metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.8D12, D6⋊C4⋊1D5, C6.11(D4×D5), D6⋊Dic5⋊2C2, (C4×Dic5)⋊8S3, C30.31(C2×D4), C2.16(D5×D12), D30⋊4C4⋊3C2, C5⋊1(C42⋊7S3), C10.11(C2×D12), (C2×C20).179D6, C15⋊2(C4.4D4), D30⋊3C4⋊24C2, (C12×Dic5)⋊20C2, C6.17(C4○D20), (C2×C12).258D10, (C2×C30).40C23, (C2×Dic3).7D10, (C3×Dic5).50D4, (C22×S3).4D10, C10.20(C4○D12), C30.105(C4○D4), C6.38(D4⋊2D5), (C2×C60).381C22, (C2×Dic5).158D6, C3⋊1(Dic5.5D4), C2.12(Dic3.D10), C2.10(D6.D10), (C6×Dic5).179C22, (C2×Dic15).45C22, (C10×Dic3).24C22, (C22×D15).19C22, (C2×C15⋊Q8)⋊2C2, (C5×D6⋊C4)⋊24C2, (C2×C4).121(S3×D5), (C2×C5⋊D12).2C2, (S3×C2×C10).4C22, C22.129(C2×S3×D5), (C2×C6).52(C22×D5), (C2×C10).52(C22×S3), SmallGroup(480,426)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.8D12
G = < a,b,c,d | a10=c12=1, b2=d2=a5, bab-1=cac-1=dad-1=a-1, bc=cb, dbd-1=a5b, dcd-1=a5c-1 >
Subgroups: 908 in 152 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C2×C10, Dic6, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C4.4D4, Dic10, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, D6⋊C4, D6⋊C4, C4×C12, C2×Dic6, C2×D12, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, S3×C10, D30, C2×C30, C4×Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C42⋊7S3, C5⋊D12, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, Dic5.5D4, D6⋊Dic5, D30⋊4C4, C12×Dic5, C5×D6⋊C4, D30⋊3C4, C2×C5⋊D12, C2×C15⋊Q8, Dic5.8D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C4.4D4, C22×D5, C2×D12, C4○D12, S3×D5, C4○D20, D4×D5, D4⋊2D5, C42⋊7S3, C2×S3×D5, Dic5.5D4, D6.D10, D5×D12, Dic3.D10, Dic5.8D12
►On 240 points
Generators in S240
(1 219 193 18 66 181 93 38 162 229)(2 230 163 39 94 182 67 19 194 220)(3 221 195 20 68 183 95 40 164 231)(4 232 165 41 96 184 69 21 196 222)(5 223 197 22 70 185 85 42 166 233)(6 234 167 43 86 186 71 23 198 224)(7 225 199 24 72 187 87 44 168 235)(8 236 157 45 88 188 61 13 200 226)(9 227 201 14 62 189 89 46 158 237)(10 238 159 47 90 190 63 15 202 228)(11 217 203 16 64 191 91 48 160 239)(12 240 161 37 92 192 65 17 204 218)(25 122 151 143 113 53 97 79 180 216)(26 205 169 80 98 54 114 144 152 123)(27 124 153 133 115 55 99 81 170 206)(28 207 171 82 100 56 116 134 154 125)(29 126 155 135 117 57 101 83 172 208)(30 209 173 84 102 58 118 136 156 127)(31 128 145 137 119 59 103 73 174 210)(32 211 175 74 104 60 120 138 146 129)(33 130 147 139 109 49 105 75 176 212)(34 213 177 76 106 50 110 140 148 131)(35 132 149 141 111 51 107 77 178 214)(36 215 179 78 108 52 112 142 150 121)
