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## G = D10⋊D12order 480 = 25·3·5

### 1st semidirect product of D10 and D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D10⋊D12
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — D10⋊D12
 Lower central C15 — C2×C30 — D10⋊D12
 Upper central C1 — C22 — C2×C4

Generators and relations for D10⋊D12
G = < a,b,c,d | a10=b2=c12=d2=1, bab=cac-1=dad=a-1, cbc-1=a8b, dbd=a3b, dcd=c-1 >

Subgroups: 1148 in 188 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×3], C10 [×3], C10, Dic3 [×2], C12 [×3], D6 [×6], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], Dic5, C20 [×2], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], D12 [×2], C2×Dic3, C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12 [×3], C22×S3, C22×S3, C22×C6, C5×S3, C3×D5 [×2], D15, C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×2], C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C4⋊Dic3, D6⋊C4, D6⋊C4, C2×D12, C2×C3⋊D4 [×2], C22×C12, C5×Dic3, C3×Dic5 [×2], Dic15, C60, C6×D5 [×2], C6×D5 [×2], S3×C10 [×3], D30 [×3], C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4 [×2], C127D4, C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], D5×C12 [×2], C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22×D15, D10⋊D4, C30.Q8, C5×D6⋊C4, D303C4, C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, D5×C2×C12, D10⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C22×D5, C2×D12, C4○D12, C2×C3⋊D4, S3×D5, C4○D20, D4×D5 [×2], C127D4, C2×S3×D5, D10⋊D4, D6.D10, D5×D12, D5×C3⋊D4, D10⋊D12

