metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊C8.6D6, C15⋊Q8.1C4, D15⋊C8⋊4C2, C5⋊3(C8○D12), C15⋊5(C8○D4), D6.F5⋊4C2, C3⋊3(D4.F5), D6.3(C2×F5), C3⋊D4.2F5, D30.3(C2×C4), C5⋊D12.1C4, C15⋊7D4.1C4, C22.1(S3×F5), Dic3.F5⋊4C2, Dic3.5(C2×F5), Dic5.3(C4×S3), C6.22(C22×F5), C15⋊8M4(2)⋊2C2, C30.22(C22×C4), C15⋊C8.3C22, (C2×Dic5).75D6, Dic15.5(C2×C4), Dic3.D10.1C2, D30.C2.7C22, (S3×Dic5).7C22, (C3×Dic5).32C23, Dic5.34(C22×S3), (C6×Dic5).143C22, (C6×C5⋊C8)⋊4C2, (C2×C5⋊C8)⋊3S3, (S3×C5⋊C8)⋊4C2, C2.23(C2×S3×F5), C10.22(S3×C2×C4), (C2×C10).1(C4×S3), (C5×C3⋊D4).1C4, (C3×C5⋊C8).6C22, (C2×C6).22(C2×F5), (S3×C10).3(C2×C4), (C2×C30).17(C2×C4), (C5×Dic3).5(C2×C4), (C3×Dic5).24(C2×C4), SmallGroup(480,1003)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 564 in 124 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×4], C2×C4 [×3], D4 [×3], Q8, D5, C10, C10 [×2], Dic3, Dic3, C12 [×2], D6, D6, C2×C6, C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5 [×2], Dic5, C20, D10, C2×C10, C2×C10, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4, C3⋊D4, C2×C12, C5×S3, D15, C30, C30, C8○D4, C5⋊C8 [×2], C5⋊C8 [×2], Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4 [×2], C5×D4, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C2×C24, C4○D12, C5×Dic3, C3×Dic5 [×2], Dic15, S3×C10, D30, C2×C30, D5⋊C8, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C22.F5 [×2], D4⋊2D5, C8○D12, C3×C5⋊C8 [×2], C15⋊C8 [×2], S3×Dic5, D30.C2, C5⋊D12, C15⋊Q8, C6×Dic5, C5×C3⋊D4, C15⋊7D4, D4.F5, S3×C5⋊C8, D15⋊C8, D6.F5, Dic3.F5, C6×C5⋊C8, C15⋊8M4(2), Dic3.D10, C5⋊C8.D6
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, F5, C4×S3 [×2], C22×S3, C8○D4, C2×F5 [×3], S3×C2×C4, C22×F5, C8○D12, S3×F5, D4.F5, C2×S3×F5, C5⋊C8.D6
Generators and relations
G = < a,b,c,d | a5=b8=1, c6=b6, d2=b2, bab-1=cac-1=dad-1=a3, bc=cb, dbd-1=b5, dcd-1=b4c5 >
►On 240 points
Generators in S240
(1 74 227 60 211)(2 61 75 212 228)(3 213 62 229 76)(4 230 214 77 63)(5 78 231 64 215)(6 65 79 216 232)(7 193 66 233 80)(8 234 194 81 67)(9 82 235 68 195)(10 69 83 196 236)(11 197 70 237 84)(12 238 198 85 71)(13 86 239 72 199)(14 49 87 200 240)(15 201 50 217 88)(16 218 202 89 51)(17 90 219 52 203)(18 53 91 204 220)(19 205 54 221 92)(20 222 206 93 55)(21 94 223 56 207)(22 57 95 208 224)(23 209 58 225 96)(24 226 210 73 59)(25 140 103 168 191)(26 145 141 192 104)(27 169 146 105 142)(28 106 170 143 147)(29 144 107 148 171)(30 149 121 172 108)(31 173 150 109 122)(32 110 174 123 151)(33 124 111 152 175)(34 153 125 176 112)(35 177 154 113 126)(36 114 178 127 155)(37 128 115 156 179)(38 157 129 180 116)(39 181 158 117 130)(40 118 182 131 159)(41 132 119 160 183)(42 161 133 184 120)(43 185 162 97 134)(44 98 186 135 163)(45 136 99 164 187)(46 165 137 188 100)(47 189 166 101 138)(48 102 190 139 167)
