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## G = Dic5⋊D12order 480 = 25·3·5

### 1st semidirect product of Dic5 and D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic5⋊D12
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×S3×Dic5 — Dic5⋊D12
 Lower central C15 — C2×C30 — Dic5⋊D12
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic5⋊D12
G = < a,b,c,d | a10=c12=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=c-1 >

Subgroups: 1100 in 188 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5, C10 [×3], C10 [×3], Dic3, C12 [×4], D6 [×2], D6 [×8], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], Dic5 [×2], C20, D10 [×3], C2×C10, C2×C10 [×7], C4×S3 [×2], D12 [×6], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3 [×2], C22×S3, C5×S3 [×3], D15, C30 [×3], C4⋊D4, C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12 [×2], C3×Dic5 [×2], C3×Dic5, Dic15, C60, S3×C10 [×2], S3×C10 [×5], D30 [×3], C2×C30, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4 [×2], D4×C10, C12⋊D4, S3×Dic5 [×2], C5⋊D12 [×4], C6×Dic5 [×2], C5×D12 [×2], C2×Dic15, C2×C60, S3×C2×C10 [×2], C22×D15, Dic5⋊D4, D6⋊Dic5, C3×C10.D4, D303C4, C2×S3×Dic5, C2×C5⋊D12 [×2], C10×D12, Dic5⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], D12 [×2], C22×S3, C4⋊D4, C5⋊D4 [×2], C22×D5, C2×D12, S3×D4, Q83S3, S3×D5, D4×D5, D42D5, C2×C5⋊D4, C12⋊D4, C2×S3×D5, Dic5⋊D4, D12⋊D5, D5×D12, S3×C5⋊D4, Dic5⋊D12

Smallest permutation representation of Dic5⋊D12
On 240 points
Generators in S240
(1 225 81 105 192 40 23 122 201 230)(2 226 82 106 181 41 24 123 202 231)(3 227 83 107 182 42 13 124 203 232)(4 228 84 108 183 43 14 125 204 233)(5 217 73 97 184 44 15 126 193 234)(6 218 74 98 185 45 16 127 194 235)(7 219 75 99 186 46 17 128 195 236)(8 220 76 100 187 47 18 129 196 237)(9 221 77 101 188 48 19 130 197 238)(10 222 78 102 189 37 20 131 198 239)(11 223 79 103 190 38 21 132 199 240)(12 224 80 104 191 39 22 121 200 229)(25 65 60 147 160 171 109 89 139 213)(26 66 49 148 161 172 110 90 140 214)(27 67 50 149 162 173 111 91 141 215)(28 68 51 150 163 174 112 92 142 216)(29 69 52 151 164 175 113 93 143 205)(30 70 53 152 165 176 114 94 144 206)(31 71 54 153 166 177 115 95 133 207)(32 72 55 154 167 178 116 96 134 208)(33 61 56 155 168 179 117 85 135 209)(34 62 57 156 157 180 118 86 136 210)(35 63 58 145 158 169 119 87 137 211)(36 64 59 146 159 170 120 88 138 212)
(1 176 40 30)(2 31 41 177)(3 178 42 32)(4 33 43 179)(5 180 44 34)(6 35 45 169)(7 170 46 36)(8 25 47 171)(9 172 48 26)(10 27 37 173)(11 174 38 28)(12 29 39 175)(13 208 227 167)(14 168 228 209)(15 210 217 157)(16 158 218 211)(17 212 219 159)(18 160 220 213)(19 214 221 161)(20 162 222 215)(21 216 223 163)(22 164 224 205)(23 206 225 165)(24 166 226 207)(49 197 90 101)(50 102 91 198)(51 199 92 103)(52 104 93 200)(53 201 94 105)(54 106 95 202)(55 203 96 107)(56 108 85 204)(57 193 86 97)(58 98 87 194)(59 195 88 99)(60 100 89 196)(61 183 117 233)(62 234 118 184)(63 185 119 235)(64 236 120 186)(65 187 109 237)(66 238 110 188)(67 189 111 239)(68 240 112 190)(69 191 113 229)(70 230 114 192)(71 181 115 231)(72 232 116 182)(73 156 126 136)(74 137 127 145)(75 146 128 138)(76 139 129 147)(77 148 130 140)(78 141 131 149)(79 150 132 142)(80 143 121 151)(81 152 122 144)(82 133 123 153)(83 154 124 134)(84 135 125 155)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 76)(74 75)(77 84)(78 83)(79 82)(80 81)(85 90)(86 89)(87 88)(91 96)(92 95)(93 94)(97 100)(98 99)(101 108)(102 107)(103 106)(104 105)(109 118)(110 117)(111 116)(112 115)(113 114)(119 120)(121 122)(123 132)(124 131)(125 130)(126 129)(127 128)(133 142)(134 141)(135 140)(136 139)(137 138)(143 144)(145 146)(147 156)(148 155)(149 154)(150 153)(151 152)(157 160)(158 159)(161 168)(162 167)(163 166)(164 165)(169 170)(171 180)(172 179)(173 178)(174 177)(175 176)(181 190)(182 189)(183 188)(184 187)(185 186)(191 192)(193 196)(194 195)(197 204)(198 203)(199 202)(200 201)(205 206)(207 216)(208 215)(209 214)(210 213)(211 212)(217 220)(218 219)(221 228)(222 227)(223 226)(224 225)(229 230)(231 240)(232 239)(233 238)(234 237)(235 236)

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

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

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

60 conjugacy classes

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

Matrix representation of Dic5⋊D12 in GL6(𝔽61)

 0 60 0 0 0 0 1 18 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
,
 15 11 0 0 0 0 46 46 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
,
 30 44 0 0 0 0 17 31 0 0 0 0 0 0 13 20 0 0 0 0 22 48 0 0 0 0 0 0 59 41 0 0 0 0 52 1
,
 30 44 0 0 0 0 17 31 0 0 0 0 0 0 13 20 0 0 0 0 16 48 0 0 0 0 0 0 59 41 0 0 0 0 52 2

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,46,0,0,0,0,11,46,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[30,17,0,0,0,0,44,31,0,0,0,0,0,0,13,22,0,0,0,0,20,48,0,0,0,0,0,0,59,52,0,0,0,0,41,1],[30,17,0,0,0,0,44,31,0,0,0,0,0,0,13,16,0,0,0,0,20,48,0,0,0,0,0,0,59,52,0,0,0,0,41,2] >;

Dic5⋊D12 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes D_{12}
% in TeX

G:=Group("Dic5:D12");
// GroupNames label

G:=SmallGroup(480,492);
// by ID

G=gap.SmallGroup(480,492);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,422,135,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^12=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

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