metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.7D12, C30.15D8, C60.77D4, D12⋊3Dic5, C30.3SD16, (C5×D12)⋊10C4, C20.33(C4×S3), C4⋊Dic5⋊10S3, (C2×D12).6D5, (C2×C20).47D6, (C2×C30).13D4, C5⋊4(C6.D8), C4.6(S3×Dic5), C15⋊5(D4⋊C4), C60.121(C2×C4), C6.13(D4⋊D5), (C10×D12).6C2, (C2×C12).49D10, C3⋊2(D4⋊Dic5), C2.2(C15⋊D8), C6.4(D4.D5), C12.1(C2×Dic5), C10.41(D6⋊C4), C10.13(D4⋊S3), C12.12(C5⋊D4), C4.21(C5⋊D12), C2.5(D6⋊Dic5), C6.4(C23.D5), C30.49(C22⋊C4), (C2×C60).182C22, C10.4(Q8⋊2S3), C2.1(C20.D6), C22.13(C15⋊D4), (C3×C4⋊Dic5)⋊7C2, (C2×C15⋊3C8)⋊14C2, (C2×C4).185(S3×D5), (C2×C6).44(C5⋊D4), (C2×C10).44(C3⋊D4), SmallGroup(480,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊Dic5
G = < a,b,c | a60=b4=1, c2=a15, bab-1=a19, cac-1=a29, cbc-1=a15b-1 >
Subgroups: 428 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, C12, C12, D6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, C2×C10, C2×C10, C3⋊C8, D12, D12, C2×C12, C2×C12, C22×S3, C5×S3, C30, D4⋊C4, C5⋊2C8, C2×Dic5, C2×C20, C5×D4, C22×C10, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C3×Dic5, C60, S3×C10, C2×C30, C2×C5⋊2C8, C4⋊Dic5, D4×C10, C6.D8, C15⋊3C8, C6×Dic5, C5×D12, C5×D12, C2×C60, S3×C2×C10, D4⋊Dic5, C3×C4⋊Dic5, C2×C15⋊3C8, C10×D12, D12⋊Dic5
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, D6⋊C4, D4⋊S3, Q8⋊2S3, S3×D5, D4⋊D5, D4.D5, C23.D5, C6.D8, S3×Dic5, C15⋊D4, C5⋊D12, D4⋊Dic5, C15⋊D8, C20.D6, D6⋊Dic5, D12⋊Dic5
►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)
(1 198 180 71)(2 217 121 90)(3 236 122 109)(4 195 123 68)(5 214 124 87)(6 233 125 106)(7 192 126 65)(8 211 127 84)(9 230 128 103)(10 189 129 62)(11 208 130 81)(12 227 131 100)(13 186 132 119)(14 205 133 78)(15 224 134 97)(16 183 135 116)(17 202 136 75)(18 221 137 94)(19 240 138 113)(20 199 139 72)(21 218 140 91)(22 237 141 110)(23 196 142 69)(24 215 143 88)(25 234 144 107)(26 193 145 66)(27 212 146 85)(28 231 147 104)(29 190 148 63)(30 209 149 82)(31 228 150 101)(32 187 151 120)(33 206 152 79)(34 225 153 98)(35 184 154 117)(36 203 155 76)(37 222 156 95)(38 181 157 114)(39 200 158 73)(40 219 159 92)(41 238 160 111)(42 197 161 70)(43 216 162 89)(44 235 163 108)(45 194 164 67)(46 213 165 86)(47 232 166 105)(48 191 167 64)(49 210 168 83)(50 229 169 102)(51 188 170 61)(52 207 171 80)(53 226 172 99)(54 185 173 118)(55 204 174 77)(56 223 175 96)(57 182 176 115)(58 201 177 74)(59 220 178 93)(60 239 179 112)
(1 213 16 228 31 183 46 198)(2 182 17 197 32 212 47 227)(3 211 18 226 33 181 48 196)(4 240 19 195 34 210 49 225)(5 209 20 224 35 239 50 194)(6 238 21 193 36 208 51 223)(7 207 22 222 37 237 52 192)(8 236 23 191 38 206 53 221)(9 205 24 220 39 235 54 190)(10 234 25 189 40 204 55 219)(11 203 26 218 41 233 56 188)(12 232 27 187 42 202 57 217)(13 201 28 216 43 231 58 186)(14 230 29 185 44 200 59 215)(15 199 30 214 45 229 60 184)(61 130 76 145 91 160 106 175)(62 159 77 174 92 129 107 144)(63 128 78 143 93 158 108 173)(64 157 79 172 94 127 109 142)(65 126 80 141 95 156 110 171)(66 155 81 170 96 125 111 140)(67 124 82 139 97 154 112 169)(68 153 83 168 98 123 113 138)(69 122 84 137 99 152 114 167)(70 151 85 166 100 121 115 136)(71 180 86 135 101 150 116 165)(72 149 87 164 102 179 117 134)(73 178 88 133 103 148 118 163)(74 147 89 162 104 177 119 132)(75 176 90 131 105 146 120 161)
