metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.69D4, C20.19D12, D6⋊Dic5⋊6C2, (C4×Dic5)⋊3S3, (C2×D12).3D5, C5⋊4(C42⋊7S3), (C12×Dic5)⋊3C2, (C10×D12).4C2, C10.57(C2×D12), C30.115(C2×D4), (C2×C20).115D6, C4.9(C5⋊D12), C15⋊7(C4.4D4), (C2×Dic30)⋊28C2, C30.39(C4○D4), (C2×C12).298D10, C12.59(C5⋊D4), C3⋊1(C20.17D4), (C2×C30).63C23, (C22×S3).9D10, C10.54(C4○D12), C6.22(D4⋊2D5), (C2×C60).142C22, (C2×Dic5).164D6, C2.11(D12⋊5D5), (C6×Dic5).186C22, (C2×Dic15).60C22, C6.11(C2×C5⋊D4), (C2×C4).155(S3×D5), C2.15(C2×C5⋊D12), (S3×C2×C10).9C22, C22.149(C2×S3×D5), (C2×C6).75(C22×D5), (C2×C10).75(C22×S3), SmallGroup(480,449)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.69D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a49, cac-1=a-1, cbc-1=a30b-1 >
Subgroups: 716 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C30, C4.4D4, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, D6⋊C4, C4×C12, C2×Dic6, C2×D12, C3×Dic5, Dic15, C60, S3×C10, C2×C30, C4×Dic5, C23.D5, C2×Dic10, D4×C10, C42⋊7S3, C6×Dic5, C5×D12, Dic30, C2×Dic15, C2×C60, S3×C2×C10, C20.17D4, D6⋊Dic5, C12×Dic5, C10×D12, C2×Dic30, C60.69D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C4.4D4, C5⋊D4, C22×D5, C2×D12, C4○D12, S3×D5, D4⋊2D5, C2×C5⋊D4, C42⋊7S3, C5⋊D12, C2×S3×D5, C20.17D4, D12⋊5D5, C2×C5⋊D12, C60.69D4
►On 240 points
Generators in S240
(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 75 186 144)(2 64 187 133)(3 113 188 122)(4 102 189 171)(5 91 190 160)(6 80 191 149)(7 69 192 138)(8 118 193 127)(9 107 194 176)(10 96 195 165)(11 85 196 154)(12 74 197 143)(13 63 198 132)(14 112 199 121)(15 101 200 170)(16 90 201 159)(17 79 202 148)(18 68 203 137)(19 117 204 126)(20 106 205 175)(21 95 206 164)(22 84 207 153)(23 73 208 142)(24 62 209 131)(25 111 210 180)(26 100 211 169)(27 89 212 158)(28 78 213 147)(29 67 214 136)(30 116 215 125)(31 105 216 174)(32 94 217 163)(33 83 218 152)(34 72 219 141)(35 61 220 130)(36 110 221 179)(37 99 222 168)(38 88 223 157)(39 77 224 146)(40 66 225 135)(41 115 226 124)(42 104 227 173)(43 93 228 162)(44 82 229 151)(45 71 230 140)(46 120 231 129)(47 109 232 178)(48 98 233 167)(49 87 234 156)(50 76 235 145)(51 65 236 134)(52 114 237 123)(53 103 238 172)(54 92 239 161)(55 81 240 150)(56 70 181 139)(57 119 182 128)(58 108 183 177)(59 97 184 166)(60 86 185 155)
(1 120 31 90)(2 119 32 89)(3 118 33 88)(4 117 34 87)(5 116 35 86)(6 115 36 85)(7 114 37 84)(8 113 38 83)(9 112 39 82)(10 111 40 81)(11 110 41 80)(12 109 42 79)(13 108 43 78)(14 107 44 77)(15 106 45 76)(16 105 46 75)(17 104 47 74)(18 103 48 73)(19 102 49 72)(20 101 50 71)(21 100 51 70)(22 99 52 69)(23 98 53 68)(24 97 54 67)(25 96 55 66)(26 95 56 65)(27 94 57 64)(28 93 58 63)(29 92 59 62)(30 91 60 61)(121 224 151 194)(122 223 152 193)(123 222 153 192)(124 221 154 191)(125 220 155 190)(126 219 156 189)(127 218 157 188)(128 217 158 187)(129 216 159 186)(130 215 160 185)(131 214 161 184)(132 213 162 183)(133 212 163 182)(134 211 164 181)(135 210 165 240)(136 209 166 239)(137 208 167 238)(138 207 168 237)(139 206 169 236)(140 205 170 235)(141 204 171 234)(142 203 172 233)(143 202 173 232)(144 201 174 231)(145 200 175 230)(146 199 176 229)(147 198 177 228)(148 197 178 227)(149 196 179 226)(150 195 180 225)
