metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.1Dic6, C3⋊C8.1F5, C12.7(C2×F5), C30.3(C4⋊C4), C4.17(S3×F5), (C6×D5).1Q8, C15⋊3C8.1C4, C20.17(C4×S3), C4.F5.1S3, C60.17(C2×C4), (C4×D5).62D6, C6.10(C4⋊F5), C3⋊2(C40.C4), C15⋊1(C8.C4), C12.F5.1C2, C5⋊1(C12.53D4), C2.6(Dic3⋊F5), (C3×Dic5).28D4, C10.3(Dic3⋊C4), (D5×C12).48C22, Dic5.16(C3⋊D4), (C5×C3⋊C8).1C4, (D5×C3⋊C8).4C2, (C3×C4.F5).1C2, SmallGroup(480,237)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.Dic6
G = < a,b,c,d | a10=b2=1, c12=a5, d2=a-1bc6, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, bd=db, dcd-1=a-1bc11 >
Subgroups: 260 in 60 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), Dic5, C20, D10, C3⋊C8, C3⋊C8, C24, C2×C12, C3×D5, C30, C8.C4, C5⋊2C8, C40, C5⋊C8, C4×D5, C2×C3⋊C8, C4.Dic3, C3×M4(2), C3×Dic5, C60, C6×D5, C8×D5, C4.F5, C4.F5, C12.53D4, C5×C3⋊C8, C15⋊3C8, C3×C5⋊C8, C15⋊C8, D5×C12, C40.C4, D5×C3⋊C8, C3×C4.F5, C12.F5, D10.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, F5, Dic6, C4×S3, C3⋊D4, C8.C4, C2×F5, Dic3⋊C4, C4⋊F5, C12.53D4, S3×F5, C40.C4, Dic3⋊F5, D10.Dic6
►On 240 points
Generators in S240
(1 82 195 157 233 13 94 207 145 221)(2 158 95 222 196 14 146 83 234 208)(3 223 147 209 96 15 235 159 197 84)(4 210 236 85 148 16 198 224 73 160)(5 86 199 161 237 17 74 211 149 225)(6 162 75 226 200 18 150 87 238 212)(7 227 151 213 76 19 239 163 201 88)(8 214 240 89 152 20 202 228 77 164)(9 90 203 165 217 21 78 215 153 229)(10 166 79 230 204 22 154 91 218 216)(11 231 155 193 80 23 219 167 205 92)(12 194 220 93 156 24 206 232 81 168)(25 52 178 129 111 37 64 190 141 99)(26 130 65 100 179 38 142 53 112 191)(27 101 143 192 66 39 113 131 180 54)(28 169 114 55 144 40 181 102 67 132)(29 56 182 133 115 41 68 170 121 103)(30 134 69 104 183 42 122 57 116 171)(31 105 123 172 70 43 117 135 184 58)(32 173 118 59 124 44 185 106 71 136)(33 60 186 137 119 45 72 174 125 107)(34 138 49 108 187 46 126 61 120 175)(35 109 127 176 50 47 97 139 188 62)(36 177 98 63 128 48 189 110 51 140)
(1 221)(2 196)(3 84)(4 148)(5 225)(6 200)(7 88)(8 152)(9 229)(10 204)(11 92)(12 156)(13 233)(14 208)(15 96)(16 160)(17 237)(18 212)(19 76)(20 164)(21 217)(22 216)(23 80)(24 168)(25 99)(26 179)(27 54)(28 144)(29 103)(30 183)(31 58)(32 124)(33 107)(34 187)(35 62)(36 128)(37 111)(38 191)(39 66)(40 132)(41 115)(42 171)(43 70)(44 136)(45 119)(46 175)(47 50)(48 140)(51 189)(52 141)(55 169)(56 121)(59 173)(60 125)(63 177)(64 129)(67 181)(68 133)(71 185)(72 137)(73 198)(74 161)(77 202)(78 165)(81 206)(82 145)(85 210)(86 149)(89 214)(90 153)(93 194)(94 157)(97 176)(100 130)(101 180)(104 134)(105 184)(108 138)(109 188)(112 142)(113 192)(116 122)(117 172)(120 126)(123 135)(127 139)(131 143)(146 234)(147 159)(150 238)(151 163)(154 218)(155 167)(158 222)(162 226)(166 230)(170 182)(174 186)(178 190)(193 219)(195 207)(197 223)(199 211)(201 227)(203 215)(205 231)(209 235)(213 239)
