metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.2D10, C40.28D6, Dic12⋊2D5, D10.15D12, C120.29C22, C60.122C23, Dic5.17D12, Dic6.20D10, D60.32C22, Dic30.34C22, C8.2(S3×D5), C6.7(D4×D5), C8⋊D5⋊4S3, (C4×D5).4D6, (C6×D5).4D4, C5⋊2C8.2D6, C30.15(C2×D4), C2.12(D5×D12), C10.7(C2×D12), C24⋊D5⋊10C2, C5⋊2(C8.D6), (C5×Dic12)⋊4C2, (D5×Dic6)⋊10C2, C3⋊1(Q16⋊D5), C5⋊Dic12⋊11C2, C15⋊2(C8.C22), (C3×Dic5).4D4, C12.28D10.2C2, Dic6⋊D5⋊11C2, C20.74(C22×S3), (D5×C12).27C22, C12.145(C22×D5), (C5×Dic6).24C22, C4.70(C2×S3×D5), (C3×C8⋊D5)⋊4C2, (C3×C5⋊2C8).20C22, SmallGroup(480,337)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.2D10
G = < a,b,c | a40=b6=1, c2=a20, bab-1=a29, cac-1=a19, cbc-1=b-1 >
Subgroups: 732 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C24, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D5, D15, C30, C8.C22, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, C5×Q8, C24⋊C2, Dic12, Dic12, C3×M4(2), C2×Dic6, C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, C8.D6, C3×C5⋊2C8, C120, D5×Dic3, D30.C2, C3⋊D20, C15⋊Q8, D5×C12, C5×Dic6, Dic30, D60, Q16⋊D5, Dic6⋊D5, C5⋊Dic12, C3×C8⋊D5, C5×Dic12, C24⋊D5, D5×Dic6, C12.28D10, C24.2D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C8.C22, C22×D5, C2×D12, S3×D5, D4×D5, C8.D6, C2×S3×D5, Q16⋊D5, D5×D12, C24.2D10
►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 84 76)(2 113 77 30 85 65)(3 102 78 19 86 54)(4 91 79 8 87 43)(5 120 80 37 88 72)(6 109 41 26 89 61)(7 98 42 15 90 50)(9 116 44 33 92 68)(10 105 45 22 93 57)(11 94 46)(12 83 47 40 95 75)(13 112 48 29 96 64)(14 101 49 18 97 53)(16 119 51 36 99 71)(17 108 52 25 100 60)(20 115 55 32 103 67)(21 104 56)(23 82 58 39 106 74)(24 111 59 28 107 63)(27 118 62 35 110 70)(31 114 66)(34 81 69 38 117 73)(121 210 193 153 218 185)(122 239 194 142 219 174)(123 228 195 131 220 163)(124 217 196 160 221 192)(125 206 197 149 222 181)(126 235 198 138 223 170)(127 224 199)(128 213 200 156 225 188)(129 202 161 145 226 177)(130 231 162 134 227 166)(132 209 164 152 229 184)(133 238 165 141 230 173)(135 216 167 159 232 191)(136 205 168 148 233 180)(137 234 169)(139 212 171 155 236 187)(140 201 172 144 237 176)(143 208 175 151 240 183)(146 215 178 158 203 190)(147 204 179)(150 211 182 154 207 186)(157 214 189)
(1 199 21 179)(2 178 22 198)(3 197 23 177)(4 176 24 196)(5 195 25 175)(6 174 26 194)(7 193 27 173)(8 172 28 192)(9 191 29 171)(10 170 30 190)(11 189 31 169)(12 168 32 188)(13 187 33 167)(14 166 34 186)(15 185 35 165)(16 164 36 184)(17 183 37 163)(18 162 38 182)(19 181 39 161)(20 200 40 180)(41 142 61 122)(42 121 62 141)(43 140 63 160)(44 159 64 139)(45 138 65 158)(46 157 66 137)(47 136 67 156)(48 155 68 135)(49 134 69 154)(50 153 70 133)(51 132 71 152)(52 151 72 131)(53 130 73 150)(54 149 74 129)(55 128 75 148)(56 147 76 127)(57 126 77 146)(58 145 78 125)(59 124 79 144)(60 143 80 123)(81 207 101 227)(82 226 102 206)(83 205 103 225)(84 224 104 204)(85 203 105 223)(86 222 106 202)(87 201 107 221)(88 220 108 240)(89 239 109 219)(90 218 110 238)(91 237 111 217)(92 216 112 236)(93 235 113 215)(94 214 114 234)(95 233 115 213)(96 212 116 232)(97 231 117 211)(98 210 118 230)(99 229 119 209)(100 208 120 228)
