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G = C37⋊C8order 296 = 23·37

The semidirect product of C37 and C8 acting via C8/C2=C4

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

Aliases: C37⋊C8, C74.C4, Dic37.2C2, C2.(C37⋊C4), SmallGroup(296,3)

Series: Derived Chief Lower central Upper central

C1C37 — C37⋊C8
C1C37C74Dic37 — C37⋊C8
C37 — C37⋊C8
C1C2

Generators and relations for C37⋊C8
 G = < a,b | a37=b8=1, bab-1=a6 >

37C4
37C8

Character table of C37⋊C8

 class 124A4B8A8B8C8D37A37B37C37D37E37F37G37H37I74A74B74C74D74E74F74G74H74I
 size 11373737373737444444444444444444
ρ111111111111111111111111111    trivial
ρ21111-1-1-1-1111111111111111111    linear of order 2
ρ311-1-1-ii-ii111111111111111111    linear of order 4
ρ411-1-1i-ii-i111111111111111111    linear of order 4
ρ51-1i-iζ83ζ8ζ87ζ85111111111-1-1-1-1-1-1-1-1-1    linear of order 8
ρ61-1-iiζ85ζ87ζ8ζ83111111111-1-1-1-1-1-1-1-1-1    linear of order 8
ρ71-1-iiζ8ζ83ζ85ζ87111111111-1-1-1-1-1-1-1-1-1    linear of order 8
ρ81-1i-iζ87ζ85ζ83ζ8111111111-1-1-1-1-1-1-1-1-1    linear of order 8
ρ944000000ζ3722372137163715ζ373337243713374ζ372837203717379ζ373437193718373ζ3727372337143710ζ3736373137637ζ372937263711378ζ373537253712372ζ37323730377375ζ37323730377375ζ3722372137163715ζ373337243713374ζ372837203717379ζ373437193718373ζ3727372337143710ζ3736373137637ζ372937263711378ζ373537253712372    orthogonal lifted from C37⋊C4
ρ1044000000ζ373537253712372ζ373437193718373ζ3722372137163715ζ37323730377375ζ372937263711378ζ3727372337143710ζ3736373137637ζ372837203717379ζ373337243713374ζ373337243713374ζ373537253712372ζ373437193718373ζ3722372137163715ζ37323730377375ζ372937263711378ζ3727372337143710ζ3736373137637ζ372837203717379    orthogonal lifted from C37⋊C4
ρ1144000000ζ3727372337143710ζ3722372137163715ζ3736373137637ζ373537253712372ζ373437193718373ζ373337243713374ζ37323730377375ζ372937263711378ζ372837203717379ζ372837203717379ζ3727372337143710ζ3722372137163715ζ3736373137637ζ373537253712372ζ373437193718373ζ373337243713374ζ37323730377375ζ372937263711378    orthogonal lifted from C37⋊C4
ρ1244000000ζ37323730377375ζ372937263711378ζ373437193718373ζ3736373137637ζ372837203717379ζ373537253712372ζ3722372137163715ζ373337243713374ζ3727372337143710ζ3727372337143710ζ37323730377375ζ372937263711378ζ373437193718373ζ3736373137637ζ372837203717379ζ373537253712372ζ3722372137163715ζ373337243713374    orthogonal lifted from C37⋊C4
ρ1344000000ζ3736373137637ζ372837203717379ζ372937263711378ζ3722372137163715ζ373337243713374ζ37323730377375ζ373437193718373ζ3727372337143710ζ373537253712372ζ373537253712372ζ3736373137637ζ372837203717379ζ372937263711378ζ3722372137163715ζ373337243713374ζ37323730377375ζ373437193718373ζ3727372337143710    orthogonal lifted from C37⋊C4
ρ1444000000ζ373437193718373ζ3727372337143710ζ373337243713374ζ372937263711378ζ373537253712372ζ3722372137163715ζ372837203717379ζ37323730377375ζ3736373137637ζ3736373137637ζ373437193718373ζ3727372337143710ζ373337243713374ζ372937263711378ζ373537253712372ζ3722372137163715ζ372837203717379ζ37323730377375    orthogonal lifted from C37⋊C4
ρ1544000000ζ372837203717379ζ37323730377375ζ373537253712372ζ373337243713374ζ3736373137637ζ372937263711378ζ3727372337143710ζ3722372137163715ζ373437193718373ζ373437193718373ζ372837203717379ζ37323730377375ζ373537253712372ζ373337243713374ζ3736373137637ζ372937263711378ζ3727372337143710ζ3722372137163715    orthogonal lifted from C37⋊C4
ρ1644000000ζ372937263711378ζ373537253712372ζ3727372337143710ζ372837203717379ζ37323730377375ζ373437193718373ζ373337243713374ζ3736373137637ζ3722372137163715ζ3722372137163715ζ372937263711378ζ373537253712372ζ3727372337143710ζ372837203717379ζ37323730377375ζ373437193718373ζ373337243713374ζ3736373137637    orthogonal lifted from C37⋊C4
ρ1744000000ζ373337243713374ζ3736373137637ζ37323730377375ζ3727372337143710ζ3722372137163715ζ372837203717379ζ373537253712372ζ373437193718373ζ372937263711378ζ372937263711378ζ373337243713374ζ3736373137637ζ37323730377375ζ3727372337143710ζ3722372137163715ζ372837203717379ζ373537253712372ζ373437193718373    orthogonal lifted from C37⋊C4
ρ184-4000000ζ3736373137637ζ372837203717379ζ372937263711378ζ3722372137163715ζ373337243713374ζ37323730377375ζ373437193718373ζ3727372337143710ζ37353725371237237353725371237237363731376373728372037173793729372637113783722372137163715373337243713374373237303773753734371937183733727372337143710    symplectic faithful, Schur index 2
ρ194-4000000ζ372937263711378ζ373537253712372ζ3727372337143710ζ372837203717379ζ37323730377375ζ373437193718373ζ373337243713374ζ3736373137637ζ372237213716371537223721371637153729372637113783735372537123723727372337143710372837203717379373237303773753734371937183733733372437133743736373137637    symplectic faithful, Schur index 2
ρ204-4000000ζ3722372137163715ζ373337243713374ζ372837203717379ζ373437193718373ζ3727372337143710ζ3736373137637ζ372937263711378ζ373537253712372ζ3732373037737537323730377375372237213716371537333724371337437283720371737937343719371837337273723371437103736373137637372937263711378373537253712372    symplectic faithful, Schur index 2
ρ214-4000000ζ3727372337143710ζ3722372137163715ζ3736373137637ζ373537253712372ζ373437193718373ζ373337243713374ζ37323730377375ζ372937263711378ζ37283720371737937283720371737937273723371437103722372137163715373637313763737353725371237237343719371837337333724371337437323730377375372937263711378    symplectic faithful, Schur index 2
ρ224-4000000ζ373537253712372ζ373437193718373ζ3722372137163715ζ37323730377375ζ372937263711378ζ3727372337143710ζ3736373137637ζ372837203717379ζ37333724371337437333724371337437353725371237237343719371837337223721371637153732373037737537293726371137837273723371437103736373137637372837203717379    symplectic faithful, Schur index 2
ρ234-4000000ζ37323730377375ζ372937263711378ζ373437193718373ζ3736373137637ζ372837203717379ζ373537253712372ζ3722372137163715ζ373337243713374ζ372737233714371037273723371437103732373037737537293726371137837343719371837337363731376373728372037173793735372537123723722372137163715373337243713374    symplectic faithful, Schur index 2
ρ244-4000000ζ372837203717379ζ37323730377375ζ373537253712372ζ373337243713374ζ3736373137637ζ372937263711378ζ3727372337143710ζ3722372137163715ζ37343719371837337343719371837337283720371737937323730377375373537253712372373337243713374373637313763737293726371137837273723371437103722372137163715    symplectic faithful, Schur index 2
ρ254-4000000ζ373437193718373ζ3727372337143710ζ373337243713374ζ372937263711378ζ373537253712372ζ3722372137163715ζ372837203717379ζ37323730377375ζ373637313763737363731376373734371937183733727372337143710373337243713374372937263711378373537253712372372237213716371537283720371737937323730377375    symplectic faithful, Schur index 2
ρ264-4000000ζ373337243713374ζ3736373137637ζ37323730377375ζ3727372337143710ζ3722372137163715ζ372837203717379ζ373537253712372ζ373437193718373ζ37293726371137837293726371137837333724371337437363731376373732373037737537273723371437103722372137163715372837203717379373537253712372373437193718373    symplectic faithful, Schur index 2