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), (1,198,180,71)(2,217,121,90)(3,236,122,109)(4,195,123,68)(5,214,124,87)(6,233,125,106)(7,192,126,65)(8,211,127,84)(9,230,128,103)(10,189,129,62)(11,208,130,81)(12,227,131,100)(13,186,132,119)(14,205,133,78)(15,224,134,97)(16,183,135,116)(17,202,136,75)(18,221,137,94)(19,240,138,113)(20,199,139,72)(21,218,140,91)(22,237,141,110)(23,196,142,69)(24,215,143,88)(25,234,144,107)(26,193,145,66)(27,212,146,85)(28,231,147,104)(29,190,148,63)(30,209,149,82)(31,228,150,101)(32,187,151,120)(33,206,152,79)(34,225,153,98)(35,184,154,117)(36,203,155,76)(37,222,156,95)(38,181,157,114)(39,200,158,73)(40,219,159,92)(41,238,160,111)(42,197,161,70)(43,216,162,89)(44,235,163,108)(45,194,164,67)(46,213,165,86)(47,232,166,105)(48,191,167,64)(49,210,168,83)(50,229,169,102)(51,188,170,61)(52,207,171,80)(53,226,172,99)(54,185,173,118)(55,204,174,77)(56,223,175,96)(57,182,176,115)(58,201,177,74)(59,220,178,93)(60,239,179,112), (1,213,16,228,31,183,46,198)(2,182,17,197,32,212,47,227)(3,211,18,226,33,181,48,196)(4,240,19,195,34,210,49,225)(5,209,20,224,35,239,50,194)(6,238,21,193,36,208,51,223)(7,207,22,222,37,237,52,192)(8,236,23,191,38,206,53,221)(9,205,24,220,39,235,54,190)(10,234,25,189,40,204,55,219)(11,203,26,218,41,233,56,188)(12,232,27,187,42,202,57,217)(13,201,28,216,43,231,58,186)(14,230,29,185,44,200,59,215)(15,199,30,214,45,229,60,184)(61,130,76,145,91,160,106,175)(62,159,77,174,92,129,107,144)(63,128,78,143,93,158,108,173)(64,157,79,172,94,127,109,142)(65,126,80,141,95,156,110,171)(66,155,81,170,96,125,111,140)(67,124,82,139,97,154,112,169)(68,153,83,168,98,123,113,138)(69,122,84,137,99,152,114,167)(70,151,85,166,100,121,115,136)(71,180,86,135,101,150,116,165)(72,149,87,164,102,179,117,134)(73,178,88,133,103,148,118,163)(74,147,89,162,104,177,119,132)(75,176,90,131,105,146,120,161)>;
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), (1,198,180,71)(2,217,121,90)(3,236,122,109)(4,195,123,68)(5,214,124,87)(6,233,125,106)(7,192,126,65)(8,211,127,84)(9,230,128,103)(10,189,129,62)(11,208,130,81)(12,227,131,100)(13,186,132,119)(14,205,133,78)(15,224,134,97)(16,183,135,116)(17,202,136,75)(18,221,137,94)(19,240,138,113)(20,199,139,72)(21,218,140,91)(22,237,141,110)(23,196,142,69)(24,215,143,88)(25,234,144,107)(26,193,145,66)(27,212,146,85)(28,231,147,104)(29,190,148,63)(30,209,149,82)(31,228,150,101)(32,187,151,120)(33,206,152,79)(34,225,153,98)(35,184,154,117)(36,203,155,76)(37,222,156,95)(38,181,157,114)(39,200,158,73)(40,219,159,92)(41,238,160,111)(42,197,161,70)(43,216,162,89)(44,235,163,108)(45,194,164,67)(46,213,165,86)(47,232,166,105)(48,191,167,64)(49,210,168,83)(50,229,169,102)(51,188,170,61)(52,207,171,80)(53,226,172,99)(54,185,173,118)(55,204,174,77)(56,223,175,96)(57,182,176,115)(58,201,177,74)(59,220,178,93)(60,239,179,112), (1,213,16,228,31,183,46,198)(2,182,17,197,32,212,47,227)(3,211,18,226,33,181,48,196)(4,240,19,195,34,210,49,225)(5,209,20,224,35,239,50,194)(6,238,21,193,36,208,51,223)(7,207,22,222,37,237,52,192)(8,236,23,191,38,206,53,221)(9,205,24,220,39,235,54,190)(10,234,25,189,40,204,55,219)(11,203,26,218,41,233,56,188)(12,232,27,187,42,202,57,217)(13,201,28,216,43,231,58,186)(14,230,29,185,44,200,59,215)(15,199,30,214,45,229,60,184)(61,130,76,145,91,160,106,175)(62,159,77,174,92,129,107,144)(63,128,78,143,93,158,108,173)(64,157,79,172,94,127,109,142)(65,126,80,141,95,156,110,171)(66,155,81,170,96,125,111,140)(67,124,82,139,97,154,112,169)(68,153,83,168,98,123,113,138)(69,122,84,137,99,152,114,167)(70,151,85,166,100,121,115,136)(71,180,86,135,101,150,116,165)(72,149,87,164,102,179,117,134)(73,178,88,133,103,148,118,163)(74,147,89,162,104,177,119,132)(75,176,90,131,105,146,120,161) );