(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 7 35 13 41 19 47)(2 28 20 46 14 40 8 34)(3 39 9 45 15 27 21 33)(4 38 22 32 16 26 10 44)(5 25 11 31 17 37 23 43)(6 48 24 42 18 36 12 30)(49 95 114 228 61 83 102 240)(50 221 103 88 62 233 115 76)(51 81 116 238 63 93 104 226)(52 231 105 74 64 219 117 86)(53 91 118 224 65 79 106 236)(54 217 107 84 66 229 119 96)(55 77 120 234 67 89 108 222)(56 227 109 94 68 239 97 82)(57 87 98 220 69 75 110 232)(58 237 111 80 70 225 99 92)(59 73 100 230 71 85 112 218)(60 223 113 90 72 235 101 78)(121 201 188 157 133 213 176 145)(122 150 177 194 134 162 189 206)(123 211 190 167 135 199 178 155)(124 160 179 204 136 148 191 216)(125 197 192 153 137 209 180 165)(126 146 181 214 138 158 169 202)(127 207 170 163 139 195 182 151)(128 156 183 200 140 168 171 212)(129 193 172 149 141 205 184 161)(130 166 185 210 142 154 173 198)(131 203 174 159 143 215 186 147)(132 152 187 196 144 164 175 208)
G:=sub<Sym(240)| (1,82,195,157,233,13,94,207,145,221)(2,158,95,222,196,14,146,83,234,208)(3,223,147,209,96,15,235,159,197,84)(4,210,236,85,148,16,198,224,73,160)(5,86,199,161,237,17,74,211,149,225)(6,162,75,226,200,18,150,87,238,212)(7,227,151,213,76,19,239,163,201,88)(8,214,240,89,152,20,202,228,77,164)(9,90,203,165,217,21,78,215,153,229)(10,166,79,230,204,22,154,91,218,216)(11,231,155,193,80,23,219,167,205,92)(12,194,220,93,156,24,206,232,81,168)(25,52,178,129,111,37,64,190,141,99)(26,130,65,100,179,38,142,53,112,191)(27,101,143,192,66,39,113,131,180,54)(28,169,114,55,144,40,181,102,67,132)(29,56,182,133,115,41,68,170,121,103)(30,134,69,104,183,42,122,57,116,171)(31,105,123,172,70,43,117,135,184,58)(32,173,118,59,124,44,185,106,71,136)(33,60,186,137,119,45,72,174,125,107)(34,138,49,108,187,46,126,61,120,175)(35,109,127,176,50,47,97,139,188,62)(36,177,98,63,128,48,189,110,51,140), (1,221)(2,196)(3,84)(4,148)(5,225)(6,200)(7,88)(8,152)(9,229)(10,204)(11,92)(12,156)(13,233)(14,208)(15,96)(16,160)(17,237)(18,212)(19,76)(20,164)(21,217)(22,216)(23,80)(24,168)(25,99)(26,179)(27,54)(28,144)(29,103)(30,183)(31,58)(32,124)(33,107)(34,187)(35,62)(36,128)(37,111)(38,191)(39,66)(40,132)(41,115)(42,171)(43,70)(44,136)(45,119)(46,175)(47,50)(48,140)(51,189)(52,141)(55,169)(56,121)(59,173)(60,125)(63,177)(64,129)(67,181)(68,133)(71,185)(72,137)(73,198)(74,161)(77,202)(78,165)(81,206)(82,145)(85,210)(86,149)(89,214)(90,153)(93,194)(94,157)(97,176)(100,130)(101,180)(104,134)(105,184)(108,138)(109,188)(112,142)(113,192)(116,122)(117,172)(120,126)(123,135)(127,139)(131,143)(146,234)(147,159)(150,238)(151,163)(154,218)(155,167)(158,222)(162,226)(166,230)(170,182)(174,186)(178,190)(193,219)(195,207)(197,223)(199,211)(201,227)(203,215)(205,231)(209,235)(213,239), (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