G:=sub<Sym(240)| (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), (1,75,186,144)(2,64,187,133)(3,113,188,122)(4,102,189,171)(5,91,190,160)(6,80,191,149)(7,69,192,138)(8,118,193,127)(9,107,194,176)(10,96,195,165)(11,85,196,154)(12,74,197,143)(13,63,198,132)(14,112,199,121)(15,101,200,170)(16,90,201,159)(17,79,202,148)(18,68,203,137)(19,117,204,126)(20,106,205,175)(21,95,206,164)(22,84,207,153)(23,73,208,142)(24,62,209,131)(25,111,210,180)(26,100,211,169)(27,89,212,158)(28,78,213,147)(29,67,214,136)(30,116,215,125)(31,105,216,174)(32,94,217,163)(33,83,218,152)(34,72,219,141)(35,61,220,130)(36,110,221,179)(37,99,222,168)(38,88,223,157)(39,77,224,146)(40,66,225,135)(41,115,226,124)(42,104,227,173)(43,93,228,162)(44,82,229,151)(45,71,230,140)(46,120,231,129)(47,109,232,178)(48,98,233,167)(49,87,234,156)(50,76,235,145)(51,65,236,134)(52,114,237,123)(53,103,238,172)(54,92,239,161)(55,81,240,150)(56,70,181,139)(57,119,182,128)(58,108,183,177)(59,97,184,166)(60,86,185,155), (1,120,31,90)(2,119,32,89)(3,118,33,88)(4,117,34,87)(5,116,35,86)(6,115,36,85)(7,114,37,84)(8,113,38,83)(9,112,39,82)(10,111,40,81)(11,110,41,80)(12,109,42,79)(13,108,43,78)(14,107,44,77)(15,106,45,76)(16,105,46,75)(17,104,47,74)(18,103,48,73)(19,102,49,72)(20,101,50,71)(21,100,51,70)(22,99,52,69)(23,98,53,68)(24,97,54,67)(25,96,55,66)(26,95,56,65)(27,94,57,64)(28,93,58,63)(29,92,59,62)(30,91,60,61)(121,224,151,194)(122,223,152,193)(123,222,153,192)(124,221,154,191)(125,220,155,190)(126,219,156,189)(127,218,157,188)(128,217,158,187)(129,216,159,186)(130,215,160,185)(131,214,161,184)(132,213,162,183)(133,212,163,182)(134,211,164,181)(135,210,165,240)(136,209,166,239)(137,208,167,238)(138,207,168,237)(139,206,169,236)(140,205,170,235)(141,204,171,234)(142,203,172,233)(143,202,173,232)(144,201,174,231)(145,200,175,230)(146,199,176,229)(147,198,177,228)(148,197,178,227)(149,196,179,226)(150,195,180,225)>;
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), (1,75,186,144)(2,64,187,133)(3,113,188,122)(4,102,189,171)(5,91,190,160)(6,80,191,149)(7,69,192,138)(8,118,193,127)(9,107,194,176)(10,96,195,165)(11,85,196,154)(12,74,197,143)(13,63,198,132)(14,112,199,121)(15,101,200,170)(16,90,201,159)(17,79,202,148)(18,68,203,137)(19,117,204,126)(20,106,205,175)(21,95,206,164)(22,84,207,153)(23,73,208,142)(24,62,209,131)(25,111,210,180)(26,100,211,169)(27,89,212,158)(28,78,213,147)(29,67,214,136)(30,116,215,125)(31,105,216,174)(32,94,217,163)(33,83,218,152)(34,72,219,141)(35,61,220,130)(36,110,221,179)(37,99,222,168)(38,88,223,157)(39,77,224,146)(40,66,225,135)(41,115,226,124)(42,104,227,173)(43,93,228,162)(44,82,229,151)(45,71,230,140)(46,120,231,129)(47,109,232,178)(48,98,233,167)(49,87,234,156)(50,76,235,145)(51,65,236,134)(52,114,237,123)(53,103,238,172)(54,92,239,161)(55,81,240,150)(56,70,181,139)(57,119,182,128)(58,108,183,177)(59,97,184,166)(60,86,185,155), (1,120,31,90)(2,119,32,89)(3,118,33,88)(4,117,34,87)(5,116,35,86)(6,115,36,85)(7,114,37,84)(8,113,38,83)(9,112,39,82)(10,111,40,81)(11,110,41,80)(12,109,42,79)(13,108,43,78)(14,107,44,77)(15,106,45,76)(16,105,46,75)(17,104,47,74)(18,103,48,73)(19,102,49,72)(20,101,50,71)(21,100,51,70)(22,99,52,69)(23,98,53,68)(24,97,54,67)(25,96,55,66)(26,95,56,65)(27,94,57,64)(28,93,58,63)(29,92,59,62)(30,91,60,61)(121,224,151,194)(122,223,152,193)(123,222,153,192)(124,221,154,191)(125,220,155,190)(126,219,156,189)(127,218,157,188)(128,217,158,187)(129,216,159,186)(130,215,160,185)(131,214,161,184)(132,213,162,183)(133,212,163,182)(134,211,164,181)(135,210,165,240)(136,209,166,239)(137,208,167,238)(138,207,168,237)(139,206,169,236)(140,205,170,235)(141,204,171,234)(142,203,172,233)(143,202,173,232)(144,201,174,231)(145,200,175,230)(146,199,176,229)(147,198,177,228)(148,197,178,227)(149,196,179,226)(150,195,180,225) );