(1 101 19 119 13 113 7 107)(2 102 20 120 14 114 8 108)(3 103 21 97 15 115 9 109)(4 104 22 98 16 116 10 110)(5 105 23 99 17 117 11 111)(6 106 24 100 18 118 12 112)(25 94 43 88 37 82 31 76)(26 95 44 89 38 83 32 77)(27 96 45 90 39 84 33 78)(28 73 46 91 40 85 34 79)(29 74 47 92 41 86 35 80)(30 75 48 93 42 87 36 81)(49 127 67 121 61 139 55 133)(50 128 68 122 62 140 56 134)(51 129 69 123 63 141 57 135)(52 130 70 124 64 142 58 136)(53 131 71 125 65 143 59 137)(54 132 72 126 66 144 60 138)(145 224 163 218 157 236 151 230)(146 225 164 219 158 237 152 231)(147 226 165 220 159 238 153 232)(148 227 166 221 160 239 154 233)(149 228 167 222 161 240 155 234)(150 229 168 223 162 217 156 235)(169 209 187 203 181 197 175 215)(170 210 188 204 182 198 176 216)(171 211 189 205 183 199 177 193)(172 212 190 206 184 200 178 194)(173 213 191 207 185 201 179 195)(174 214 192 208 186 202 180 196)
(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)
(1 18 19 12 13 6 7 24)(2 11 20 5 14 23 8 17)(3 4 21 22 15 16 9 10)(25 38 43 32 37 26 31 44)(27 48 45 42 39 36 33 30)(28 41 46 35 40 29 34 47)(49 58 67 52 61 70 55 64)(50 51 68 69 62 63 56 57)(53 54 71 72 65 66 59 60)(73 74 91 92 85 86 79 80)(75 84 93 78 87 96 81 90)(76 77 94 95 88 89 82 83)(97 110 115 104 109 98 103 116)(99 120 117 114 111 108 105 102)(100 113 118 107 112 101 106 119)(121 142 139 136 133 130 127 124)(122 135 140 129 134 123 128 141)(125 138 143 132 137 126 131 144)(145 150 163 168 157 162 151 156)(146 167 164 161 158 155 152 149)(147 160 165 154 159 148 153 166)(169 190 187 184 181 178 175 172)(170 183 188 177 182 171 176 189)(173 186 191 180 185 174 179 192)(193 210 211 204 205 198 199 216)(194 203 212 197 206 215 200 209)(195 196 213 214 207 208 201 202)(217 218 235 236 229 230 223 224)(219 228 237 222 231 240 225 234)(220 221 238 239 232 233 226 227)
G:=sub<Sym(240)| (1,74,227,60,211)(2,61,75,212,228)(3,213,62,229,76)(4,230,214,77,63)(5,78,231,64,215)(6,65,79,216,232)(7,193,66,233,80)(8,234,194,81,67)(9,82,235,68,195)(10,69,83,196,236)(11,197,70,237,84)(12,238,198,85,71)(13,86,239,72,199)(14,49,87,200,240)(15,201,50,217,88)(16,218,202,89,51)(17,90,219,52,203)(18,53,91,204,220)(19,205,54,221,92)(20,222,206,93,55)(21,94,223,56,207)(22,57,95,208,224)(23,209,58,225,96)(24,226,210,73,59)(25,140,103,168,191)(26,145,141,192,104)(27,169,146,105,142)(28,106,170,143,147)(29,144,107,148,171)(30,149,121,172,108)(31,173,150,109,122)(32,110,174,123,151)(33,124,111,152,175)(34,153,125,176,112)(35,177,154,113,126)(36,114,178,127,155)(37,128,115,156,179)(38,157,129,180,116)(39,181,158,117,130)(40,118,182,131,159)(41,132,119,160,183)(42,161,133,184,120)(43,185,162,97,134)(44,98,186,135,163)(45,136,99,164,187)(46,165,137,188,100)(47,189,166,101,138)(48,102,190,139,167), (1,101,19,119,13,113,7,107)(2,102,20,120,14,114,8,108)(3,103,21,97,15,115,9,109)(4,104,22,98,16,116,10,110)(5,105,23,99,17,117,11,111)(6,106,24,100,18,118,12,112)(25,94,43,88,37,82,31,76)(26,95,44,89,38,83,32,77)(27,96,45,90,39,84,33,78)(28,73,46,91,40,85,34,79)(29,74,47,92,41,86,35,80)(30,75,48,93,42,87,36,81)(49,127,67,121,61,139,55,133)(50,128,68,122,62,140,56,134)(51,129,69,123,63,141,57,135)(52,130,70,124,64,142,58,136)(53,131,71,125,65,143,59,137)(54,132,72,126,66,144,60,138)(145,224,163,218,157,236,151,230)(146,225,164,219,158,237,152,231)(147,226,165,220,159,238,153,232)(148,227,166,221,160,239,154,233)(149,228,167,222,161,240,155,234)(150,229,168