), (1,29,7,35,13,41,19,47)(2,28,20,46,14,40,8,34)(3,39,9,45,15,27,21,33)(4,38,22,32,16,26,10,44)(5,25,11,31,17,37,23,43)(6,48,24,42,18,36,12,30)(49,95,114,228,61,83,102,240)(50,221,103,88,62,233,115,76)(51,81,116,238,63,93,104,226)(52,231,105,74,64,219,117,86)(53,91,118,224,65,79,106,236)(54,217,107,84,66,229,119,96)(55,77,120,234,67,89,108,222)(56,227,109,94,68,239,97,82)(57,87,98,220,69,75,110,232)(58,237,111,80,70,225,99,92)(59,73,100,230,71,85,112,218)(60,223,113,90,72,235,101,78)(121,201,188,157,133,213,176,145)(122,150,177,194,134,162,189,206)(123,211,190,167,135,199,178,155)(124,160,179,204,136,148,191,216)(125,197,192,153,137,209,180,165)(126,146,181,214,138,158,169,202)(127,207,170,163,139,195,182,151)(128,156,183,200,140,168,171,212)(129,193,172,149,141,205,184,161)(130,166,185,210,142,154,173,198)(131,203,174,159,143,215,186,147)(132,152,187,196,144,164,175,208)>;
G:=Group( (1,82,195,157,233,13,94,207,145,221)(2,158,95,222,196,14,146,83,234,208)(3,223,147,209,96,15,235,159,197,84)(4,210,236,85,148,16,198,224,73,160)(5,86,199,161,237,17,74,211,149,225)(6,162,75,226,200,18,150,87,238,212)(7,227,151,213,76,19,239,163,201,88)(8,214,240,89,152,20,202,228,77,164)(9,90,203,165,217,21,78,215,153,229)(10,166,79,230,204,22,154,91,218,216)(11,231,155,193,80,23,219,167,205,92)(12,194,220,93,156,24,206,232,81,168)(25,52,178,129,111,37,64,190,141,99)(26,130,65,100,179,38,142,53,112,191)(27,101,143,192,66,39,113,131,180,54)(28,169,114,55,144,40,181,102,67,132)(29,56,182,133,115,41,68,170,121,103)(30,134,69,104,183,42,122,57,116,171)(31,105,123,172,70,43,117,135,184,58)(32,173,118,59,124,44,185,106,71,136)(33,60,186,137,119,45,72,174,125,107)(34,138,49,108,187,46,126,61,120,175)(35,109,127,176,50,47,97,139,188,62)(36,177,98,63,128,48,189,110,51,140), (1,221)(2,196)(3,84)(4,148)(5,225)(6,200)(7,88)(8,152)(9,229)(10,204)(11,92)(12,156)(13,233)(14,208)(15,96)(16,160)(17,237)(18,212)(19,76)(20,164)(21,217)(22,216)(23,80)(24,168)(25,99)(26,179)(27,54)(28,144)(29,103)(30,183)(31,58)(32,124)(33,107)(34,187)(35,62)(36,128)(37,111)(38,191)(39,66)(40,132)(41,115)(42,171)(43,70)(44,136)(45,119)(46,175)(47,50)(48,140)(51,189)(52,141)(55,169)(56,121)(59,173)(60,125)(63,177)(64,129)(67,181)(68,133)(71,185)(72,137)(73,198)(74,161)(77,202)(78,165)(81,206)(82,145)(85,210)(86,149)(89,214)(90,153)(93,194)(94,157)(97,176)(100,130)(101,180)(104,134)(105,184)(108,138)(109,188)(112,142)(113,192)(116,122)(117,172)(120,126)(123,135)(127,139)(131,143)(146,234)(147,159)(150,238)(151,163)(154,218)(155,167)(158,222)(162,226)(166,230)(170,182)(174,186)(178,190)(193,219)(195,207)(197,223)(199,211)(201,227)(203,215)(205,231)(209,235)(213,239), (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), (1,29,7,35,13,41,19,47)(2,28,20,46,14,40,8,34)(3,39,9,45,15,27,21,33)(4