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), (1,84,76)(2,113,77,30,85,65)(3,102,78,19,86,54)(4,91,79,8,87,43)(5,120,80,37,88,72)(6,109,41,26,89,61)(7,98,42,15,90,50)(9,116,44,33,92,68)(10,105,45,22,93,57)(11,94,46)(12,83,47,40,95,75)(13,112,48,29,96,64)(14,101,49,18,97,53)(16,119,51,36,99,71)(17,108,52,25,100,60)(20,115,55,32,103,67)(21,104,56)(23,82,58,39,106,74)(24,111,59,28,107,63)(27,118,62,35,110,70)(31,114,66)(34,81,69,38,117,73)(121,210,193,153,218,185)(122,239,194,142,219,174)(123,228,195,131,220,163)(124,217,196,160,221,192)(125,206,197,149,222,181)(126,235,198,138,223,170)(127,224,199)(128,213,200,156,225,188)(129,202,161,145,226,177)(130,231,162,134,227,166)(132,209,164,152,229,184)(133,238,165,141,230,173)(135,216,167,159,232,191)(136,205,168,148,233,180)(137,234,169)(139,212,171,155,236,187)(140,201,172,144,237,176)(143,208,175,151,240,183)(146,215,178,158,203,190)(147,204,179)(150,211,182,154,207,186)(157,214,189), (1,199,21,179)(2,178,22,198)(3,197,23,177)(4,176,24,196)(5,195,25,175)(6,174,26,194)(7,193,27,173)(8,172,28,192)(9,191,29,171)(10,170,30,190)(11,189,31,169)(12,168,32,188)(13,187,33,167)(14,166,34,186)(15,185,35,165)(16,164,36,184)(17,183,37,163)(18,162,38,182)(19,181,39,161)(20,200,40,180)(41,142,61,122)(42,121,62,141)(43,140,63,160)(44,159,64,139)(45,138,65,158)(46,157,66,137)(47,136,67,156)(48,155,68,135)(49,134,69,154)(50,153,70,133)(51,132,71,152)(52,151,72,131)(53,130,73,150)(54,149,74,129)(55,128,75,148)(56,147,76,127)(57,126,77,146)(58,145,78,125)(59,124,79,144)(60,143,80,123)(81,207,101,227)(82,226,102,206)(83,205,103,225)(84,224,104,204)(85,203,105,223)(86,222,106,202)(87,201,107,221)(88,220,108,240)(89,239,109,219)(90,218,110,238)(91,237,111,217)(92,216,112,236)(93,235,113,215)(94,214,114,234)(95,233,115,213)(96,212,116,232)(97,231,117,211)(98,210,118,230)(99,229,119,209)(100,208,120,228)>;
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), (1,84,76)(2,113,77,30,85,65)(3,102,78,19,86,54)(4,91,79,8,87,43)(5,120,80,37,88,72)(6,109,41,26,89,61)(7,98,42,15,90,50)(9,116,44,33,92,68)(10,105,45,22,93,57)(11,94,46)(12,83,47,40,95,75)(13,112,48,29,96,64)(14,101,49,18,97,53)(16,119,51,36,99,71)(17,108,52,25,100,60)(20,115,55,32,103,67)(21,104,56)(23,82,58,39,106,74)(24,111,59,28,107,63)(27,118,62,35,110,70)(31,114,66)(34,81,69,38,117,73)(121,210,193,153,218,185)(122,239,194,142,219,174)(123,228,195,131,220,163)(124,217,196,160,221,192)(125,206,197,149,222