,223,162,217,156,235)(169,209,187,203,181,197,175,215)(170,210,188,204,182,198,176,216)(171,211,189,205,183,199,177,193)(172,212,190,206,184,200,178,194)(173,213,191,207,185,201,179,195)(174,214,192,208,186,202,180,196), (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), (1,18,19,12,13,6,7,24)(2,11,20,5,14,23,8,17)(3,4,21,22,15,16,9,10)(25,38,43,32,37,26,31,44)(27,48,45,42,39,36,33,30)(28,41,46,35,40,29,34,47)(49,58,67,52,61,70,55,64)(50,51,68,69,62,63,56,57)(53,54,71,72,65,66,59,60)(73,74,91,92,85,86,79,80)(75,84,93,78,87,96,81,90)(76,77,94,95,88,89,82,83)(97,110,115,104,109,98,103,116)(99,120,117,114,111,108,105,102)(100,113,118,107,112,101,106,119)(121,142,139,136,133,130,127,124)(122,135,140,129,134,123,128,141)(125,138,143,132,137,126,131,144)(145,150,163,168,157,162,151,156)(146,167,164,161,158,155,152,149)(147,160,165,154,159,148,153,166)(169,190,187,184,181,178,175,172)(170,183,188,177,182,171,176,189)(173,186,191,180,185,174,179,192)(193,210,211,204,205,198,199,216)(194,203,212,197,206,215,200,209)(195,196,213,214,207,208,201,202)(217,218,235,236,229,230,223,224)(219,228,237,222,231,240,225,234)(220,221,238,239,232,233,226,227)>;
G:=Group( (1,74,227,60,211)(2,61,75,212,228)(3,213,62,229,76)(4,230,214,77,63)(5,78,231,64,215)(6,65,79,216,232)(7,193,66,233,80)(8,234,194,81,67)(9,82,235,68,195)(10,69,83,196,236)(11,197,70,237,84)(12,238,198,85,71)(13,86,239,72,199)(14,49,87,200,240)(15,201,50,217,88)(16,218,202,89,51)(17,90,219,52,203)(18,53,91,204,220)(19,205,54,221,92)(20,222,206,93,55)(21,94,223,56,207)(22,57,95,208,224)(23,209,58,225,96)(24,226,210,73,59)(25,140,103,168,191)(26,145,141,192,104)(27,169,146,105,142)(28,106,170,143,147)(29,144,107,148,171)(30,149,121,172,108)(31,173,150,109,122)(32,110,174,123,151)(33,124,111,152,175)(34,153,125,176,112)(35,177,154,113,126)(36,114,178,127,155)(37,128,115,156,179)(38,157,129,180,116)(39,181,158,117,130)(40,118,182,131,159)(41,132,119,160,183)(42,161,133,184,120)(43,185,162,97,134)(44,98,186,135,163)(45,136,99,164,187)(46,165,137,188,100)(47,189,166,101,138)(48,102,190,139,167), (1,101,19,119,13,113,7,107)(2,102,20,120,14,114,8,108)(3,103,21,97,15,115,9,109)(4,104,22,98,16,116,10,110)(5,105,23,99,17,117,11,111)(6,106,24,100,18,118,12,112)(25,94,43,88,37,82,31,76)(26,95,44,89,38,83,32,77)(27,96,45,90,39,84,33,78)(28,73,46,91,40,85,34,79)(29,74,47,92,41,86,35,80)(30,75,48,93,42,87,36,81)(49,127,67,121,61,139,55,133)(50,128,68,122,62,140,56,134)(51,129,69,123,63,141,57,135)(52,130,70,124,64,142,58,136)(53,131,71,125,65,143,59,137)(54,132,72,126,66,144,60,138)(145,224,163,218,157,236,151,230)(146,225,164,219,158,237,152,231)(147,226,165,220,159,238,153,232)(148,227,166,221,160,239,154,233)(149,228,167,222,161,240,155,234)(150,229,168,223,162,217,156,235)(169,209,187,203,181,197,175,215)(170,210,188,204,182,198,176,216)(171,211,189,205,183,199,177,193)(172,212,190,206,184,200,178,194)(173,213,191,207,185,201,179,195)(174,214,192,208,186,202,180,196), (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), (1,18,19,12,13,6,7,24)(2,11,20,5,14,23,8,17)(3,4,21,22,15,16,9,10)(25,38,43