metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.3D6, C8.1D30, C4.15D60, C60.13D4, C24.3D10, Dic60⋊2C2, C20.25D12, C12.25D20, C22.6D60, M4(2)⋊2D15, C120.1C22, C60.249C23, D60.37C22, Dic30.39C22, (C2×C30).6D4, C24⋊D5⋊2C2, C6.42(C2×D20), (C2×C6).11D20, C2.16(C2×D60), (C2×C4).15D30, C5⋊3(C8.D6), C30.270(C2×D4), (C2×C20).143D6, (C2×C10).11D12, C10.43(C2×D12), C3⋊3(C8.D10), (C3×M4(2))⋊2D5, (C5×M4(2))⋊2S3, (C2×Dic30)⋊11C2, (C2×C12).142D10, C15⋊25(C8.C22), (C15×M4(2))⋊2C2, (C2×C60).69C22, C4.30(C22×D15), C20.220(C22×S3), C12.222(C22×D5), D60⋊11C2.10C2, SmallGroup(480,874)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.D30
G = < a,b,c | a8=1, b30=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b29 >
Subgroups: 788 in 120 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, D15, C30, C30, C8.C22, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, C4○D12, Dic15, C60, D30, C2×C30, C40⋊C2, Dic20, C5×M4(2), C2×Dic10, C4○D20, C8.D6, C120, Dic30, Dic30, Dic30, C4×D15, D60, C2×Dic15, C15⋊7D4, C2×C60, C8.D10, C24⋊D5, Dic60, C15×M4(2), C2×Dic30, D60⋊11C2, C8.D30
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, D15, C8.C22, D20, C22×D5, C2×D12, D30, C2×D20, C8.D6, D60, C22×D15, C8.D10, C2×D60, C8.D30
►On 240 points
Generators in S240
(1 61 16 106 31 91 46 76)(2 92 17 77 32 62 47 107)(3 63 18 108 33 93 48 78)(4 94 19 79 34 64 49 109)(5 65 20 110 35 95 50 80)(6 96 21 81 36 66 51 111)(7 67 22 112 37 97 52 82)(8 98 23 83 38 68 53 113)(9 69 24 114 39 99 54 84)(10 100 25 85 40 70 55 115)(11 71 26 116 41 101 56 86)(12 102 27 87 42 72 57 117)(13 73 28 118 43 103 58 88)(14 104 29 89 44 74 59 119)(15 75 30 120 45 105 60 90)(121 198 166 213 151 228 136 183)(122 229 167 184 152 199 137 214)(123 200 168 215 153 230 138 185)(124 231 169 186 154 201 139 216)(125 202 170 217 155 232 140 187)(126 233 171 188 156 203 141 218)(127 204 172 219 157 234 142 189)(128 235 173 190 158 205 143 220)(129 206 174 221 159 236 144 191)(130 237 175 192 160 207 145 222)(131 208 176 223 161 238 146 193)(132 239 177 194 162 209 147 224)(133 210 178 225 163 240 148 195)(134 181 179 196 164 211 149 226)(135 212 180 227 165 182 150 197)
(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)
(1 128 31 158)(2 157 32 127)(3 126 33 156)(4 155 34 125)(5 124 35 154)(6 153 36 123)(7 122 37 152)(8 151 38 121)(9 180 39 150)(10 149 40 179)(11 178 41 148)(12 147 42 177)(13 176 43 146)(14 145 44 175)(15 174 45 144)(16 143 46 173)(17 172 47 142)(18 141 48 171)(19 170 49 140)(20 139 50 169)(21 168 51 138)(22 137 52 167)(23 166 53 136)(24 135 54 165)(25 164 55 134)(26 133 56 163)(27 162 57 132)(28 131 58 161)(29 160 59 130)(30 129 60 159)(61 220 91 190)(62 189 92 219)(63 218 93 188)(64 187 94 217)(65 216 95 186)(66 185 96 215)(67 214 97 184)(68 183 98 213)(69 212 99 182)(70 181 100 211)(71 210 101 240)(72 239 102 209)(73 208 103 238)(74 237 104 207)(75 206 105 236)(76 235 106 205)(77 204 107 234)(78 233 108 203)(79 202 109 232)(80 231 110 201)(81 200 111 230)(82 229 112 199)(83 198 113 228)(84 227 114 197)(85 196 115 226)(86 225 116 195)(87 194 117 224)(88 223 118 193)(89 192 119 222)(90 221 120 191)
