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## G = D6⋊Dic10order 480 = 25·3·5

### 1st semidirect product of D6 and Dic10 acting via Dic10/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D6⋊Dic10
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — D6⋊Dic5 — D6⋊Dic10
 Lower central C15 — C2×C30 — D6⋊Dic10
 Upper central C1 — C22 — C2×C4

Generators and relations for D6⋊Dic10
G = < a,b,c,d | a6=b2=c20=1, d2=c10, bab=cac-1=dad-1=a-1, cbc-1=a4b, dbd-1=ab, dcd-1=c-1 >

Subgroups: 652 in 148 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C22⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, S3×C10, C2×C30, C10.D4, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, D6⋊Q8, C15⋊Q8, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C20.48D4, D6⋊Dic5, C6.Dic10, C3×C10.D4, C30.4Q8, C2×C15⋊Q8, S3×C2×C20, D6⋊Dic10
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C22×S3, C22⋊Q8, Dic10, C5⋊D4, C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, C2×Dic10, C4○D20, C2×C5⋊D4, D6⋊Q8, C2×S3×D5, C20.48D4, S3×Dic10, D6.D10, S3×C5⋊D4, D6⋊Dic10

Smallest permutation representation of D6⋊Dic10
On 240 points
Generators in S240
(1 24 98 160 222 216)(2 217 223 141 99 25)(3 26 100 142 224 218)(4 219 225 143 81 27)(5 28 82 144 226 220)(6 201 227 145 83 29)(7 30 84 146 228 202)(8 203 229 147 85 31)(9 32 86 148 230 204)(10 205 231 149 87 33)(11 34 88 150 232 206)(12 207 233 151 89 35)(13 36 90 152 234 208)(14 209 235 153 91 37)(15 38 92 154 236 210)(16 211 237 155 93 39)(17 40 94 156 238 212)(18 213 239 157 95 21)(19 22 96 158 240 214)(20 215 221 159 97 23)(41 169 71 104 191 135)(42 136 192 105 72 170)(43 171 73 106 193 137)(44 138 194 107 74 172)(45 173 75 108 195 139)(46 140 196 109 76 174)(47 175 77 110 197 121)(48 122 198 111 78 176)(49 177 79 112 199 123)(50 124 200 113 80 178)(51 179 61 114 181 125)(52 126 182 115 62 180)(53 161 63 116 183 127)(54 128 184 117 64 162)(55 163 65 118 185 129)(56 130 186 119 66 164)(57 165 67 120 187 131)(58 132 188 101 68 166)(59 167 69 102 189 133)(60 134 190 103 70 168)
(1 88)(2 233)(3 90)(4 235)(5 92)(6 237)(7 94)(8 239)(9 96)(10 221)(11 98)(12 223)(13 100)(14 225)(15 82)(16 227)(17 84)(18 229)(19 86)(20 231)(21 147)(22 32)(23 149)(24 34)(25 151)(26 36)(27 153)(28 38)(29 155)(30 40)(31 157)(33 159)(35 141)(37 143)(39 145)(41 125)(42 180)(43 127)(44 162)(45 129)(46 164)(47 131)(48 166)(49 133)(50 168)(51 135)(52 170)(53 137)(54 172)(55 139)(56 174)(57 121)(58 176)(59 123)(60 178)(61 104)(62 136)(63 106)(64 138)(65 108)(66 140)(67 110)(68 122)(69 112)(70 124)(71 114)(72 126)(73 116)(74 128)(75 118)(76 130)(77 120)(78 132)(79 102)(80 134)(81 91)(83 93)(85 95)(87 97)(89 99)(101 198)(103 200)(105 182)(107 184)(109 186)(111 188)(113 190)(115 192)(117 194)(119 196)(142 208)(144 210)(146 212)(148 214)(150 216)(152 218)(154 220)(156 202)(158 204)(160 206)(161 193)(163 195)(165 197)(167 199)(169 181)(171 183)(173 185)(175 187)(177 189)(179 191)(201 211)(203 213)(205 215)(207 217)(209 219)(222 232)(224 234)(226 236)(228 238)(230 240)
(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)
(1 109 11 119)(2 108 12 118)(3 107 13 117)(4 106 14 116)(5 105 15 115)(6 104 16 114)(7 103 17 113)(8 102 18 112)(9 101 19 111)(10 120 20 110)(21 199 31 189)(22 198 32 188)(23 197 33 187)(24 196 34 186)(25 195 35 185)(26 194 36 184)(27 193 37 183)(28 192 38 182)(29 191 39 181)(30 190 40 200)(41 155 51 145)(42 154 52 144)(43 153 53 143)(44 152 54 142)(45 151 55 141)(46 150 56 160)(47 149 57 159)(48 148 58 158)(49 147 59 157)(50 146 60 156)(61 201 71 211)(62 220 72 210)(63 219 73 209)(64 218 74 208)(65 217 75 207)(66 216 76 206)(67 215 77 205)(68 214 78 204)(69 213 79 203)(70 212 80 202)(81 137 91 127)(82 136 92 126)(83 135 93 125)(84 134 94 124)(85 133 95 123)(86 132 96 122)(87 131 97 121)(88 130 98 140)(89 129 99 139)(90 128 100 138)(161 225 171 235)(162 224 172 234)(163 223 173 233)(164 222 174 232)(165 221 175 231)(166 240 176 230)(167 239 177 229)(168 238 178 228)(169 237 179 227)(170 236 180 226)

