζ4140413241941ζ4125412141204116ζ413541284113416ζ4129412641154112ζ4130412441174111ζ41374136415414ζ413441224119417ζ413341314110418ζ413941234118412ζ413841274114413413841274114413414041324194141254121412041164135412841134164129412641154112413041244117411141374136415414413441224119417413341314110418413941234118412    symplectic faithful, Schur index 2
ρ274-4000000ζ413841274114413ζ413441224119417ζ413941234118412ζ41374136415414ζ413341314110418ζ4129412641154112ζ4125412141204116ζ4130412441174111ζ413541284113416ζ4140413241941414041324194141384127411441341344122411941741394123411841241374136415414413341314110418412941264115411241254121412041164130412441174111413541284113416    symplectic faithful, Schur index 2
ρ284-4000000ζ4125412141204116ζ413341314110418ζ413841274114413ζ413541284113416ζ4129412641154112ζ413941234118412ζ4130412441174111ζ41374136415414ζ4140413241941ζ413441224119417413441224119417412541214120411641334131411041841384127411441341354128411341641294126411541124139412341184124130412441174111413741364154144140413241941    symplectic faithful, Schur index 2

Smallest permutation representation of C412C8
Regular action on 328 points
Generators in S328
(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)(288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328)
(1 324 124 225 74 249 83 204)(2 315 164 234 75 281 123 172)(3 306 163 243 76 272 122 181)(4 297 162 211 77 263 121 190)(5 288 161 220 78 254 120 199)(6 320 160 229 79 286 119 167)(7 311 159 238 80 277 118 176)(8 302 158 206 81 268 117 185)(9 293 157 215 82 259 116 194)(10 325 156 224 42 250 115 203)(11 316 155 233 43 282 114 171)(12 307 154 242 44 273 113 180)(13 298 153 210 45 264 112 189)(14 289 152 219 46 255 111 198)(15 321 151 228 47 287 110 166)(16 312 150 237 48 278 109 175)(17 303 149 246 49 269 108 184)(18 294 148 214 50 260 107 193)(19 326 147 223 51 251 106 202)(20 317 146 232 52 283 105 170)(21 308 145 241 53 274 104 179)(22 299 144 209 54 265 103 188)(23 290 143 218 55 256 102 197)(24 322 142 227 56 247 101 165)(25 313 141 236 57 279 100 174)(26 304 140 245 58 270 99 183)(27 295 139 213 59 261 98 192)(28 327 138 222 60 252 97 201)(29 318 137 231 61 284 96 169)(30 309 136 240 62 275 95 178)(31 300 135 208 63 266 94 187)(32 291 134 217 64 257 93 196)(33 323 133 226 65 248 92 205)(34 314 132 235 66 280 91 173)(35 305 131 244 67 271 90 182)(36 296 130 212 68 262 89 191)(37 328 129 221 69 253 88 200)(38 319 128 230 70 285 87 168)(39 310 127 239 71 276 86 177)(40 301 126 207 72 267 85 186)(41 292 125 216 73 258 84 195)

G:=sub<Sym(328)| (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)(288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328), (1,324,124,225,74,249,83,204)(2,315,164,234,75,281,123,172)(3,306,163,243,76,272,122,181)(4,297,162,211,77,263,121,190)(5,288,161,220,78,254,120,199)(6,320,160,229,79,286,119,167)(7,311,159,238,80,277,118,176)(8,302,158,206,81,268,117,185)(9,293,157,215,82,259,116,194)(10,325,156,224,42,250,115,203)(11,316,155,233,43,282,114,171)(12,307,154,242,44,273,113,180)(13,298,153,210,45,264,112,189)(14,289,152,219,46,255,111,198)(15,321,151,228,47,287,110,166)(16,312,150,237,48,278,109,175)(17,303,149,246,49,269,108,184)(18,294,148,214,50,260,107,193)(19,326,147,223,51,251,106,202)(20,317,146,232,52,283,105,170)(21,308,145,241,53,274,104,179)(22,299,144,209,54,265,103,188)(23,290,143,218,55,256,102,197)(24,322,142,227,56,247,101,165)(25,313,141,236,57,279,100,174)(26,304,140,245,58,270,99,183)(27,295,139,213,59,261,98,192)(28,327,138,222,60,252,97,201)(29,318,137,231,61,284,96,169)(30,309,136,240,62,275,95,178)(31,300,135,208,63,266,94,187)(32,291,134,217,64,257,93,196)(33,323,133,226,65,248,92,205)(34,314,132,235,66,280,91,173)(35,305,131,244,67,271,90,182)(36,296,130,212,68,262,89,191)(37,328,129,221,69,253,88,200)(38,319,128,230,70,285,87,168)(39,310,127,239,71,276,86,177)(40,301,126,207,72,267,85,186)(41,292,125,216,73,258,84,195)>;