Smallest permutation representation of C37⋊C8
Regular action on 296 points
Generators in S296
(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)
(1 266 112 202 74 252 75 164)(2 260 148 208 38 246 111 170)(3 291 147 214 39 240 110 176)(4 285 146 220 40 234 109 182)(5 279 145 189 41 228 108 151)(6 273 144 195 42 259 107 157)(7 267 143 201 43 253 106 163)(8 261 142 207 44 247 105 169)(9 292 141 213 45 241 104 175)(10 286 140 219 46 235 103 181)(11 280 139 188 47 229 102 150)(12 274 138 194 48 223 101 156)(13 268 137 200 49 254 100 162)(14 262 136 206 50 248 99 168)(15 293 135 212 51 242 98 174)(16 287 134 218 52 236 97 180)(17 281 133 187 53 230 96 149)(18 275 132 193 54 224 95 155)(19 269 131 199 55 255 94 161)(20 263 130 205 56 249 93 167)(21 294 129 211 57 243 92 173)(22 288 128 217 58 237 91 179)(23 282 127 186 59 231 90 185)(24 276 126 192 60 225 89 154)(25 270 125 198 61 256 88 160)(26 264 124 204 62 250 87 166)(27 295 123 210 63 244 86 172)(28 289 122 216 64 238 85 178)(29 283 121 222 65 232 84 184)(30 277 120 191 66 226 83 153)(31 271 119 197 67 257 82 159)(32 265 118 203 68 251 81 165)(33 296 117 209 69 245 80 171)(34 290 116 215 70 239 79 177)(35 284 115 221 71 233 78 183)(36 278 114 190 72 227 77 152)(37 272 113 196 73 258 76 158)

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

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

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

Matrix representation of C37⋊C8 in GL4(𝔽593) generated by

0100
0010
0001
592570157570
,
111207135562
475382118390
49012037388
3867846863
G:=sub<GL(4,GF(593))| [0,0,0,592,1,0,0,570,0,1,0,157,0,0,1,570],[111,475,490,386,207,382,120,78,135,118,37,468,562,390,388,63] >;

C37⋊C8 in GAP, Magma, Sage, TeX

C_{37}\rtimes C_8
% in TeX

G:=Group("C37:C8");
// GroupNames label

G:=SmallGroup(296,3);
// by ID

G=gap.SmallGroup(296,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-37,8,21,3971,2311]);
// Polycyclic

G:=Group<a,b|a^37=b^8=1,b*a*b^-1=a^6>;
// generators/relations

Export

Subgroup lattice of C37⋊C8 in TeX
Character table of C37⋊C8 in TeX

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