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## G = D10⋊4Dic6order 480 = 25·3·5

### 4th semidirect product of D10 and Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D10⋊4Dic6
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — D10⋊4Dic6
 Lower central C15 — C2×C30 — D10⋊4Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for D104Dic6
G = < a,b,c,d | a10=b2=c12=1, d2=c6, bab=dad-1=a-1, ac=ca, cbc-1=a5b, dbd-1=a8b, dcd-1=c-1 >

Subgroups: 748 in 148 conjugacy classes, 50 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, 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, C3×C22⋊C4, C2×Dic6, C22×Dic3, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C6×D5, C6×D5, C2×C30, C10.D4, D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, Dic3.D4, D5×Dic3, C6×Dic5, C10×Dic3, Dic30, C2×Dic15, C2×C60, D5×C2×C6, D10⋊Q8, D10⋊Dic3, C30.Q8, Dic155C4, C3×D10⋊C4, C5×Dic3⋊C4, C2×D5×Dic3, C2×Dic30, D104Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, Dic6, C22×S3, C22⋊Q8, C22×D5, C2×Dic6, S3×D4, D42S3, S3×D5, C4○D20, D4×D5, Q8×D5, Dic3.D4, C2×S3×D5, D10⋊Q8, D5×Dic6, D205S3, D10⋊D6, D104Dic6

Smallest permutation representation of D104Dic6
On 240 points
Generators in S240
(1 128 177 186 38 53 100 147 219 193)(2 129 178 187 39 54 101 148 220 194)(3 130 179 188 40 55 102 149 221 195)(4 131 180 189 41 56 103 150 222 196)(5 132 169 190 42 57 104 151 223 197)(6 121 170 191 43 58 105 152 224 198)(7 122 171 192 44 59 106 153 225 199)(8 123 172 181 45 60 107 154 226 200)(9 124 173 182 46 49 108 155 227 201)(10 125 174 183 47 50 97 156 228 202)(11 126 175 184 48 51 98 145 217 203)(12 127 176 185 37 52 99 146 218 204)(13 162 77 238 140 71 209 96 119 27)(14 163 78 239 141 72 210 85 120 28)(15 164 79 240 142 61 211 86 109 29)(16 165 80 229 143 62 212 87 110 30)(17 166 81 230 144 63 213 88 111 31)(18 167 82 231 133 64 214 89 112 32)(19 168 83 232 134 65 215 90 113 33)(20 157 84 233 135 66 216 91 114 34)(21 158 73 234 136 67 205 92 115 35)(22 159 74 235 137 68 206 93 116 36)(23 160 75 236 138 69 207 94 117 25)(24 161 76 237 139 70 208 95 118 26)
(1 193)(2 39)(3 195)(4 41)(5 197)(6 43)(7 199)(8 45)(9 201)(10 47)(11 203)(12 37)(13 238)(14 120)(15 240)(16 110)(17 230)(18 112)(19 232)(20 114)(21 234)(22 116)(23 236)(24 118)(25 138)(27 140)(29 142)(31 144)(33 134)(35 136)(38 53)(40 55)(42 57)(44 59)(46 49)(48 51)(50 202)(52 204)(54 194)(56 196)(58 198)(60 200)(61 109)(62 229)(63 111)(64 231)(65 113)(66 233)(67 115)(68 235)(69 117)(70 237)(71 119)(72 239)(73 158)(74 206)(75 160)(76 208)(77 162)(78 210)(79 164)(80 212)(81 166)(82 214)(83 168)(84 216)(85 163)(86 211)(87 165)(88 213)(89 167)(90 215)(91 157)(92 205)(93 159)(94 207)(95 161)(96 209)(97 228)(98 184)(99 218)(100 186)(101 220)(102 188)(103 222)(104 190)(105 224)(106 192)(107 226)(108 182)(121 191)(122 225)(123 181)(124 227)(125 183)(126 217)(127 185)(128 219)(129 187)(130 221)(131 189)(132 223)(145 175)(147 177)(149 179)(151 169)(153 171)(155 173)
(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 73 7 79)(2 84 8 78)(3 83 9 77)(4 82 10 76)(5 81 11 75)(6 80 12 74)(13 179 19 173)(14 178 20 172)(15 177 21 171)(16 176 22 170)(17 175 23 169)(18 174 24 180)(25 190 31 184)(26 189 32 183)(27 188 33 182)(28 187 34 181)(29 186 35 192)(30 185 36 191)(37 116 43 110)(38 115 44 109)(39 114 45 120)(40 113 46 119)(41 112 47 118)(42 111 48 117)(49 96 55 90)(50 95 56 89)(51 94 57 88)(52 93 58 87)(53 92 59 86)(54 91 60 85)(61 147 67 153)(62 146 68 152)(63 145 69 151)(64 156 70 150)(65 155 71 149)(66 154 72 148)(97 208 103 214)(98 207 104 213)(99 206 105 212)(100 205 106 211)(101 216 107 210)(102 215 108 209)(121 165 127 159)(122 164 128 158)(123 163 129 157)(124 162 130 168)(125 161 131 167)(126 160 132 166)(133 228 139 222)(134 227 140 221)(135 226 141 220)(136 225 142 219)(137 224 143 218)(138 223 144 217)(193 234 199 240)(194 233 200 239)(195 232 201 238)(196 231 202 237)(197 230 203 236)(198 229 204 235)

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

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

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

60 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 10A ··· 10F 12A 12B 12C 12D 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 10 ··· 10 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 2 4 6 6 12 20 30 30 60 2 2 2 2 2 20 20 2 ··· 2 4 4 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 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 C2 S3 D4 Q8 D5 D6 D6 D6 C4○D4 D10 D10 Dic6 C4○D20 S3×D4 D4⋊2S3 S3×D5 D4×D5 Q8×D5 C2×S3×D5 D5×Dic6 D20⋊5S3 D10⋊D6 kernel D10⋊4Dic6 D10⋊Dic3 C30.Q8 Dic15⋊5C4 C3×D10⋊C4 C5×Dic3⋊C4 C2×D5×Dic3 C2×Dic30 D10⋊C4 Dic15 C6×D5 Dic3⋊C4 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 D10 C6 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 2 4 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of D104Dic6 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 1 0 0 0 0 0 0 1
,
 18 18 0 0 0 0 60 43 0 0 0 0 0 0 60 0 0 0 0 0 46 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 54 5 0 0 0 0 51 7 0 0 0 0 0 0 0 21 0 0 0 0 29 8
,
 60 0 0 0 0 0 18 1 0 0 0 0 0 0 50 0 0 0 0 0 18 11 0 0 0 0 0 0 11 0 0 0 0 0 10 50

G:=sub<GL(6,GF(61))| [18,43,0,0,0,0,18,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,60,0,0,0,0,18,43,0,0,0,0,0,0,60,46,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,54,51,0,0,0,0,5,7,0,0,0,0,0,0,0,29,0,0,0,0,21,8],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,50,18,0,0,0,0,0,11,0,0,0,0,0,0,11,10,0,0,0,0,0,50] >;

D104Dic6 in GAP, Magma, Sage, TeX

D_{10}\rtimes_4{\rm Dic}_6
% in TeX

G:=Group("D10:4Dic6");
// GroupNames label

G:=SmallGroup(480,507);
// by ID

G=gap.SmallGroup(480,507);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=c^6,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b,d*b*d^-1=a^8*b,d*c*d^-1=c^-1>;
// generators/relations

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