(1 52 181 36)(2 53 182 25)(3 54 183 26)(4 55 184 27)(5 56 185 28)(6 57 186 29)(7 58 187 30)(8 59 188 31)(9 60 189 32)(10 49 190 33)(11 50 191 34)(12 51 192 35)(13 174 157 137)(14 175 158 138)(15 176 159 139)(16 177 160 140)(17 178 161 141)(18 179 162 142)(19 180 163 143)(20 169 164 144)(21 170 165 133)(22 171 166 134)(23 172 167 135)(24 173 168 136)(37 149 204 77)(38 150 193 78)(39 151 194 79)(40 152 195 80)(41 153 196 81)(42 154 197 82)(43 155 198 83)(44 156 199 84)(45 145 200 73)(46 146 201 74)(47 147 202 75)(48 148 203 76)(61 210 236 119)(62 211 237 120)(63 212 238 109)(64 213 239 110)(65 214 240 111)(66 215 229 112)(67 216 230 113)(68 205 231 114)(69 206 232 115)(70 207 233 116)(71 208 234 117)(72 209 235 118)(85 125 223 100)(86 126 224 101)(87 127 225 102)(88 128 226 103)(89 129 227 104)(90 130 228 105)(91 131 217 106)(92 132 218 107)(93 121 219 108)(94 122 220 97)(95 123 221 98)(96 124 222 99)
(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 29 181 57)(2 56 182 28)(3 27 183 55)(4 54 184 26)(5 25 185 53)(6 52 186 36)(7 35 187 51)(8 50 188 34)(9 33 189 49)(10 60 190 32)(11 31 191 59)(12 58 192 30)(13 148 157 76)(14 75 158 147)(15 146 159 74)(16 73 160 145)(17 156 161 84)(18 83 162 155)(19 154 163 82)(20 81 164 153)(21 152 165 80)(22 79 166 151)(23 150 167 78)(24 77 168 149)(37 173 204 136)(38 135 193 172)(39 171 194 134)(40 133 195 170)(41 169 196 144)(42 143 197 180)(43 179 198 142)(44 141 199 178)(45 177 200 140)(46 139 201 176)(47 175 202 138)(48 137 203 174)(61 131 236 106)(62 105 237 130)(63 129 238 104)(64 103 239 128)(65 127 240 102)(66 101 229 126)(67 125 230 100)(68 99 231 124)(69 123 232 98)(70 97 233 122)(71 121 234 108)(72 107 235 132)(85 113 223 216)(86 215 224 112)(87 111 225 214)(88 213 226 110)(89 109 227 212)(90 211 228 120)(91 119 217 210)(92 209 218 118)(93 117 219 208)(94 207 220 116)(95 115 221 206)(96 205 222 114)
G:=sub<Sym(240)| (1,219,193,18,66,181,93,38,162,229)(2,230,163,39,94,182,67,19,194,220)(3,221,195,20,68,183,95,40,164,231)(4,232,165,41,96,184,69,21,196,222)(5,223,197,22,70,185,85,42,166,233)(6,234,167,43,86,186,71,23,198,224)(7,225,199,24,72,187,87,44,168,235)(8,236,157,45,88,188,61,13,200,226)(9,227,201,14,62,189,89,46,158,237)(10,238,159,47,90,190,63,15,202,228)(11,217,203,16,64,191,91,48,160,239)(12,240,161,37,92,192,65,17,204,218)(25,122,151,143,113,53,97,79,180,216)(26,205,169,80,98,54,114,144,152,123)(27,124,153,133,115,55,99,81,170,206)(28,207,171,82,100,56,116,134,154,125)(29,126,155,135,117,57,101,83,172,208)(30,209,173,84,102,58,118,136,156,127)(31,128,145,137,119,59,103,73,174,210)(32,211,175,74,104,60,120,138,146,129)(33,130,147,139,109,49,105,75,176,212)(34,213,177,76,106,50,110,140,148,131)(35,132,149,141,111,51,107,77,178,214)(36,215,179,78,108,52,112,142,150,121), (1,52,181,36)(2,53,182,25)(3,54,183,26)(4,55,184,27)(5,56,185,28)(6,57,186,29)(7,58,187,30)(8,59,188,31)(9,60,189,32)(10,49,190,33)(11,50,191,34)(12,51,192,35)(13,174,157,137)(14,175,158,138)(15,176,159,139)(16,177,160,140)(17,178,161,141)(18,179,162,142)(19,180,163,143)(20,169,164,144)(21,170,165,133)(22,171,166,134)(23,172,167,135)(24,173,168,136)(37,149,204,77)(38,150,193,78)(39,151,194,79)(40,152,195,80)(41,153,196,81)(42,154,197,82)(43,155,198,83)(44,156,199,84)(45,145,200,73)(46,146,201,74)(47,147,202,75)(48,148,203,76)(61,210