G:=sub<Sym(240)| (1,24,98,160,222,216)(2,217,223,141,99,25)(3,26,100,142,224,218)(4,219,225,143,81,27)(5,28,82,144,226,220)(6,201,227,145,83,29)(7,30,84,146,228,202)(8,203,229,147,85,31)(9,32,86,148,230,204)(10,205,231,149,87,33)(11,34,88,150,232,206)(12,207,233,151,89,35)(13,36,90,152,234,208)(14,209,235,153,91,37)(15,38,92,154,236,210)(16,211,237,155,93,39)(17,40,94,156,238,212)(18,213,239,157,95,21)(19,22,96,158,240,214)(20,215,221,159,97,23)(41,169,71,104,191,135)(42,136,192,105,72,170)(43,171,73,106,193,137)(44,138,194,107,74,172)(45,173,75,108,195,139)(46,140,196,109,76,174)(47,175,77,110,197,121)(48,122,198,111,78,176)(49,177,79,112,199,123)(50,124,200,113,80,178)(51,179,61,114,181,125)(52,126,182,115,62,180)(53,161,63,116,183,127)(54,128,184,117,64,162)(55,163,65,118,185,129)(56,130,186,119,66,164)(57,165,67,120,187,131)(58,132,188,101,68,166)(59,167,69,102,189,133)(60,134,190,103,70,168), (1,88)(2,233)(3,90)(4,235)(5,92)(6,237)(7,94)(8,239)(9,96)(10,221)(11,98)(12,223)(13,100)(14,225)(15,82)(16,227)(17,84)(18,229)(19,86)(20,231)(21,147)(22,32)(23,149)(24,34)(25,151)(26,36)(27,153)(28,38)(29,155)(30,40)(31,157)(33,159)(35,141)(37,143)(39,145)(41,125)(42,180)(43,127)(44,162)(45,129)(46,164)(47,131)(48,166)(49,133)(50,168)(51,135)(52,170)(53,137)(54,172)(55,139)(56,174)(57,121)(58,176)(59,123)(60,178)(61,104)(62,136)(63,106)(64,138)(65,108)(66,140)(67,110)(68,122)(69,112)(70,124)(71,114)(72,126)(73,116)(74,128)(75,118)(76,130)(77,120)(78,132)(79,102)(80,134)(81,91)(83,93)(85,95)(87,97)(89,99)(101,198)(103,200)(105,182)(107,184)(109,186)(111,188)(113,190)(115,192)(117,194)(119,196)(142,208)(144,210)(146,212)(148,214)(150,216)(152,218)(154,220)(156,202)(158,204)(160,206)(161,193)(163,195)(165,197)(167,199)(169,181)(171,183)(173,185)(175,187)(177,189)(179,191)(201,211)(203,213)(205,215)(207,217)(209,219)(222,232)(224,234)(226,236)(228,238)(230,240), (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), (1,109,11,119)(2,108,12,118)(3,107,13,117)(4,106,14,116)(5,105,15,115)(6,104,16,114)(7,103,17,113)(8,102,18,112)(9,101,19,111)(10,120,20,110)(21,199,31,189)(22,198,32,188)(23,197,33,187)(24,196,34,186)(25,195,35,185)(26,194,36,184)(27,193,37,183)(28,192,38,182)(29,191,39,181)(30,190,40,200)(41,155,51,145)(42,154,52,144)(43,153,53,143)(44,152,54,142)(45,151,55,141)(46,150,56,160)(47,149,57,159)(48,148,58,158)(49,147,59,157)(50,146,60,156)(61,201,71,211)(62,220,72,210)(63,219,73,209)(64,218,74,208)(65,217,75,207)(66,216,76,206)(67,215,77,205)(68,214,78,204)(69,213,79,203)(70,212,80,202)(81,137,91,127)(82,136,92,126)(83,135,93,125)(84,134,94,124)(85,133,95,123)(86,132,96,122)(87,131,97,121)(88,130,98,140)(89,129,99,139)(90,128,100,138)(161,225,171,235)(162,224,172,234)(163,223,173,233)(164,222,174,232)(165,221,175,231)(166,240,176,230)(167,239,177,229)(168,238,178,228)(169,237,179,227)(170,236,180,226)>;

