metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.55D4, C30.16D8, D12⋊1Dic5, C10.10D24, C30.4SD16, (C5×D12)⋊7C4, C20.34(C4×S3), C6.5(D4⋊D5), (C2×D12).1D5, C60⋊5C4⋊28C2, (C2×C20).48D6, (C2×C30).14D4, C5⋊4(C2.D24), C4.1(S3×Dic5), C15⋊6(D4⋊C4), C60.105(C2×C4), (C10×D12).2C2, (C2×C10).29D12, C3⋊1(D4⋊Dic5), C2.1(C5⋊D24), C6.1(D4.D5), C10.7(C24⋊C2), C10.42(D6⋊C4), (C2×C12).283D10, C20.10(C3⋊D4), C4.14(C15⋊D4), C12.80(C5⋊D4), C2.6(D6⋊Dic5), C12.16(C2×Dic5), C6.5(C23.D5), C30.50(C22⋊C4), (C2×C60).127C22, C2.1(D12.D5), C22.12(C5⋊D12), (C6×C5⋊2C8)⋊4C2, (C2×C5⋊2C8)⋊1S3, (C2×C4).133(S3×D5), (C2×C6).24(C5⋊D4), SmallGroup(480,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.D24
G = < a,b,c | a10=b24=1, c2=a5, bab-1=cac-1=a-1, cbc-1=a5b-1 >
Subgroups: 476 in 100 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C8, C2×C4, C2×C4, D4, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, C2×C10, C2×C10, C24, D12, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, D4⋊C4, C5⋊2C8, C2×Dic5, C2×C20, C5×D4, C22×C10, C4⋊Dic3, C2×C24, C2×D12, Dic15, C60, S3×C10, C2×C30, C2×C5⋊2C8, C4⋊Dic5, D4×C10, C2.D24, C3×C5⋊2C8, C5×D12, C5×D12, C2×Dic15, C2×C60, S3×C2×C10, D4⋊Dic5, C6×C5⋊2C8, C60⋊5C4, C10×D12, C10.D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D8, SD16, Dic5, D10, C4×S3, D12, C3⋊D4, D4⋊C4, C2×Dic5, C5⋊D4, C24⋊C2, D24, D6⋊C4, S3×D5, D4⋊D5, D4.D5, C23.D5, C2.D24, S3×Dic5, C15⋊D4, C5⋊D12, D4⋊Dic5, C5⋊D24, D12.D5, D6⋊Dic5, C10.D24
►On 240 points
Generators in S240
(1 98 80 39 219 173 127 50 207 152)(2 153 208 51 128 174 220 40 81 99)(3 100 82 41 221 175 129 52 209 154)(4 155 210 53 130 176 222 42 83 101)(5 102 84 43 223 177 131 54 211 156)(6 157 212 55 132 178 224 44 85 103)(7 104 86 45 225 179 133 56 213 158)(8 159 214 57 134 180 226 46 87 105)(9 106 88 47 227 181 135 58 215 160)(10 161 216 59 136 182 228 48 89 107)(11 108 90 25 229 183 137 60 193 162)(12 163 194 61 138 184 230 26 91 109)(13 110 92 27 231 185 139 62 195 164)(14 165 196 63 140 186 232 28 93 111)(15 112 94 29 233 187 141 64 197 166)(16 167 198 65 142 188 234 30 95 113)(17 114 96 31 235 189 143 66 199 168)(18 145 200 67 144 190 236 32 73 115)(19 116 74 33 237 191 121 68 201 146)(20 147 202 69 122 192 238 34 75 117)(21 118 76 35 239 169 123 70 203 148)(22 149 204 71 124 170 240 36 77 119)(23 120 78 37 217 171 125 72 205 150)(24 151 206 49 126 172 218 38 79 97)
(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 24 173 172)(2 171 174 23)(3 22 175 170)(4 169 176 21)(5 20 177 192)(6 191 178 19)(7 18 179 190)(8 189 180 17)(9 16 181 188)(10 187 182 15)(11 14 183 186)(12 185 184 13)(25 28 193 196)(26 195 194 27)(29 48 197 216)(30 215 198 47)(31 46 199 214)(32 213 200 45)(33 44 201 212)(34 211 202 43)(35 42 203 210)(36 209 204 41)(37 40 205 208)(38 207 206 39)(49 80 79 50)(51 78 81 72)(52 71 82 77)(53 76 83 70)(54 69 84 75)(55 74 85 68)(56 67 86 73)(57 96 87 66)(58 65 88 95)(59 94 89 64)(60 63 90 93)(61 92 91 62)(97 127 126 98)(99 125 128 120)(100 119 129 124)(101 123 130 118)(102 117 131 122)(103 121 132 116)(104 115 133 144)(105 143 134 114)(106 113 135 142)(107 141 136 112)(108 111 137 140)(109 139 138 110)(145 225 236 158)(146 157 237 224)(147 223 238 156)(148 155 239 222)(149 221 240 154)(150 153 217 220)(151 219 218 152)(159 235 226 168)(160 167 227 234)(161 233 228 166)(162 165 229 232)(163 231 230 164)