,236,119)(62,211,237,120)(63,212,238,109)(64,213,239,110)(65,214,240,111)(66,215,229,112)(67,216,230,113)(68,205,231,114)(69,206,232,115)(70,207,233,116)(71,208,234,117)(72,209,235,118)(85,125,223,100)(86,126,224,101)(87,127,225,102)(88,128,226,103)(89,129,227,104)(90,130,228,105)(91,131,217,106)(92,132,218,107)(93,121,219,108)(94,122,220,97)(95,123,221,98)(96,124,222,99), (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), (1,29,181,57)(2,56,182,28)(3,27,183,55)(4,54,184,26)(5,25,185,53)(6,52,186,36)(7,35,187,51)(8,50,188,34)(9,33,189,49)(10,60,190,32)(11,31,191,59)(12,58,192,30)(13,148,157,76)(14,75,158,147)(15,146,159,74)(16,73,160,145)(17,156,161,84)(18,83,162,155)(19,154,163,82)(20,81,164,153)(21,152,165,80)(22,79,166,151)(23,150,167,78)(24,77,168,149)(37,173,204,136)(38,135,193,172)(39,171,194,134)(40,133,195,170)(41,169,196,144)(42,143,197,180)(43,179,198,142)(44,141,199,178)(45,177,200,140)(46,139,201,176)(47,175,202,138)(48,137,203,174)(61,131,236,106)(62,105,237,130)(63,129,238,104)(64,103,239,128)(65,127,240,102)(66,101,229,126)(67,125,230,100)(68,99,231,124)(69,123,232,98)(70,97,233,122)(71,121,234,108)(72,107,235,132)(85,113,223,216)(86,215,224,112)(87,111,225,214)(88,213,226,110)(89,109,227,212)(90,211,228,120)(91,119,217,210)(92,209,218,118)(93,117,219,208)(94,207,220,116)(95,115,221,206)(96,205,222,114)>;
G:=Group( (1,219,193,18,66,181,93,38,162,229)(2,230,163,39,94,182,67,19,194,220)(3,221,195,20,68,183,95,40,164,231)(4,232,165,41,96,184,69,21,196,222)(5,223,197,22,70,185,85,42,166,233)(6,234,167,43,86,186,71,23,198,224)(7,225,199,24,72,187,87,44,168,235)(8,236,157,45,88,188,61,13,200,226)(9,227,201,14,62,189,89,46,158,237)(10,238,159,47,90,190,63,15,202,228)(11,217,203,16,64,191,91,48,160,239)(12,240,161,37,92,192,65,17,204,218)(25,122,151,143,113,53,97,79,180,216)(26,205,169,80,98,54,114,144,152,123)(27,124,153,133,115,55,99,81,170,206)(28,207,171,82,100,56,116,134,154,125)(29,126,155,135,117,57,101,83,172,208)(30,209,173,84,102,58,118,136,156,127)(31,128,145,137,119,59,103,73,174,210)(32,211,175,74,104,60,120,138,146,129)(33,130,147,139,109,49,105,75,176,212)(34,213,177,76,106,50,110,140,148,131)(35,132,149,141,111,51,107,77,178,214)(36,215,179,78,108,52,112,142,150,121), (1,52,181,36)(2,53,182,25)(3,54,183,26)(4,55,184,27)(5,56,185,28)(6,57,186,29)(7,58,187,30)(8,59,188,31)(9,60,189,32)(10,49,190,33)(11,50,191,34)(12,51,192,35)(13,174,157,137)(14,175,158,138)(15,176,159,139)(16,177,160,140)(17,178,161,141)(18,179,162,142)(19,180,163,143)(20,169,164,144)(21,170,165,133)(22,171,166,134)(23,172,167,135)(24,173,168,136)(37,149,204,77)(38,150,193,78)(39,151,194,79)(40,152,195,80)(41,153,196,81)(42,154,197,82)(43,155,198,83)(44,156,199,84)(45,145,200,73)(46,146,201,74)(47,147,202,75)(48,148,203,76)(61,210,236,119)(62,211,237,120)(63,212,238,109)(64,213,239,110)(65,214,240,111)(66,215,229,112)(67,216,230,113)(68,205,231,114)(69,206