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)], [(1,75,186,144),(2,64,187,133),(3,113,188,122),(4,102,189,171),(5,91,190,160),(6,80,191,149),(7,69,192,138),(8,118,193,127),(9,107,194,176),(10,96,195,165),(11,85,196,154),(12,74,197,143),(13,63,198,132),(14,112,199,121),(15,101,200,170),(16,90,201,159),(17,79,202,148),(18,68,203,137),(19,117,204,126),(20,106,205,175),(21,95,206,164),(22,84,207,153),(23,73,208,142),(24,62,209,131),(25,111,210,180),(26,100,211,169),(27,89,212,158),(28,78,213,147),(29,67,214,136),(30,116,215,125),(31,105,216,174),(32,94,217,163),(33,83,218,152),(34,72,219,141),(35,61,220,130),(36,110,221,179),(37,99,222,168),(38,88,223,157),(39,77,224,146),(40,66,225,135),(41,115,226,124),(42,104,227,173),(43,93,228,162),(44,82,229,151),(45,71,230,140),(46,120,231,129),(47,109,232,178),(48,98,233,167),(49,87,234,156),(50,76,235,145),(51,65,236,134),(52,114,237,123),(53,103,238,172),(54,92,239,161),(55,81,240,150),(56,70,181,139),(57,119,182,128),(58,108,183,177),(59,97,184,166),(60,86,185,155)], [(1,120,31,90),(2,119,32,89),(3,118,33,88),(4,117,34,87),(5,116,35,86),(6,115,36,85),(7,114,37,84),(8,113,38,83),(9,112,39,82),(10,111,40,81),(11,110,41,80),(12,109,42,79),(13,108,43,78),(14,107,44,77),(15,106,45,76),(16,105,46,75),(17,104,47,74),(18,103,48,73),(19,102,49,72),(20,101,50,71),(21,100,51,70),(22,99,52,69),(23,98,53,68),(24,97,54,67),(25,96,55,66),(26,95,56,65),(27,94,57,64),(28,93,58,63),(29,92,59,62),(30,91,60,61),(121,224,151,194),(122,223,152,193),(123,222,153,192),(124,221,154,191),(125,220,155,190),(126,219,156,189),(127,218,157,188),(128,217,158,187),(129,216,159,186),(130,215,160,185),(131,214,161,184),(132,213,162,183),(133,212,163,182),(134,211,164,181),(135,210,165,240),(136,209,166,239),(137,208,167,238),(138,207,168,237),(139,206,169,236),(140,205,170,235),(141,204,171,234),(142,203,172,233),(143,202,173,232),(144,201,174,231),(145,200,175,230),(146,199,176,229),(147,198,177,228),(148,197,178,227),(149,196,179,226),(150,195,180,225)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D12 | C5⋊D4 | C4○D12 | S3×D5 | D4⋊2D5 | C5⋊D12 | C2×S3×D5 | D12⋊5D5 |
| kernel | C60.69D4 | D6⋊Dic5 | C12×Dic5 | C10×D12 | C2×Dic30 | C4×Dic5 | C60 | C2×D12 | C2×Dic5 | C2×C20 | C30 | C2×C12 | C22×S3 | C20 | C12 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 8 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C60.69D4 ►in GL6(𝔽61)
| 60 | 46 | 0 | 0 | 0 | 0 |
| 53 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 2 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 19 |
| 0 | 0 | 0 | 0 | 48 | 2 |
| 50 | 18 | 0 | 0 | 0 | 0 |
| 34 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 59 | 0 | 0 |
| 0 | 0 | 60 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 34 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 2 | 0 | 0 |
| 0 | 0 | 2 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 48 | 1 |
G:=sub<GL(6,GF(61))| [60,53,0,0,0,0,46,1,0,0,0,0,0,0,9,2,0,0,0,0,0,34,0,0,0,0,0,0,60,48,0,0,0,0,19,2],[50,34,0,0,0,0,18,11,0,0,0,0,0,0,34,60,0,0,0,0,59,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,34,0,0,0,0,0,11,0,0,0,0,0,0,27,2,0,0,0,0,2,34,0,0,0,0,0,0,60,48,0,0,0,0,0,1] >;
C60.69D4 in GAP, Magma, Sage, TeX
C_{60}._{69}D_4 % in TeX
G:=Group("C60.69D4"); // GroupNames label
G:=SmallGroup(480,449);
// by ID
G=gap.SmallGroup(480,449);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^49,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;
// generators/relations