Smallest permutation representation of D10⋊D12
On 240 points
Generators in S240
```(1 35 118 163 17 209 126 171 62 225)(2 226 63 172 127 210 18 164 119 36)(3 25 120 165 19 211 128 173 64 227)(4 228 65 174 129 212 20 166 109 26)(5 27 110 167 21 213 130 175 66 217)(6 218 67 176 131 214 22 168 111 28)(7 29 112 157 23 215 132 177 68 219)(8 220 69 178 121 216 24 158 113 30)(9 31 114 159 13 205 122 179 70 221)(10 222 71 180 123 206 14 160 115 32)(11 33 116 161 15 207 124 169 72 223)(12 224 61 170 125 208 16 162 117 34)(37 77 188 50 98 197 238 91 145 143)(38 144 146 92 239 198 99 51 189 78)(39 79 190 52 100 199 240 93 147 133)(40 134 148 94 229 200 101 53 191 80)(41 81 192 54 102 201 230 95 149 135)(42 136 150 96 231 202 103 55 181 82)(43 83 182 56 104 203 232 85 151 137)(44 138 152 86 233 204 105 57 183 84)(45 73 184 58 106 193 234 87 153 139)(46 140 154 88 235 194 107 59 185 74)(47 75 186 60 108 195 236 89 155 141)(48 142 156 90 237 196 97 49 187 76)
(1 23)(2 121)(3 13)(4 123)(5 15)(6 125)(7 17)(8 127)(9 19)(10 129)(11 21)(12 131)(14 109)(16 111)(18 113)(20 115)(22 117)(24 119)(25 159)(26 206)(27 161)(28 208)(29 163)(30 210)(31 165)(32 212)(33 167)(34 214)(35 157)(36 216)(37 85)(38 138)(39 87)(40 140)(41 89)(42 142)(43 91)(44 144)(45 93)(46 134)(47 95)(48 136)(49 231)(50 104)(51 233)(52 106)(53 235)(54 108)(55 237)(56 98)(57 239)(58 100)(59 229)(60 102)(61 67)(62 132)(63 69)(64 122)(65 71)(66 124)(68 126)(70 128)(72 130)(73 240)(74 148)(75 230)(76 150)(77 232)(78 152)(79 234)(80 154)(81 236)(82 156)(83 238)(84 146)(86 189)(88 191)(90 181)(92 183)(94 185)(96 187)(97 202)(99 204)(101 194)(103 196)(105 198)(107 200)(110 116)(112 118)(114 120)(133 153)(135 155)(137 145)(139 147)(141 149)(143 151)(158 164)(160 166)(162 168)(169 175)(170 218)(171 177)(172 220)(173 179)(174 222)(176 224)(178 226)(180 228)(182 197)(184 199)(186 201)(188 203)(190 193)(192 195)(205 227)(207 217)(209 219)(211 221)(213 223)(215 225)
(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)
(1 101)(2 100)(3 99)(4 98)(5 97)(6 108)(7 107)(8 106)(9 105)(10 104)(11 103)(12 102)(13 152)(14 151)(15 150)(16 149)(17 148)(18 147)(19 146)(20 145)(21 156)(22 155)(23 154)(24 153)(25 198)(26 197)(27 196)(28 195)(29 194)(30 193)(31 204)(32 203)(33 202)(34 201)(35 200)(36 199)(37 129)(38 128)(39 127)(40 126)(41 125)(42 124)(43 123)(44 122)(45 121)(46 132)(47 131)(48 130)(49 217)(50 228)(51 227)(52 226)(53 225)(54 224)(55 223)(56 222)(57 221)(58 220)(59 219)(60 218)(61 192)(62 191)(63 190)(64 189)(65 188)(66 187)(67 186)(68 185)(69 184)(70 183)(71 182)(72 181)(73 178)(74 177)(75 176)(76 175)(77 174)(78 173)(79 172)(80 171)(81 170)(82 169)(83 180)(84 179)(85 160)(86 159)(87 158)(88 157)(89 168)(90 167)(91 166)(92 165)(93 164)(94 163)(95 162)(96 161)(109 238)(110 237)(111 236)(112 235)(113 234)(114 233)(115 232)(116 231)(117 230)(118 229)(119 240)(120 239)(133 210)(134 209)(135 208)(136 207)(137 206)(138 205)(139 216)(140 215)(141 214)(142 213)(143 212)(144 211)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 10 10 10 12 12 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 10 10 12 60 2 2 2 10 10 12 60 2 2 2 2 2 10 10 10 10 2 ··· 2 12 12 12 12 2 2 2 2 10 10 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 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 C4○D4 D10 D10 D10 C3⋊D4 D12 C4○D12 C4○D20 S3×D5 D4×D5 C2×S3×D5 D6.D10 D5×D12 D5×C3⋊D4 kernel D10⋊D12 C30.Q8 C5×D6⋊C4 D30⋊3C4 C2×C15⋊D4 C2×C3⋊D20 C2×C5⋊D12 D5×C2×C12 C2×C4×D5 C3×Dic5 C6×D5 D6⋊C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C22×S3 Dic5 D10 C10 C6 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 2 2 2 2 4 4 4 8 2 4 2 4 4 4

Matrix representation of D10⋊D12 in GL6(𝔽61)

 18 18 0 0 0 0 43 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 18 18 0 0 0 0 60 43 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 53 60
,
 1 0 0 0 0 0 43 60 0 0 0 0 0 0 21 0 0 0 0 0 0 32 0 0 0 0 0 0 47 0 0 0 0 0 47 13
,
 1 0 0 0 0 0 43 60 0 0 0 0 0 0 0 32 0 0 0 0 21 0 0 0 0 0 0 0 13 49 0 0 0 0 14 48

`G:=sub<GL(6,GF(61))| [18,43,0,0,0,0,18,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,60,0,0,0,0,18,43,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,53,0,0,0,0,0,60],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,21,0,0,0,0,0,0,32,0,0,0,0,0,0,47,47,0,0,0,0,0,13],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,0,21,0,0,0,0,32,0,0,0,0,0,0,0,13,14,0,0,0,0,49,48] >;`

D10⋊D12 in GAP, Magma, Sage, TeX

`D_{10}\rtimes D_{12}`
`% in TeX`

`G:=Group("D10:D12");`
`// GroupNames label`

`G:=SmallGroup(480,524);`
`// by ID`

`G=gap.SmallGroup(480,524);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,142,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^12=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^8*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