,232,115)(70,207,233,116)(71,208,234,117)(72,209,235,118)(85,125,223,100)(86,126,224,101)(87,127,225,102)(88,128,226,103)(89,129,227,104)(90,130,228,105)(91,131,217,106)(92,132,218,107)(93,121,219,108)(94,122,220,97)(95,123,221,98)(96,124,222,99), (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), (1,29,181,57)(2,56,182,28)(3,27,183,55)(4,54,184,26)(5,25,185,53)(6,52,186,36)(7,35,187,51)(8,50,188,34)(9,33,189,49)(10,60,190,32)(11,31,191,59)(12,58,192,30)(13,148,157,76)(14,75,158,147)(15,146,159,74)(16,73,160,145)(17,156,161,84)(18,83,162,155)(19,154,163,82)(20,81,164,153)(21,152,165,80)(22,79,166,151)(23,150,167,78)(24,77,168,149)(37,173,204,136)(38,135,193,172)(39,171,194,134)(40,133,195,170)(41,169,196,144)(42,143,197,180)(43,179,198,142)(44,141,199,178)(45,177,200,140)(46,139,201,176)(47,175,202,138)(48,137,203,174)(61,131,236,106)(62,105,237,130)(63,129,238,104)(64,103,239,128)(65,127,240,102)(66,101,229,126)(67,125,230,100)(68,99,231,124)(69,123,232,98)(70,97,233,122)(71,121,234,108)(72,107,235,132)(85,113,223,216)(86,215,224,112)(87,111,225,214)(88,213,226,110)(89,109,227,212)(90,211,228,120)(91,119,217,210)(92,209,218,118)(93,117,219,208)(94,207,220,116)(95,115,221,206)(96,205,222,114) );
G=PermutationGroup([[(1,219,193,18,66,181,93,38,162,229),(2,230,163,39,94,182,67,19,194,220),(3,221,195,20,68,183,95,40,164,231),(4,232,165,41,96,184,69,21,196,222),(5,223,197,22,70,185,85,42,166,233),(6,234,167,43,86,186,71,23,198,224),(7,225,199,24,72,187,87,44,168,235),(8,236,157,45,88,188,61,13,200,226),(9,227,201,14,62,189,89,46,158,237),(10,238,159,47,90,190,63,15,202,228),(11,217,203,16,64,191,91,48,160,239),(12,240,161,37,92,192,65,17,204,218),(25,122,151,143,113,53,97,79,180,216),(26,205,169,80,98,54,114,144,152,123),(27,124,153,133,115,55,99,81,170,206),(28,207,171,82,100,56,116,134,154,125),(29,126,155,135,117,57,101,83,172,208),(30,209,173,84,102,58,118,136,156,127),(31,128,145,137,119,59,103,73,174,210),(32,211,175,74,104,60,120,138,146,129),(33,130,147,139,109,49,105,75,176,212),(34,213,177,76,106,50,110,140,148,131),(35,132,149,141,111,51,107,77,178,214),(36,215,179,78,108,52,112,142,150,121)], [(1,52,181,36),(2,53,182,25),(3,54,183,26),(4,55,184,27),(5,56,185,28),(6,57,186,29),(7,58,187,30),(8,59,188,31),(9,60,189,32),(10,49,190,33),(11,50,191,34),(12,51,192,35),(13,174,157,137),(14,175,158,138),(15,176,159,139),(16,177,160,140),(17,178,161,141),(18,179,162,142),(19,180,163,143),(20,169,164,144),(21,170,165,133),(22,171,166,134),(23,172,167,135),(24,173,168,136),(37,149,204,77),(38,150,193,78),(39,151,194,79),(40,152,195,80),(41,153,196,81),(42,154,197,82),(43,155,198,83),(44,156,199,84),(45,145,200,73),(46,146,201,74),(47,147,202,75),(48,148,203,76),(61,210,236,119),(62,211,237,120),(63,212,238,109),(64,213,239,110),(65,214,240,111),(66,215,229,112),(67,216,230,113),(68,205,231,114),(69,206,232,115),(70,207,233,116),(71,208,234,117),(72