G:=Group( (1,24,98,160,222,216)(2,217,223,141,99,25)(3,26,100,142,224,218)(4,219,225,143,81,27)(5,28,82,144,226,220)(6,201,227,145,83,29)(7,30,84,146,228,202)(8,203,229,147,85,31)(9,32,86,148,230,204)(10,205,231,149,87,33)(11,34,88,150,232,206)(12,207,233,151,89,35)(13,36,90,152,234,208)(14,209,235,153,91,37)(15,38,92,154,236,210)(16,211,237,155,93,39)(17,40,94,156,238,212)(18,213,239,157,95,21)(19,22,96,158,240,214)(20,215,221,159,97,23)(41,169,71,104,191,135)(42,136,192,105,72,170)(43,171,73,106,193,137)(44,138,194,107,74,172)(45,173,75,108,195,139)(46,140,196,109,76,174)(47,175,77,110,197,121)(48,122,198,111,78,176)(49,177,79,112,199,123)(50,124,200,113,80,178)(51,179,61,114,181,125)(52,126,182,115,62,180)(53,161,63,116,183,127)(54,128,184,117,64,162)(55,163,65,118,185,129)(56,130,186,119,66,164)(57,165,67,120,187,131)(58,132,188,101,68,166)(59,167,69,102,189,133)(60,134,190,103,70,168), (1,88)(2,233)(3,90)(4,235)(5,92)(6,237)(7,94)(8,239)(9,96)(10,221)(11,98)(12,223)(13,100)(14,225)(15,82)(16,227)(17,84)(18,229)(19,86)(20,231)(21,147)(22,32)(23,149)(24,34)(25,151)(26,36)(27,153)(28,38)(29,155)(30,40)(31,157)(33,159)(35,141)(37,143)(39,145)(41,125)(42,180)(43,127)(44,162)(45,129)(46,164)(47,131)(48,166)(49,133)(50,168)(51,135)(52,170)(53,137)(54,172)(55,139)(56,174)(57,121)(58,176)(59,123)(60,178)(61,104)(62,136)(63,106)(64,138)(65,108)(66,140)(67,110)(68,122)(69,112)(70,124)(71,114)(72,126)(73,116)(74,128)(75,118)(76,130)(77,120)(78,132)(79,102)(80,134)(81,91)(83,93)(85,95)(87,97)(89,99)(101,198)(103,200)(105,182)(107,184)(109,186)(111,188)(113,190)(115,192)(117,194)(119,196)(142,208)(144,210)(146,212)(148,214)(150,216)(152,218)(154,220)(156,202)(158,204)(160,206)(161,193)(163,195)(165,197)(167,199)(169,181)(171,183)(173,185)(175,187)(177,189)(179,191)(201,211)(203,213)(205,215)(207,217)(209,219)(222,232)(224,234)(226,236)(228,238)(230,240), (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), (1,109,11,119)(2,108,12,118)(3,107,13,117)(4,106,14,116)(5,105,15,115)(6,104,16,114)(7,103,17,113)(8,102,18,112)(9,101,19,111)(10,120,20,110)(21,199,31,189)(22,198,32,188)(23,197,33,187)(24,196,34,186)(25,195,35,185)(26,194,36,184)(27,193,37,183)(28,192,38,182)(29,191,39,181)(30,190,40,200)(41,155,51,145)(42,154,52,144)(43,153,53,143)(44,152,54,142)(45,151,55,141)(46,150,56,160)(47,149,57,159)(48,148,58,158)(49,147,59,157)(50,146,60,156)(61,201,71,211)(62,220,72,210)(63,219,73,209)(64,218,74,208)(65,217,75,207)(66,216,76,206)(67,215,77,205)(68,214,78,204)(69,213,79,203)(70,212,80,202)(81,137,91,127)(82,136,92,126)(83,135,93,125)(84,134,94,124)(85,133,95,123)(86,132,96,122)(87,131,97,121)(88,130,98,140)(89,129,99,139)(90,128,100,138)(161,225,171,235)(162,224,172,234)(163,223,173,233)(164,222,174,232)(165,221,175,231)(166,240,176,230)(167,239,177,229)(168,238,178,228)(169,237,179,227)(170,236,180,226) );