,181)(126,235,198,138,223,170)(127,224,199)(128,213,200,156,225,188)(129,202,161,145,226,177)(130,231,162,134,227,166)(132,209,164,152,229,184)(133,238,165,141,230,173)(135,216,167,159,232,191)(136,205,168,148,233,180)(137,234,169)(139,212,171,155,236,187)(140,201,172,144,237,176)(143,208,175,151,240,183)(146,215,178,158,203,190)(147,204,179)(150,211,182,154,207,186)(157,214,189), (1,199,21,179)(2,178,22,198)(3,197,23,177)(4,176,24,196)(5,195,25,175)(6,174,26,194)(7,193,27,173)(8,172,28,192)(9,191,29,171)(10,170,30,190)(11,189,31,169)(12,168,32,188)(13,187,33,167)(14,166,34,186)(15,185,35,165)(16,164,36,184)(17,183,37,163)(18,162,38,182)(19,181,39,161)(20,200,40,180)(41,142,61,122)(42,121,62,141)(43,140,63,160)(44,159,64,139)(45,138,65,158)(46,157,66,137)(47,136,67,156)(48,155,68,135)(49,134,69,154)(50,153,70,133)(51,132,71,152)(52,151,72,131)(53,130,73,150)(54,149,74,129)(55,128,75,148)(56,147,76,127)(57,126,77,146)(58,145,78,125)(59,124,79,144)(60,143,80,123)(81,207,101,227)(82,226,102,206)(83,205,103,225)(84,224,104,204)(85,203,105,223)(86,222,106,202)(87,201,107,221)(88,220,108,240)(89,239,109,219)(90,218,110,238)(91,237,111,217)(92,216,112,236)(93,235,113,215)(94,214,114,234)(95,233,115,213)(96,212,116,232)(97,231,117,211)(98,210,118,230)(99,229,119,209)(100,208,120,228) );
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)], [(1,84,76),(2,113,77,30,85,65),(3,102,78,19,86,54),(4,91,79,8,87,43),(5,120,80,37,88,72),(6,109,41,26,89,61),(7,98,42,15,90,50),(9,116,44,33,92,68),(10,105,45,22,93,57),(11,94,46),(12,83,47,40,95,75),(13,112,48,29,96,64),(14,101,49,18,97,53),(16,119,51,36,99,71),(17,108,52,25,100,60),(20,115,55,32,103,67),(21,104,56),(23,82,58,39,106,74),(24,111,59,28,107,63),(27,118,62,35,110,70),(31,114,66),(34,81,69,38,117,73),(121,210,193,153,218,185),(122,239,194,142,219,174),(123,228,195,131,220,163),(124,217,196,160,221,192),(125,206,197,149,222,181),(126,235,198,138,223,170),(127,224,199),(128,213,200,156,225,188),(129,202,161,145,226,177),(130,231,162,134,227,166),(132,209,164,152,229,184),(133,238,165,141,230,173),(135,216,167,159,232,191),(136,205,168,148,233,180),(137,234,169),(139,212,171,155,236,187),(140,201,172,144,237,176),(143,208,175,151,240,183),(146,215,178,158,203,190),(147,204,179),(150,211,182,154,207,186),(157,214,189)], [(1,199,21,179),(2,178,22,198),(3,197,23,177),(4,176,24,196),(5,195,25,175),(6,174,26,194),(7,193,27,173),(8,172,28,192),(9,191,29,171),(10,170,30,190),(11,189,31,169),(12,168,32,188),(13,187,33,167),(14,166,34,186),(15,185,35,165),(16,164,36,184),(17,183,37,163),(18,162,38,182),(19,181,39,161),(20,200,40,180),(41,142,61,122),(42,121,62,141),(43,140,63,160),(44,159,64,139),(45,138,65,158),(46,157,66,137),(47,136,67,156),(48,155,68,135),(49,134,69,154),(50,153,70,133),(51,132,71,152),(52,151,72