,32,37,26,31,44)(27,48,45,42,39,36,33,30)(28,41,46,35,40,29,34,47)(49,58,67,52,61,70,55,64)(50,51,68,69,62,63,56,57)(53,54,71,72,65,66,59,60)(73,74,91,92,85,86,79,80)(75,84,93,78,87,96,81,90)(76,77,94,95,88,89,82,83)(97,110,115,104,109,98,103,116)(99,120,117,114,111,108,105,102)(100,113,118,107,112,101,106,119)(121,142,139,136,133,130,127,124)(122,135,140,129,134,123,128,141)(125,138,143,132,137,126,131,144)(145,150,163,168,157,162,151,156)(146,167,164,161,158,155,152,149)(147,160,165,154,159,148,153,166)(169,190,187,184,181,178,175,172)(170,183,188,177,182,171,176,189)(173,186,191,180,185,174,179,192)(193,210,211,204,205,198,199,216)(194,203,212,197,206,215,200,209)(195,196,213,214,207,208,201,202)(217,218,235,236,229,230,223,224)(219,228,237,222,231,240,225,234)(220,221,238,239,232,233,226,227) );
G=PermutationGroup([(1,74,227,60,211),(2,61,75,212,228),(3,213,62,229,76),(4,230,214,77,63),(5,78,231,64,215),(6,65,79,216,232),(7,193,66,233,80),(8,234,194,81,67),(9,82,235,68,195),(10,69,83,196,236),(11,197,70,237,84),(12,238,198,85,71),(13,86,239,72,199),(14,49,87,200,240),(15,201,50,217,88),(16,218,202,89,51),(17,90,219,52,203),(18,53,91,204,220),(19,205,54,221,92),(20,222,206,93,55),(21,94,223,56,207),(22,57,95,208,224),(23,209,58,225,96),(24,226,210,73,59),(25,140,103,168,191),(26,145,141,192,104),(27,169,146,105,142),(28,106,170,143,147),(29,144,107,148,171),(30,149,121,172,108),(31,173,150,109,122),(32,110,174,123,151),(33,124,111,152,175),(34,153,125,176,112),(35,177,154,113,126),(36,114,178,127,155),(37,128,115,156,179),(38,157,129,180,116),(39,181,158,117,130),(40,118,182,131,159),(41,132,119,160,183),(42,161,133,184,120),(43,185,162,97,134),(44,98,186,135,163),(45,136,99,164,187),(46,165,137,188,100),(47,189,166,101,138),(48,102,190,139,167)], [(1,101,19,119,13,113,7,107),(2,102,20,120,14,114,8,108),(3,103,21,97,15,115,9,109),(4,104,22,98,16,116,10,110),(5,105,23,99,17,117,11,111),(6,106,24,100,18,118,12,112),(25,94,43,88,37,82,31,76),(26,95,44,89,38,83,32,77),(27,96,45,90,39,84,33,78),(28,73,46,91,40,85,34,79),(29,74,47,92,41,86,35,80),(30,75,48,93,42,87,36,81),(49,127,67,121,61,139,55,133),(50,128,68,122,62,140,56,134),(51,129,69,123,63,141,57,135),(52,130,70,124,64,142,58,136),(53,131,71,125,65,143,59,137),(54,132,72,126,66,144,60,138),(145,224,163,218,157,236,151,230),(146,225,164,219,158,237,152,231),(147,226,165,220,159,238,153,232),(148,227,166,221,160,239,154,233),(149,228,167,222,161,240,155,234),(150,229,168,223,162,217,156,235),(169,209,187,203,181,197,175,215),(170,210,188,204,182,198,176,216),(171,211,189,205,183,199,177,193),(172,212,190,206,184,200,178,194),(173,213,191,207,185,201,179,195),(174,214,192,208,186,202,180,196)], [(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)], [(1,18,19,12,13,6,7,24),(2,11,20,5,14,23,8,17),(3,4,21,22,15,16,9,10),(25,38,43,32,37,26,31,44),(27,48,45,42,39,36,33,30),(28,41,46,35,40,29,34,47),(49,58,67,52,61,70,55,64),(50,51,68,69,62,63,56,57),(53,54,71,72,65,66,59,60),(73,74,91,92,85,86,79,80),(75,84,93,78,87,96,81,90),(76,77,94,95,88,89,82,83),(97,110,115,104,109,98,103,116),