G:=sub<Sym(240)| (1,61,16,106,31,91,46,76)(2,92,17,77,32,62,47,107)(3,63,18,108,33,93,48,78)(4,94,19,79,34,64,49,109)(5,65,20,110,35,95,50,80)(6,96,21,81,36,66,51,111)(7,67,22,112,37,97,52,82)(8,98,23,83,38,68,53,113)(9,69,24,114,39,99,54,84)(10,100,25,85,40,70,55,115)(11,71,26,116,41,101,56,86)(12,102,27,87,42,72,57,117)(13,73,28,118,43,103,58,88)(14,104,29,89,44,74,59,119)(15,75,30,120,45,105,60,90)(121,198,166,213,151,228,136,183)(122,229,167,184,152,199,137,214)(123,200,168,215,153,230,138,185)(124,231,169,186,154,201,139,216)(125,202,170,217,155,232,140,187)(126,233,171,188,156,203,141,218)(127,204,172,219,157,234,142,189)(128,235,173,190,158,205,143,220)(129,206,174,221,159,236,144,191)(130,237,175,192,160,207,145,222)(131,208,176,223,161,238,146,193)(132,239,177,194,162,209,147,224)(133,210,178,225,163,240,148,195)(134,181,179,196,164,211,149,226)(135,212,180,227,165,182,150,197), (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), (1,128,31,158)(2,157,32,127)(3,126,33,156)(4,155,34,125)(5,124,35,154)(6,153,36,123)(7,122,37,152)(8,151,38,121)(9,180,39,150)(10,149,40,179)(11,178,41,148)(12,147,42,177)(13,176,43,146)(14,145,44,175)(15,174,45,144)(16,143,46,173)(17,172,47,142)(18,141,48,171)(19,170,49,140)(20,139,50,169)(21,168,51,138)(22,137,52,167)(23,166,53,136)(24,135,54,165)(25,164,55,134)(26,133,56,163)(27,162,57,132)(28,131,58,161)(29,160,59,130)(30,129,60,159)(61,220,91,190)(62,189,92,219)(63,218,93,188)(64,187,94,217)(65,216,95,186)(66,185,96,215)(67,214,97,184)(68,183,98,213)(69,212,99,182)(70,181,100,211)(71,210,101,240)(72,239,102,209)(73,208,103,238)(74,237,104,207)(75,206,105,236)(76,235,106,205)(77,204,107,234)(78,233,108,203)(79,202,109,232)(80,231,110,201)(81,200,111,230)(82,229,112,199)(83,198,113,228)(84,227,114,197)(85,196,115,226)(86,225,116,195)(87,194,117,224)(88,223,118,193)(89,192,119,222)(90,221,120,191)>;
G:=Group( (1,61,16,106,31,91,46,76)(2,92,17,77,32,62,47,107)(3,63,18,108,33,93,48,78)(4,94,19,79,34,64,49,109)(5,65,20,110,35,95,50,80)(6,96,21,81,36,66,51,111)(7,67,22,112,37,97,52,82)(8,98,23,83,38,68,53,113)(9,69,24,114,39,99,54,84)(10,100,25,85,40,70,55,115)(11,71,26,116,41,101,56,86)(12,102,27,87,42,72,57,117)(13,73,28,118,43,103,58,88)(14,104,29,89,44,74,59,119)(15,75,30,120,45,105,60,90)(121,198,166,213,151,228,136,183)(122,229,167,184,152,199,137,214)(123,200,168,215,153,230,138,185)(124,231,169,186,154,201,139,216)(125,202,170,217,155,232,140,187)(126,233,171,188,156,203,141,218)(127,204,172,219,157,234,142,189)(128,235,173,190,158,205,143,220)(129,206,174,221,159,236,144,191)(130,237,175,192,160,207,145,222)(131,208,176,223,161,238,146,193)(132,239,177,194,162,209,147,224)(133,210,178,225,163,240,148,195)(134,181,179,196,164,211,149,226)(135,212,180,227,165,182,150,197), (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), (1,128,31,158)(2,157,32,127)(3,126,33,156)(4,155,34,125)(5,124,35,154)(6,153,36,123)(7,122,37,152)(8,151,38,121)(9,180,39,150)(10,149,40,179)(11,178,41,148)(12,147,42,177)(13,176,43,146)(14,145,44,175)(15,174,45,144)(16,143,46,173)(17,172,47,142)(18,141,48,171)(19,170,49,140)(20,139,50,169)(21,168,51,138)(22,137,52,167)(23,166,53,136)(24,135,54,165)(25,164,55,134)(26,133,56,163)(27,162,57,132)(28,131,58,161)(29,160,59,130)(30,129,60,159)(61,220,91,190)(62,189,92,219)(63,218,93,188)(64,187,94,217)(65,216,95,186)(66,185,96,215)(67,214,97,184)(68,183,98,213)(69,212,99,182)(70,181,100,211)(71,210,101,240)(72,239,102,209)(73,208,103,238)(74,237,104,207)(75,206,105,236)(76,235,106,205)(77,204,107,234)(78,233,108,203)(79,202,109,232)(80,231,110,201)(81,200,111,230)(82,229,112,199)(83,198,113,228)(84,227,114,197)(85,196,115,226)(86,225,116,195)(87,194,117,224)(88,223,118,193)(89,192,119,222)(90,221,120,191) );