,209,235,118),(85,125,223,100),(86,126,224,101),(87,127,225,102),(88,128,226,103),(89,129,227,104),(90,130,228,105),(91,131,217,106),(92,132,218,107),(93,121,219,108),(94,122,220,97),(95,123,221,98),(96,124,222,99)], [(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)], [(1,29,181,57),(2,56,182,28),(3,27,183,55),(4,54,184,26),(5,25,185,53),(6,52,186,36),(7,35,187,51),(8,50,188,34),(9,33,189,49),(10,60,190,32),(11,31,191,59),(12,58,192,30),(13,148,157,76),(14,75,158,147),(15,146,159,74),(16,73,160,145),(17,156,161,84),(18,83,162,155),(19,154,163,82),(20,81,164,153),(21,152,165,80),(22,79,166,151),(23,150,167,78),(24,77,168,149),(37,173,204,136),(38,135,193,172),(39,171,194,134),(40,133,195,170),(41,169,196,144),(42,143,197,180),(43,179,198,142),(44,141,199,178),(45,177,200,140),(46,139,201,176),(47,175,202,138),(48,137,203,174),(61,131,236,106),(62,105,237,130),(63,129,238,104),(64,103,239,128),(65,127,240,102),(66,101,229,126),(67,125,230,100),(68,99,231,124),(69,123,232,98),(70,97,233,122),(71,121,234,108),(72,107,235,132),(85,113,223,216),(86,215,224,112),(87,111,225,214),(88,213,226,110),(89,109,227,212),(90,211,228,120),(91,119,217,210),(92,209,218,118),(93,117,219,208),(94,207,220,116),(95,115,221,206),(96,205,222,114)]])
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 | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 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 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 60 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 12 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D12 | C4○D12 | C4○D20 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | D6.D10 | D5×D12 | Dic3.D10 |
| kernel | Dic5.8D12 | D6⋊Dic5 | D30⋊4C4 | C12×Dic5 | C5×D6⋊C4 | D30⋊3C4 | C2×C5⋊D12 | C2×C15⋊Q8 | C4×Dic5 | C3×Dic5 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | Dic5 | C10 | C6 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic5.8D12 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 47 | 16 | 0 | 0 | 0 | 0 |
| 22 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 7 | 0 | 0 | 0 | 0 |
| 2 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 53 |
| 0 | 0 | 0 | 0 | 46 | 1 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 57 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 8 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[47,22,0,0,0,0,16,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,2,0,0,0,0,7,29,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,60,46,0,0,0,0,53,1],[50,57,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,8,60] >;
Dic5.8D12 in GAP, Magma, Sage, TeX
{\rm Dic}_5._8D_{12} % in TeX
G:=Group("Dic5.8D12"); // GroupNames label
G:=SmallGroup(480,426);
// by ID
G=gap.SmallGroup(480,426);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^12=1,b^2=d^2=a^5,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^-1>;
// generators/relations