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)], [(1,198,180,71),(2,217,121,90),(3,236,122,109),(4,195,123,68),(5,214,124,87),(6,233,125,106),(7,192,126,65),(8,211,127,84),(9,230,128,103),(10,189,129,62),(11,208,130,81),(12,227,131,100),(13,186,132,119),(14,205,133,78),(15,224,134,97),(16,183,135,116),(17,202,136,75),(18,221,137,94),(19,240,138,113),(20,199,139,72),(21,218,140,91),(22,237,141,110),(23,196,142,69),(24,215,143,88),(25,234,144,107),(26,193,145,66),(27,212,146,85),(28,231,147,104),(29,190,148,63),(30,209,149,82),(31,228,150,101),(32,187,151,120),(33,206,152,79),(34,225,153,98),(35,184,154,117),(36,203,155,76),(37,222,156,95),(38,181,157,114),(39,200,158,73),(40,219,159,92),(41,238,160,111),(42,197,161,70),(43,216,162,89),(44,235,163,108),(45,194,164,67),(46,213,165,86),(47,232,166,105),(48,191,167,64),(49,210,168,83),(50,229,169,102),(51,188,170,61),(52,207,171,80),(53,226,172,99),(54,185,173,118),(55,204,174,77),(56,223,175,96),(57,182,176,115),(58,201,177,74),(59,220,178,93),(60,239,179,112)], [(1,213,16,228,31,183,46,198),(2,182,17,197,32,212,47,227),(3,211,18,226,33,181,48,196),(4,240,19,195,34,210,49,225),(5,209,20,224,35,239,50,194),(6,238,21,193,36,208,51,223),(7,207,22,222,37,237,52,192),(8,236,23,191,38,206,53,221),(9,205,24,220,39,235,54,190),(10,234,25,189,40,204,55,219),(11,203,26,218,41,233,56,188),(12,232,27,187,42,202,57,217),(13,201,28,216,43,231,58,186),(14,230,29,185,44,200,59,215),(15,199,30,214,45,229,60,184),(61,130,76,145,91,160,106,175),(62,159,77,174,92,129,107,144),(63,128,78,143,93,158,108,173),(64,157,79,172,94,127,109,142),(65,126,80,141,95,156,110,171),(66,155,81,170,96,125,111,140),(67,124,82,139,97,154,112,169),(68,153,83,168,98,123,113,138),(69,122,84,137,99,152,114,167),(70,151,85,166,100,121,115,136),(71,180,86,135,101,150,116,165),(72,149,87,164,102,179,117,134),(73,178,88,133,103,148,118,163),(74,147,89,162,104,177,119,132),(75,176,90,131,105,146,120,161)]])
60 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 | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | D12 | C3⋊D4 | C5⋊D4 | C5⋊D4 | D4⋊S3 | Q8⋊2S3 | S3×D5 | D4⋊D5 | D4.D5 | S3×Dic5 | C5⋊D12 | C15⋊D4 | C15⋊D8 | C20.D6 |
| kernel | D12⋊Dic5 | C3×C4⋊Dic5 | C2×C15⋊3C8 | C10×D12 | C5×D12 | C4⋊Dic5 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of D12⋊Dic5 ►in GL6(𝔽241)
| 2 | 49 | 0 | 0 | 0 | 0 |
| 177 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 51 | 0 | 0 |
| 0 | 0 | 190 | 51 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 164 |
| 0 | 0 | 0 | 0 | 144 | 1 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 166 | 0 | 0 |
| 0 | 0 | 104 | 150 | 0 | 0 |
| 0 | 0 | 0 | 0 | 203 | 224 |
| 0 | 0 | 0 | 0 | 85 | 38 |
| 177 | 238 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 150 | 75 | 0 | 0 |
| 0 | 0 | 137 | 91 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 224 |
| 0 | 0 | 0 | 0 | 85 | 38 |
G:=sub<GL(6,GF(241))| [2,177,0,0,0,0,49,240,0,0,0,0,0,0,1,190,0,0,0,0,51,51,0,0,0,0,0,0,240,144,0,0,0,0,164,1],[177,0,0,0,0,0,0,177,0,0,0,0,0,0,91,104,0,0,0,0,166,150,0,0,0,0,0,0,203,85,0,0,0,0,224,38],[177,0,0,0,0,0,238,64,0,0,0,0,0,0,150,137,0,0,0,0,75,91,0,0,0,0,0,0,0,85,0,0,0,0,224,38] >;
D12⋊Dic5 in GAP, Magma, Sage, TeX
D_{12}\rtimes {\rm Dic}_5 % in TeX
G:=Group("D12:Dic5"); // GroupNames label
G:=SmallGroup(480,42);
// by ID
G=gap.SmallGroup(480,42);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^15,b*a*b^-1=a^19,c*a*c^-1=a^29,c*b*c^-1=a^15*b^-1>;
// generators/relations