G=PermutationGroup([[(1,61,16,106,31,91,46,76),(2,92,17,77,32,62,47,107),(3,63,18,108,33,93,48,78),(4,94,19,79,34,64,49,109),(5,65,20,110,35,95,50,80),(6,96,21,81,36,66,51,111),(7,67,22,112,37,97,52,82),(8,98,23,83,38,68,53,113),(9,69,24,114,39,99,54,84),(10,100,25,85,40,70,55,115),(11,71,26,116,41,101,56,86),(12,102,27,87,42,72,57,117),(13,73,28,118,43,103,58,88),(14,104,29,89,44,74,59,119),(15,75,30,120,45,105,60,90),(121,198,166,213,151,228,136,183),(122,229,167,184,152,199,137,214),(123,200,168,215,153,230,138,185),(124,231,169,186,154,201,139,216),(125,202,170,217,155,232,140,187),(126,233,171,188,156,203,141,218),(127,204,172,219,157,234,142,189),(128,235,173,190,158,205,143,220),(129,206,174,221,159,236,144,191),(130,237,175,192,160,207,145,222),(131,208,176,223,161,238,146,193),(132,239,177,194,162,209,147,224),(133,210,178,225,163,240,148,195),(134,181,179,196,164,211,149,226),(135,212,180,227,165,182,150,197)], [(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)], [(1,128,31,158),(2,157,32,127),(3,126,33,156),(4,155,34,125),(5,124,35,154),(6,153,36,123),(7,122,37,152),(8,151,38,121),(9,180,39,150),(10,149,40,179),(11,178,41,148),(12,147,42,177),(13,176,43,146),(14,145,44,175),(15,174,45,144),(16,143,46,173),(17,172,47,142),(18,141,48,171),(19,170,49,140),(20,139,50,169),(21,168,51,138),(22,137,52,167),(23,166,53,136),(24,135,54,165),(25,164,55,134),(26,133,56,163),(27,162,57,132),(28,131,58,161),(29,160,59,130),(30,129,60,159),(61,220,91,190),(62,189,92,219),(63,218,93,188),(64,187,94,217),(65,216,95,186),(66,185,96,215),(67,214,97,184),(68,183,98,213),(69,212,99,182),(70,181,100,211),(71,210,101,240),(72,239,102,209),(73,208,103,238),(74,237,104,207),(75,206,105,236),(76,235,106,205),(77,204,107,234),(78,233,108,203),(79,202,109,232),(80,231,110,201),(81,200,111,230),(82,229,112,199),(83,198,113,228),(84,227,114,197),(85,196,115,226),(86,225,116,195),(87,194,117,224),(88,223,118,193),(89,192,119,222),(90,221,120,191)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 60 | 2 | 2 | 2 | 60 | 60 | 60 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D12 | D12 | D15 | D20 | D20 | D30 | D30 | D60 | D60 | C8.C22 | C8.D6 | C8.D10 | C8.D30 |
| kernel | C8.D30 | C24⋊D5 | Dic60 | C15×M4(2) | C2×Dic30 | D60⋊11C2 | C5×M4(2) | C60 | C2×C30 | C3×M4(2) | C40 | C2×C20 | C24 | C2×C12 | C20 | C2×C10 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C8.D30 ►in GL4(𝔽241) generated by
| 105 | 2 | 162 | 115 |
| 0 | 101 | 194 | 47 |
| 112 | 182 | 138 | 239 |
| 119 | 171 | 239 | 138 |
| 191 | 113 | 0 | 0 |
| 107 | 57 | 0 | 0 |
| 94 | 141 | 71 | 128 |
| 66 | 194 | 113 | 163 |
| 125 | 78 | 12 | 203 |
| 31 | 2 | 191 | 219 |
| 212 | 204 | 194 | 165 |
| 212 | 8 | 163 | 161 |
G:=sub<GL(4,GF(241))| [105,0,112,119,2,101,182,171,162,194,138,239,115,47,239,138],[191,107,94,66,113,57,141,194,0,0,71,113,0,0,128,163],[125,31,212,212,78,2,204,8,12,191,194,163,203,219,165,161] >;
C8.D30 in GAP, Magma, Sage, TeX
C_8.D_{30} % in TeX
G:=Group("C8.D30"); // GroupNames label
G:=SmallGroup(480,874);
// by ID
G=gap.SmallGroup(480,874);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,58,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^30=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^29>;
// generators/relations