G=PermutationGroup([[(1,24,98,160,222,216),(2,217,223,141,99,25),(3,26,100,142,224,218),(4,219,225,143,81,27),(5,28,82,144,226,220),(6,201,227,145,83,29),(7,30,84,146,228,202),(8,203,229,147,85,31),(9,32,86,148,230,204),(10,205,231,149,87,33),(11,34,88,150,232,206),(12,207,233,151,89,35),(13,36,90,152,234,208),(14,209,235,153,91,37),(15,38,92,154,236,210),(16,211,237,155,93,39),(17,40,94,156,238,212),(18,213,239,157,95,21),(19,22,96,158,240,214),(20,215,221,159,97,23),(41,169,71,104,191,135),(42,136,192,105,72,170),(43,171,73,106,193,137),(44,138,194,107,74,172),(45,173,75,108,195,139),(46,140,196,109,76,174),(47,175,77,110,197,121),(48,122,198,111,78,176),(49,177,79,112,199,123),(50,124,200,113,80,178),(51,179,61,114,181,125),(52,126,182,115,62,180),(53,161,63,116,183,127),(54,128,184,117,64,162),(55,163,65,118,185,129),(56,130,186,119,66,164),(57,165,67,120,187,131),(58,132,188,101,68,166),(59,167,69,102,189,133),(60,134,190,103,70,168)], [(1,88),(2,233),(3,90),(4,235),(5,92),(6,237),(7,94),(8,239),(9,96),(10,221),(11,98),(12,223),(13,100),(14,225),(15,82),(16,227),(17,84),(18,229),(19,86),(20,231),(21,147),(22,32),(23,149),(24,34),(25,151),(26,36),(27,153),(28,38),(29,155),(30,40),(31,157),(33,159),(35,141),(37,143),(39,145),(41,125),(42,180),(43,127),(44,162),(45,129),(46,164),(47,131),(48,166),(49,133),(50,168),(51,135),(52,170),(53,137),(54,172),(55,139),(56,174),(57,121),(58,176),(59,123),(60,178),(61,104),(62,136),(63,106),(64,138),(65,108),(66,140),(67,110),(68,122),(69,112),(70,124),(71,114),(72,126),(73,116),(74,128),(75,118),(76,130),(77,120),(78,132),(79,102),(80,134),(81,91),(83,93),(85,95),(87,97),(89,99),(101,198),(103,200),(105,182),(107,184),(109,186),(111,188),(113,190),(115,192),(117,194),(119,196),(142,208),(144,210),(146,212),(148,214),(150,216),(152,218),(154,220),(156,202),(158,204),(160,206),(161,193),(163,195),(165,197),(167,199),(169,181),(171,183),(173,185),(175,187),(177,189),(179,191),(201,211),(203,213),(205,215),(207,217),(209,219),(222,232),(224,234),(226,236),(228,238),(230,240)], [(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)], [(1,109,11,119),(2,108,12,118),(3,107,13,117),(4,106,14,116),(5,105,15,115),(6,104,16,114),(7,103,17,113),(8,102,18,112),(9,101,19,111),(10,120,20,110),(21,199,31,189),(22,198,32,188),(23,197,33,187),(24,196,34,186),(25,195,35,185),(26,194,36,184),(27,193,37,183),(28,192,38,182),(29,191,39,181),(30,190,40,200),(41,155,51,145),(42,154,52,144),(43,153,53,143),(44,152,54,142),(45,151,55,141),(46,150,56,160),(47,149,57,159),(48,148,58,158),(49,147,59,157),(50,146,60,156),(61,201,71,211),(62,220,72,210),(63,219,73,209),(64,218,74,208),(65,217,75,207),(66,216,76,206),(67,215,77,205),(68,214,78,204),(69,213,79,203),(70,212,80,202),(81,137,91,127),(82,136,92,126),(83,135,93,125),(84,134,94,124),(85,133,95,123),(86,132,96,122),(87,131,97,121),(88,130,98,140),(89,129,99,139),(90,128,100,138),(161,225,171,235),(162,224,172,234),(163,223,173,233),(164,222,174,232),(165,221,175,231),(166,240,176,230),(167,239,177,229),(168,238,178,228),(169,237,179,227),(170,236,180,226)]])