,38,22,32,16,26,10,44)(5,25,11,31,17,37,23,43)(6,48,24,42,18,36,12,30)(49,95,114,228,61,83,102,240)(50,221,103,88,62,233,115,76)(51,81,116,238,63,93,104,226)(52,231,105,74,64,219,117,86)(53,91,118,224,65,79,106,236)(54,217,107,84,66,229,119,96)(55,77,120,234,67,89,108,222)(56,227,109,94,68,239,97,82)(57,87,98,220,69,75,110,232)(58,237,111,80,70,225,99,92)(59,73,100,230,71,85,112,218)(60,223,113,90,72,235,101,78)(121,201,188,157,133,213,176,145)(122,150,177,194,134,162,189,206)(123,211,190,167,135,199,178,155)(124,160,179,204,136,148,191,216)(125,197,192,153,137,209,180,165)(126,146,181,214,138,158,169,202)(127,207,170,163,139,195,182,151)(128,156,183,200,140,168,171,212)(129,193,172,149,141,205,184,161)(130,166,185,210,142,154,173,198)(131,203,174,159,143,215,186,147)(132,152,187,196,144,164,175,208) );
G=PermutationGroup([[(1,82,195,157,233,13,94,207,145,221),(2,158,95,222,196,14,146,83,234,208),(3,223,147,209,96,15,235,159,197,84),(4,210,236,85,148,16,198,224,73,160),(5,86,199,161,237,17,74,211,149,225),(6,162,75,226,200,18,150,87,238,212),(7,227,151,213,76,19,239,163,201,88),(8,214,240,89,152,20,202,228,77,164),(9,90,203,165,217,21,78,215,153,229),(10,166,79,230,204,22,154,91,218,216),(11,231,155,193,80,23,219,167,205,92),(12,194,220,93,156,24,206,232,81,168),(25,52,178,129,111,37,64,190,141,99),(26,130,65,100,179,38,142,53,112,191),(27,101,143,192,66,39,113,131,180,54),(28,169,114,55,144,40,181,102,67,132),(29,56,182,133,115,41,68,170,121,103),(30,134,69,104,183,42,122,57,116,171),(31,105,123,172,70,43,117,135,184,58),(32,173,118,59,124,44,185,106,71,136),(33,60,186,137,119,45,72,174,125,107),(34,138,49,108,187,46,126,61,120,175),(35,109,127,176,50,47,97,139,188,62),(36,177,98,63,128,48,189,110,51,140)], [(1,221),(2,196),(3,84),(4,148),(5,225),(6,200),(7,88),(8,152),(9,229),(10,204),(11,92),(12,156),(13,233),(14,208),(15,96),(16,160),(17,237),(18,212),(19,76),(20,164),(21,217),(22,216),(23,80),(24,168),(25,99),(26,179),(27,54),(28,144),(29,103),(30,183),(31,58),(32,124),(33,107),(34,187),(35,62),(36,128),(37,111),(38,191),(39,66),(40,132),(41,115),(42,171),(43,70),(44,136),(45,119),(46,175),(47,50),(48,140),(51,189),(52,141),(55,169),(56,121),(59,173),(60,125),(63,177),(64,129),(67,181),(68,133),(71,185),(72,137),(73,198),(74,161),(77,202),(78,165),(81,206),(82,145),(85,210),(86,149),(89,214),(90,153),(93,194),(94,157),(97,176),(100,130),(101,180),(104,134),(105,184),(108,138),(109,188),(112,142),(113,192),(116,122),(117,172),(120,126),(123,135),(127,139),(131,143),(146,234),(147,159),(150,238),(151,163),(154,218),(155,167),(158,222),(162,226),(166,230),(170,182),(174,186),(178,190),(193,219),(195,207),(197,223),(199,211),(201,227),(203,215),(205,231),(209,235),(213,239)], [(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)], [(1,29,7,35,13,41,19