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## G = D10⋊Dic6order 480 = 25·3·5

### 1st semidirect product of D10 and Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D10⋊Dic6
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — D10⋊Dic3 — D10⋊Dic6
 Lower central C15 — C2×C30 — D10⋊Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for D10⋊Dic6
G = < a,b,c,d | a10=b2=c12=1, d2=c6, bab=cac-1=dad-1=a-1, cbc-1=a8b, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 700 in 148 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, D10, D10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, C2×C30, C10.D4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C12.48D4, C15⋊Q8, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, D10⋊Q8, D10⋊Dic3, C30.Q8, C5×Dic3⋊C4, C30.4Q8, C2×C15⋊Q8, D5×C2×C12, D10⋊Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C22×D5, C2×Dic6, C4○D12, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, Q8×D5, C12.48D4, C2×S3×D5, D10⋊Q8, D5×Dic6, D6.D10, D5×C3⋊D4, D10⋊Dic6

Smallest permutation representation of D10⋊Dic6
On 240 points
Generators in S240
(1 229 94 108 81 129 181 154 43 194)(2 195 44 155 182 130 82 97 95 230)(3 231 96 98 83 131 183 156 45 196)(4 197 46 145 184 132 84 99 85 232)(5 233 86 100 73 121 185 146 47 198)(6 199 48 147 186 122 74 101 87 234)(7 235 88 102 75 123 187 148 37 200)(8 201 38 149 188 124 76 103 89 236)(9 237 90 104 77 125 189 150 39 202)(10 203 40 151 190 126 78 105 91 238)(11 239 92 106 79 127 191 152 41 204)(12 193 42 153 192 128 80 107 93 240)(13 111 224 209 141 66 57 161 179 26)(14 27 180 162 58 67 142 210 225 112)(15 113 226 211 143 68 59 163 169 28)(16 29 170 164 60 69 144 212 227 114)(17 115 228 213 133 70 49 165 171 30)(18 31 172 166 50 71 134 214 217 116)(19 117 218 215 135 72 51 167 173 32)(20 33 174 168 52 61 136 216 219 118)(21 119 220 205 137 62 53 157 175 34)(22 35 176 158 54 63 138 206 221 120)(23 109 222 207 139 64 55 159 177 36)(24 25 178 160 56 65 140 208 223 110)
(1 75)(2 188)(3 77)(4 190)(5 79)(6 192)(7 81)(8 182)(9 83)(10 184)(11 73)(12 186)(13 167)(14 33)(15 157)(16 35)(17 159)(18 25)(19 161)(20 27)(21 163)(22 29)(23 165)(24 31)(26 173)(28 175)(30 177)(32 179)(34 169)(36 171)(37 181)(38 44)(39 183)(40 46)(41 185)(42 48)(43 187)(45 189)(47 191)(49 109)(50 208)(51 111)(52 210)(53 113)(54 212)(55 115)(56 214)(57 117)(58 216)(59 119)(60 206)(61 142)(62 226)(63 144)(64 228)(65 134)(66 218)(67 136)(68 220)(69 138)(70 222)(71 140)(72 224)(74 93)(76 95)(78 85)(80 87)(82 89)(84 91)(86 92)(88 94)(90 96)(97 103)(98 237)(99 105)(100 239)(101 107)(102 229)(104 231)(106 233)(108 235)(110 172)(112 174)(114 176)(116 178)(118 180)(120 170)(121 204)(122 240)(123 194)(124 230)(125 196)(126 232)(127 198)(128 234)(129 200)(130 236)(131 202)(132 238)(133 207)(135 209)(137 211)(139 213)(141 215)(143 205)(145 203)(146 152)(147 193)(148 154)(149 195)(150 156)(151 197)(153 199)(155 201)(158 227)(160 217)(162 219)(164 221)(166 223)(168 225)
(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 25 7 31)(2 36 8 30)(3 35 9 29)(4 34 10 28)(5 33 11 27)(6 32 12 26)(13 234 19 240)(14 233 20 239)(15 232 21 238)(16 231 22 237)(17 230 23 236)(18 229 24 235)(37 166 43 160)(38 165 44 159)(39 164 45 158)(40 163 46 157)(41 162 47 168)(42 161 48 167)(49 155 55 149)(50 154 56 148)(51 153 57 147)(52 152 58 146)(53 151 59 145)(54 150 60 156)(61 191 67 185)(62 190 68 184)(63 189 69 183)(64 188 70 182)(65 187 71 181)(66 186 72 192)(73 216 79 210)(74 215 80 209)(75 214 81 208)(76 213 82 207)(77 212 83 206)(78 211 84 205)(85 119 91 113)(86 118 92 112)(87 117 93 111)(88 116 94 110)(89 115 95 109)(90 114 96 120)(97 222 103 228)(98 221 104 227)(99 220 105 226)(100 219 106 225)(101 218 107 224)(102 217 108 223)(121 136 127 142)(122 135 128 141)(123 134 129 140)(124 133 130 139)(125 144 131 138)(126 143 132 137)(169 197 175 203)(170 196 176 202)(171 195 177 201)(172 194 178 200)(173 193 179 199)(174 204 180 198)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 20E ··· 20L 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 6 6 6 6 10 ··· 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 2 2 2 10 10 12 12 60 60 2 2 2 2 2 10 10 10 10 2 ··· 2 2 2 2 2 10 10 10 10 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 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 D6 C4○D4 D10 D10 C3⋊D4 Dic6 C4○D12 C4○D20 S3×D5 D4×D5 Q8×D5 C2×S3×D5 D5×Dic6 D6.D10 D5×C3⋊D4 kernel D10⋊Dic6 D10⋊Dic3 C30.Q8 C5×Dic3⋊C4 C30.4Q8 C2×C15⋊Q8 D5×C2×C12 C2×C4×D5 C3×Dic5 C6×D5 Dic3⋊C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 Dic5 D10 C10 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 1 2 2 2 1 1 1 2 4 2 4 4 4 8 2 2 2 2 4 4 4

Matrix representation of D10⋊Dic6 in GL6(𝔽61)

 43 43 0 0 0 0 18 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 43 43 0 0 0 0 1 18 0 0 0 0 0 0 60 47 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 11 0 0 0 0 0 46 50 0 0 0 0 0 0 29 45 0 0 0 0 0 40 0 0 0 0 0 0 50 41 0 0 0 0 0 11
,
 30 44 0 0 0 0 53 31 0 0 0 0 0 0 7 50 0 0 0 0 60 54 0 0 0 0 0 0 2 30 0 0 0 0 10 59

G:=sub<GL(6,GF(61))| [43,18,0,0,0,0,43,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[43,1,0,0,0,0,43,18,0,0,0,0,0,0,60,0,0,0,0,0,47,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[11,46,0,0,0,0,0,50,0,0,0,0,0,0,29,0,0,0,0,0,45,40,0,0,0,0,0,0,50,0,0,0,0,0,41,11],[30,53,0,0,0,0,44,31,0,0,0,0,0,0,7,60,0,0,0,0,50,54,0,0,0,0,0,0,2,10,0,0,0,0,30,59] >;

D10⋊Dic6 in GAP, Magma, Sage, TeX

D_{10}\rtimes {\rm Dic}_6
% in TeX

G:=Group("D10:Dic6");
// GroupNames label

G:=SmallGroup(480,425);
// by ID

G=gap.SmallGroup(480,425);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=c^6,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^8*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

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