(99,120,117,114,111,108,105,102),(100,113,118,107,112,101,106,119),(121,142,139,136,133,130,127,124),(122,135,140,129,134,123,128,141),(125,138,143,132,137,126,131,144),(145,150,163,168,157,162,151,156),(146,167,164,161,158,155,152,149),(147,160,165,154,159,148,153,166),(169,190,187,184,181,178,175,172),(170,183,188,177,182,171,176,189),(173,186,191,180,185,174,179,192),(193,210,211,204,205,198,199,216),(194,203,212,197,206,215,200,209),(195,196,213,214,207,208,201,202),(217,218,235,236,229,230,223,224),(219,228,237,222,231,240,225,234),(220,221,238,239,232,233,226,227)])
Matrix representation ►G ⊆ GL8(𝔽241)
| 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 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 240 |
| 30 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 211 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 160 | 165 | 55 |
| 0 | 0 | 0 | 0 | 210 | 215 | 222 | 100 |
| 0 | 0 | 0 | 0 | 26 | 19 | 141 | 24 |
| 0 | 0 | 0 | 0 | 186 | 184 | 196 | 81 |
| 211 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 211 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 160 | 165 | 55 |
| 0 | 0 | 0 | 0 | 210 | 215 | 222 | 100 |
| 0 | 0 | 0 | 0 | 26 | 19 | 141 | 24 |
| 0 | 0 | 0 | 0 | 186 | 184 | 196 | 81 |
| 211 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 196 | 81 | 76 | 186 |
| 0 | 0 | 0 | 0 | 31 | 26 | 19 | 141 |
| 0 | 0 | 0 | 0 | 215 | 222 | 100 | 217 |
| 0 | 0 | 0 | 0 | 55 | 57 | 45 | 160 |
G:=sub<GL(8,GF(241))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[30,60,0,0,0,0,0,0,0,211,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,45,210,26,186,0,0,0,0,160,215,19,184,0,0,0,0,165,222,141,196,0,0,0,0,55,100,24,81],[211,0,0,0,0,0,0,0,0,211,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,45,210,26,186,0,0,0,0,160,215,19,184,0,0,0,0,165,222,141,196,0,0,0,0,55,100,24,81],[211,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,196,31,215,55,0,0,0,0,81,26,222,57,0,0,0,0,76,19,100,45,0,0,0,0,186,141,217,160] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20 | 24A | ··· | 24H | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 24 | ··· | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 2 | 6 | 30 | 2 | 5 | 5 | 6 | 10 | 30 | 4 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 8 | 24 | 10 | 10 | 10 | 10 | 8 | 24 | 10 | ··· | 10 | 8 | 8 | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | F5 | C2×F5 | C2×F5 | C2×F5 | S3×F5 | D4.F5 | C2×S3×F5 | C5⋊C8.D6 |
| kernel | C5⋊C8.D6 | S3×C5⋊C8 | D15⋊C8 | D6.F5 | Dic3.F5 | C6×C5⋊C8 | C15⋊8M4(2) | Dic3.D10 | C5⋊D12 | C15⋊Q8 | C5×C3⋊D4 | C15⋊7D4 | C2×C5⋊C8 | C5⋊C8 | C2×Dic5 | Dic5 | C2×C10 | C15 | C5 | C3⋊D4 | Dic3 | D6 | C2×C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_5\rtimes C_8.D_6
% in TeX
G:=Group("C5:C8.D6"); // GroupNames label
G:=SmallGroup(480,1003);
// by ID
G=gap.SmallGroup(480,1003);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,80,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=1,c^6=b^6,d^2=b^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^4*c^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