,47),(2,28,20,46,14,40,8,34),(3,39,9,45,15,27,21,33),(4,38,22,32,16,26,10,44),(5,25,11,31,17,37,23,43),(6,48,24,42,18,36,12,30),(49,95,114,228,61,83,102,240),(50,221,103,88,62,233,115,76),(51,81,116,238,63,93,104,226),(52,231,105,74,64,219,117,86),(53,91,118,224,65,79,106,236),(54,217,107,84,66,229,119,96),(55,77,120,234,67,89,108,222),(56,227,109,94,68,239,97,82),(57,87,98,220,69,75,110,232),(58,237,111,80,70,225,99,92),(59,73,100,230,71,85,112,218),(60,223,113,90,72,235,101,78),(121,201,188,157,133,213,176,145),(122,150,177,194,134,162,189,206),(123,211,190,167,135,199,178,155),(124,160,179,204,136,148,191,216),(125,197,192,153,137,209,180,165),(126,146,181,214,138,158,169,202),(127,207,170,163,139,195,182,151),(128,156,183,200,140,168,171,212),(129,193,172,149,141,205,184,161),(130,166,185,210,142,154,173,198),(131,203,174,159,143,215,186,147),(132,152,187,196,144,164,175,208)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 12A | 12B | 12C | 15 | 20A | 20B | 24A | 24B | 24C | 24D | 30 | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 10 | 2 | 2 | 5 | 5 | 4 | 2 | 20 | 6 | 6 | 20 | 20 | 30 | 30 | 60 | 60 | 4 | 4 | 10 | 10 | 8 | 4 | 4 | 20 | 20 | 20 | 20 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | - | + | - | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | D6 | C3⋊D4 | C4×S3 | Dic6 | C8.C4 | F5 | C2×F5 | C4⋊F5 | C12.53D4 | C40.C4 | S3×F5 | Dic3⋊F5 | D10.Dic6 |
| kernel | D10.Dic6 | D5×C3⋊C8 | C3×C4.F5 | C12.F5 | C5×C3⋊C8 | C15⋊3C8 | C4.F5 | C3×Dic5 | C6×D5 | C4×D5 | Dic5 | C20 | D10 | C15 | C3⋊C8 | C12 | C6 | C5 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of D10.Dic6 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 177 | 64 | 0 | 0 | 0 | 0 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 211 | 181 | 214 | 208 |
| 0 | 0 | 33 | 27 | 60 | 30 |
| 0 | 0 | 33 | 3 | 214 | 6 |
| 0 | 0 | 211 | 3 | 238 | 30 |
| 137 | 34 | 0 | 0 | 0 | 0 |
| 171 | 104 | 0 | 0 | 0 | 0 |
| 0 | 0 | 178 | 0 | 77 | 77 |
| 0 | 0 | 164 | 101 | 164 | 0 |
| 0 | 0 | 0 | 164 | 101 | 164 |
| 0 | 0 | 77 | 77 | 0 | 178 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,240,0,0,0,0,1,0,240,0,0,0,1,0,0,0,0,240,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0],[177,177,0,0,0,0,64,0,0,0,0,0,0,0,211,33,33,211,0,0,181,27,3,3,0,0,214,60,214,238,0,0,208,30,6,30],[137,171,0,0,0,0,34,104,0,0,0,0,0,0,178,164,0,77,0,0,0,101,164,77,0,0,77,164,101,0,0,0,77,0,164,178] >;
D10.Dic6 in GAP, Magma, Sage, TeX
D_{10}.{\rm Dic}_6 % in TeX
G:=Group("D10.Dic6"); // GroupNames label
G:=SmallGroup(480,237);
// by ID
G=gap.SmallGroup(480,237);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,675,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=1,c^12=a^5,d^2=a^-1*b*c^6,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=a^-1*b*c^11>;
// generators/relations