G:=sub<Sym(240)| (1,98,80,39,219,173,127,50,207,152)(2,153,208,51,128,174,220,40,81,99)(3,100,82,41,221,175,129,52,209,154)(4,155,210,53,130,176,222,42,83,101)(5,102,84,43,223,177,131,54,211,156)(6,157,212,55,132,178,224,44,85,103)(7,104,86,45,225,179,133,56,213,158)(8,159,214,57,134,180,226,46,87,105)(9,106,88,47,227,181,135,58,215,160)(10,161,216,59,136,182,228,48,89,107)(11,108,90,25,229,183,137,60,193,162)(12,163,194,61,138,184,230,26,91,109)(13,110,92,27,231,185,139,62,195,164)(14,165,196,63,140,186,232,28,93,111)(15,112,94,29,233,187,141,64,197,166)(16,167,198,65,142,188,234,30,95,113)(17,114,96,31,235,189,143,66,199,168)(18,145,200,67,144,190,236,32,73,115)(19,116,74,33,237,191,121,68,201,146)(20,147,202,69,122,192,238,34,75,117)(21,118,76,35,239,169,123,70,203,148)(22,149,204,71,124,170,240,36,77,119)(23,120,78,37,217,171,125,72,205,150)(24,151,206,49,126,172,218,38,79,97), (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), (1,24,173,172)(2,171,174,23)(3,22,175,170)(4,169,176,21)(5,20,177,192)(6,191,178,19)(7,18,179,190)(8,189,180,17)(9,16,181,188)(10,187,182,15)(11,14,183,186)(12,185,184,13)(25,28,193,196)(26,195,194,27)(29,48,197,216)(30,215,198,47)(31,46,199,214)(32,213,200,45)(33,44,201,212)(34,211,202,43)(35,42,203,210)(36,209,204,41)(37,40,205,208)(38,207,206,39)(49,80,79,50)(51,78,81,72)(52,71,82,77)(53,76,83,70)(54,69,84,75)(55,74,85,68)(56,67,86,73)(57,96,87,66)(58,65,88,95)(59,94,89,64)(60,63,90,93)(61,92,91,62)(97,127,126,98)(99,125,128,120)(100,119,129,124)(101,123,130,118)(102,117,131,122)(103,121,132,116)(104,115,133,144)(105,143,134,114)(106,113,135,142)(107,141,136,112)(108,111,137,140)(109,139,138,110)(145,225,236,158)(146,157,237,224)(147,223,238,156)(148,155,239,222)(149,221,240,154)(150,153,217,220)(151,219,218,152)(159,235,226,168)(160,167,227,234)(161,233,228,166)(162,165,229,232)(163,231,230,164)>;
G:=Group( (1,98,80,39,219,173,127,50,207,152)(2,153,208,51,128,174,220,40,81,99)(3,100,82,41,221,175,129,52,209,154)(4,155,210,53,130,176,222,42,83,101)(5,102,84,43,223,177,131,54,211,156)(6,157,212,55,132,178,224,44,85,103)(7,104,86,45,225,179,133,56,213,158)(8,159,214,57,134,180,226,46,87,105)(9,106,88,47,227,181,135,58,215,160)(10,161,216,59,136,182,228,48,89,107)(11,108,90,25,229,183,137,60,193,162)(12,163,194,61,138,184,230,26,91,109)(13,110,92,27,231,185,139,62,195,164)(14,165,196,63,140,186,232,28,93,111)(15,112,94,29,233,187,141,64,197,166)(16,167,198,65,142,188,234,30,95,113)(17,114,96,31,235,189,143,66,199,168)(18,145,200,67,144,190,236,32,73,115)(19,116,74,33,237,191,121,68,201,146)(20,147,202,69,122,192,238,34,75,117)(21,118,76,35,239,169,123,70,203,148)(22,149,204,71,124,170,240,36,77,119)(23,120,78,37,217,171,125,72,205,150)(24,151,206,49,126,172,218,38,79,97), (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), (1,24,173,172)(2,171,174,23)(3,22,175,170)(4,169,176,21)(5,20,177,192)(6,191,178,19)(7,18,179,190)(8,189,180,17)(9,16,181,188)(10,187,182,15)(11,14,183,186)(12,185,184,13)(25,28,193,196)(26,195,194,27)(29,48,197,216)(30,215,198,47)(31,46,199,214)(32,213,200,45)(33,44,201,212)(34,211,202,43)(35,42,203,210)(36,209,204,41)(37,40,205,208)(38,207,206,39)(49,80,79,50)(51,78,81,72)(52,71,82,77)(53,76,83,70)(54,69,84,75)(55,74,85,68)(56,67,86,73)(57,96,87,66)(58,65,88,95)(59,94,89,64)(60,63,90,93)(61,92,91,62)(97,127,126,98)(99,125,128,120)(100,119,129,124)(101,123,130,118)(102,117,131,122)(103,121,132,116)(104,115,133,144)(105,143,134,114)(106,113,135,142)(107,141,136,112)(108,111,137,140)(109,139,138,110)(145,225,236,158)(146,157,237,224)(147,223,238,156)(148,155,239,222)(149,221,240,154)(150,153,217,220)(151,219,218,152)(159,235,226,168)(160,167,227,234)(161,233,228,166)(162,165,229,232)(163,231,230,164) );