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)(288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328), (1,324,124,225,74,249,83,204)(2,315,164,234,75,281,123,172)(3,306,163,243,76,272,122,181)(4,297,162,211,77,263,121,190)(5,288,161,220,78,254,120,199)(6,320,160,229,79,286,119,167)(7,311,159,238,80,277,118,176)(8,302,158,206,81,268,117,185)(9,293,157,215,82,259,116,194)(10,325,156,224,42,250,115,203)(11,316,155,233,43,282,114,171)(12,307,154,242,44,273,113,180)(13,298,153,210,45,264,112,189)(14,289,152,219,46,255,111,198)(15,321,151,228,47,287,110,166)(16,312,150,237,48,278,109,175)(17,303,149,246,49,269,108,184)(18,294,148,214,50,260,107,193)(19,326,147,223,51,251,106,202)(20,317,146,232,52,283,105,170)(21,308,145,241,53,274,104,179)(22,299,144,209,54,265,103,188)(23,290,143,218,55,256,102,197)(24,322,142,227,56,247,101,165)(25,313,141,236,57,279,100,174)(26,304,140,245,58,270,99,183)(27,295,139,213,59,261,98,192)(28,327,138,222,60,252,97,201)(29,318,137,231,61,284,96,169)(30,309,136,240,62,275,95,178)(31,300,135,208,63,266,94,187)(32,291,134,217,64,257,93,196)(33,323,133,226,65,248,92,205)(34,314,132,235,66,280,91,173)(35,305,131,244,67,271,90,182)(36,296,130,212,68,262,89,191)(37,328,129,221,69,253,88,200)(38,319,128,230,70,285,87,168)(39,310,127,239,71,276,86,177)(40,301,126,207,72,267,85,186)(41,292,125,216,73,258,84,195) );

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),(288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328)], [(1,324,124,225,74,249,83,204),(2,315,164,234,75,281,123,172),(3,306,163,243,76,272,122,181),(4,297,162,211,77,263,121,190),(5,288,161,220,78,254,120,199),(6,320,160,229,79,286,119,167),(7,311,159,238,80,277,118,176),(8,302,158,206,81,268,117,185),(9,293,157,215,82,259,116,194),(10,325,156,224,42,250,115,203),(11,316,155,233,43,282,114,171),(12,307,154,242,44,273,113,180),(13,298,153,210,45,264,112,189),(14,289,152,219,46,255,111,198),(15,321,151,228,47,287,110,166),(16,312,150,237,48,278,109,175),(17,303,149,246,49,269,108,184),(18,294,148,214,50,260,107,193),(19,326,147,223,51,251,106,202),(20,317,146,232,52,283,105,170),(21,308,145,241,53,274,104,179),(22,299,144,209,54,265,103,188),(23,290,143,218,55,256,102,197),(24,322,142,227,56,247,101,165),(25,313,141,236,57,279,100,174),(26,304,140,245,58,270,99,183),(27,295,139,213,59,261,98,192),(28,327,138,222,60,252,97,201),(29,318,137,231,61,284,96,169),(30,309,136,240,62,275,95,178),(31,300,135,208,63,266,94,187),(32,291,134,217,64,257,93,196),(33,323,133,226,65,248,92,205),(34,314,132,235,66,280,91,173),(35,305,131,244,67,271,90,182),(36,296,130,212,68,262,89,191),(37,328,129,221,69,253,88,200),(38,319,128,230,70,285,87,168),(39,310,127,239,71,276,86,177),(40,301,126,207,72,267,85,186),(41,292,125,216,73,258,84,195)])

Matrix representation of C412C8 in GL5(𝔽2297)

10000
02296100
02296010
02296001
020541653644242
,
13240000
0755138315422237
086216467311804
027515072522234
05236033961941

G:=sub<GL(5,GF(2297))| [1,0,0,0,0,0,2296,2296,2296,2054,0,1,0,0,1653,0,0,1,0,644,0,0,0,1,242],[1324,0,0,0,0,0,755,862,275,523,0,1383,1646,1507,603,0,1542,731,252,396,0,2237,1804,2234,1941] >;

C412C8 in GAP, Magma, Sage, TeX

C_{41}\rtimes_2C_8
% in TeX

G:=Group("C41:2C8");
// GroupNames label

G:=SmallGroup(328,3);
// by ID

G=gap.SmallGroup(328,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-41,8,21,4099,2567]);
// Polycyclic

G:=Group<a,b|a^41=b^8=1,b*a*b^-1=a^9>;
// generators/relations

Export

Subgroup lattice of C412C8 in TeX
Character table of C412C8 in TeX

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