,131),(53,130,73,150),(54,149,74,129),(55,128,75,148),(56,147,76,127),(57,126,77,146),(58,145,78,125),(59,124,79,144),(60,143,80,123),(81,207,101,227),(82,226,102,206),(83,205,103,225),(84,224,104,204),(85,203,105,223),(86,222,106,202),(87,201,107,221),(88,220,108,240),(89,239,109,219),(90,218,110,238),(91,237,111,217),(92,216,112,236),(93,235,113,215),(94,214,114,234),(95,233,115,213),(96,212,116,232),(97,231,117,211),(98,210,118,230),(99,229,119,209),(100,208,120,228)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 10 | 60 | 2 | 2 | 10 | 12 | 12 | 60 | 2 | 2 | 2 | 20 | 4 | 20 | 2 | 2 | 2 | 2 | 20 | 4 | 4 | 4 | 4 | 24 | 24 | 24 | 24 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D12 | D12 | C8.C22 | S3×D5 | D4×D5 | C8.D6 | C2×S3×D5 | Q16⋊D5 | D5×D12 | C24.2D10 |
| kernel | C24.2D10 | Dic6⋊D5 | C5⋊Dic12 | C3×C8⋊D5 | C5×Dic12 | C24⋊D5 | D5×Dic6 | C12.28D10 | C8⋊D5 | C3×Dic5 | C6×D5 | Dic12 | C5⋊2C8 | C40 | C4×D5 | C24 | Dic6 | Dic5 | D10 | C15 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C24.2D10 ►in GL8(𝔽241)
| 0 | 12 | 0 | 24 | 0 | 0 | 0 | 0 |
| 87 | 87 | 174 | 174 | 0 | 0 | 0 | 0 |
| 0 | 217 | 0 | 229 | 0 | 0 | 0 | 0 |
| 67 | 67 | 154 | 154 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 142 | 98 | 5 | 0 |
| 0 | 0 | 0 | 0 | 143 | 44 | 0 | 5 |
| 0 | 0 | 0 | 0 | 125 | 9 | 99 | 143 |
| 0 | 0 | 0 | 0 | 232 | 116 | 98 | 197 |
| 52 | 51 | 52 | 51 | 0 | 0 | 0 | 0 |
| 188 | 189 | 188 | 189 | 0 | 0 | 0 | 0 |
| 189 | 190 | 0 | 0 | 0 | 0 | 0 | 0 |
| 53 | 52 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 114 | 105 | 1 | 1 |
| 0 | 0 | 0 | 0 | 136 | 9 | 240 | 0 |
| 77 | 6 | 192 | 40 | 0 | 0 | 0 | 0 |
| 93 | 164 | 138 | 49 | 0 | 0 | 0 | 0 |
| 115 | 34 | 164 | 235 | 0 | 0 | 0 | 0 |
| 45 | 126 | 148 | 77 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 148 | 95 | 0 | 0 |
| 0 | 0 | 0 | 0 | 188 | 93 | 0 | 0 |
| 0 | 0 | 0 | 0 | 130 | 20 | 180 | 137 |
| 0 | 0 | 0 | 0 | 131 | 111 | 198 | 61 |
G:=sub<GL(8,GF(241))| [0,87,0,67,0,0,0,0,12,87,217,67,0,0,0,0,0,174,0,154,0,0,0,0,24,174,229,154,0,0,0,0,0,0,0,0,142,143,125,232,0,0,0,0,98,44,9,116,0,0,0,0,5,0,99,98,0,0,0,0,0,5,143,197],[52,188,189,53,0,0,0,0,51,189,190,52,0,0,0,0,52,188,0,0,0,0,0,0,51,189,0,0,0,0,0,0,0,0,0,0,240,1,114,136,0,0,0,0,240,0,105,9,0,0,0,0,0,0,1,240,0,0,0,0,0,0,1,0],[77,93,115,45,0,0,0,0,6,164,34,126,0,0,0,0,192,138,164,148,0,0,0,0,40,49,235,77,0,0,0,0,0,0,0,0,148,188,130,131,0,0,0,0,95,93,20,111,0,0,0,0,0,0,180,198,0,0,0,0,0,0,137,61] >;
C24.2D10 in GAP, Magma, Sage, TeX
C_{24}._2D_{10} % in TeX
G:=Group("C24.2D10"); // GroupNames label
G:=SmallGroup(480,337);
// by ID
G=gap.SmallGroup(480,337);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,135,142,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^29,c*a*c^-1=a^19,c*b*c^-1=b^-1>;
// generators/relations