G=PermutationGroup([[(1,98,80,39,219,173,127,50,207,152),(2,153,208,51,128,174,220,40,81,99),(3,100,82,41,221,175,129,52,209,154),(4,155,210,53,130,176,222,42,83,101),(5,102,84,43,223,177,131,54,211,156),(6,157,212,55,132,178,224,44,85,103),(7,104,86,45,225,179,133,56,213,158),(8,159,214,57,134,180,226,46,87,105),(9,106,88,47,227,181,135,58,215,160),(10,161,216,59,136,182,228,48,89,107),(11,108,90,25,229,183,137,60,193,162),(12,163,194,61,138,184,230,26,91,109),(13,110,92,27,231,185,139,62,195,164),(14,165,196,63,140,186,232,28,93,111),(15,112,94,29,233,187,141,64,197,166),(16,167,198,65,142,188,234,30,95,113),(17,114,96,31,235,189,143,66,199,168),(18,145,200,67,144,190,236,32,73,115),(19,116,74,33,237,191,121,68,201,146),(20,147,202,69,122,192,238,34,75,117),(21,118,76,35,239,169,123,70,203,148),(22,149,204,71,124,170,240,36,77,119),(23,120,78,37,217,171,125,72,205,150),(24,151,206,49,126,172,218,38,79,97)], [(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)], [(1,24,173,172),(2,171,174,23),(3,22,175,170),(4,169,176,21),(5,20,177,192),(6,191,178,19),(7,18,179,190),(8,189,180,17),(9,16,181,188),(10,187,182,15),(11,14,183,186),(12,185,184,13),(25,28,193,196),(26,195,194,27),(29,48,197,216),(30,215,198,47),(31,46,199,214),(32,213,200,45),(33,44,201,212),(34,211,202,43),(35,42,203,210),(36,209,204,41),(37,40,205,208),(38,207,206,39),(49,80,79,50),(51,78,81,72),(52,71,82,77),(53,76,83,70),(54,69,84,75),(55,74,85,68),(56,67,86,73),(57,96,87,66),(58,65,88,95),(59,94,89,64),(60,63,90,93),(61,92,91,62),(97,127,126,98),(99,125,128,120),(100,119,129,124),(101,123,130,118),(102,117,131,122),(103,121,132,116),(104,115,133,144),(105,143,134,114),(106,113,135,142),(107,141,136,112),(108,111,137,140),(109,139,138,110),(145,225,236,158),(146,157,237,224),(147,223,238,156),(148,155,239,222),(149,221,240,154),(150,153,217,220),(151,219,218,152),(159,235,226,168),(160,167,227,234),(161,233,228,166),(162,165,229,232),(163,231,230,164)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | D6 | D8 | SD16 | Dic5 | D10 | C4×S3 | C3⋊D4 | D12 | C5⋊D4 | C5⋊D4 | C24⋊C2 | D24 | S3×D5 | D4⋊D5 | D4.D5 | S3×Dic5 | C15⋊D4 | C5⋊D12 | C5⋊D24 | D12.D5 |
| kernel | C10.D24 | C6×C5⋊2C8 | C60⋊5C4 | C10×D12 | C5×D12 | C2×C5⋊2C8 | C60 | C2×C30 | C2×D12 | C2×C20 | C30 | C30 | D12 | C2×C12 | C20 | C20 | C2×C10 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C10.D24 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 190 | 189 |
| 0 | 0 | 51 | 0 |
| 127 | 105 | 0 | 0 |
| 136 | 232 | 0 | 0 |
| 0 | 0 | 182 | 181 |
| 0 | 0 | 50 | 59 |
| 136 | 232 | 0 | 0 |
| 127 | 105 | 0 | 0 |
| 0 | 0 | 59 | 60 |
| 0 | 0 | 191 | 182 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,190,51,0,0,189,0],[127,136,0,0,105,232,0,0,0,0,182,50,0,0,181,59],[136,127,0,0,232,105,0,0,0,0,59,191,0,0,60,182] >;
C10.D24 in GAP, Magma, Sage, TeX
C_{10}.D_{24} % in TeX
G:=Group("C10.D24"); // GroupNames label
G:=SmallGroup(480,43);
// by ID
G=gap.SmallGroup(480,43);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,197,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^10=b^24=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^-1>;
// generators/relations