72 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 ··· 10N 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 20I ··· 20P 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 12 12 12 12 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 2 2 2 6 6 20 20 60 60 2 2 2 2 2 2 ··· 2 6 ··· 6 4 4 20 20 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 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 type + + + + + + + + + - + + + + + + - + - + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D5 D6 D6 C4○D4 D10 D10 D10 C5⋊D4 Dic10 C4○D12 C4○D20 S3×D4 S3×Q8 S3×D5 C2×S3×D5 S3×Dic10 D6.D10 S3×C5⋊D4 kernel D6⋊Dic10 D6⋊Dic5 C6.Dic10 C3×C10.D4 C30.4Q8 C2×C15⋊Q8 S3×C2×C20 C10.D4 C5×Dic3 S3×C10 S3×C2×C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C22×S3 Dic3 D6 C10 C6 C10 C10 C2×C4 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 1 2 2 2 2 1 2 2 2 2 8 8 4 8 1 1 2 2 4 4 4

Matrix representation of D6⋊Dic10 in GL4(𝔽61) generated by

 1 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 1 1 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 50 0 0 0 11 11 0 0 0 0 36 32 0 0 2 34
,
 52 43 0 0 52 9 0 0 0 0 7 29 0 0 13 54
G:=sub<GL(4,GF(61))| [1,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,60,0,0,0,0,60,0,0,0,0,60],[50,11,0,0,0,11,0,0,0,0,36,2,0,0,32,34],[52,52,0,0,43,9,0,0,0,0,7,13,0,0,29,54] >;

D6⋊Dic10 in GAP, Magma, Sage, TeX

D_6\rtimes {\rm Dic}_{10}
% in TeX

G:=Group("D6:Dic10");
// GroupNames label

G:=SmallGroup(480,428);
// by ID

G=gap.SmallGroup(480,428);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=c^10,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

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