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G = D63Dic10order 480 = 25·3·5

3rd semidirect product of D6 and Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D63Dic10, Dic15.8D4, D6⋊C4.4D5, (S3×C10)⋊3Q8, C6.61(D4×D5), C53(D6⋊Q8), C10.63(S3×D4), (C2×C20).24D6, C30.49(C2×Q8), C10.35(S3×Q8), (C2×Dic30)⋊2C2, C10.D47S3, C30.138(C2×D4), (C2×C12).24D10, C1520(C22⋊Q8), C30.71(C4○D4), D6⋊Dic5.13C2, (C2×C60).11C22, C6.Dic1023C2, Dic155C421C2, (C2×Dic5).40D6, C2.17(S3×Dic10), C6.17(C2×Dic10), C10.58(C4○D12), C6.26(D42D5), (C2×C30).122C23, (C2×Dic3).40D10, (C22×S3).42D10, C2.14(D10⋊D6), C2.14(D125D5), C32(Dic5.14D4), (C6×Dic5).75C22, (C10×Dic3).76C22, (C2×Dic15).99C22, (C5×D6⋊C4).4C2, (C2×C4).55(S3×D5), (C2×S3×Dic5).7C2, C22.185(C2×S3×D5), (C3×C10.D4)⋊7C2, (S3×C2×C10).23C22, (C2×C6).134(C22×D5), (C2×C10).134(C22×S3), SmallGroup(480,508)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D63Dic10
C1C5C15C30C2×C30C6×Dic5C2×S3×Dic5 — D63Dic10
C15C2×C30 — D63Dic10
C1C22C2×C4

Generators and relations for D63Dic10
 G = < a,b,c,d | a6=b2=c20=1, d2=c10, bab=dad-1=a-1, ac=ca, cbc-1=a3b, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 716 in 148 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×4], C12 [×3], D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×5], C20 [×2], C2×C10, C2×C10 [×4], Dic6 [×2], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C22×C10, Dic3⋊C4 [×2], D6⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C3×Dic5 [×2], Dic15 [×2], Dic15, C60, S3×C10 [×2], S3×C10 [×2], C2×C30, C10.D4, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, D6⋊Q8, S3×Dic5 [×2], C6×Dic5 [×2], C10×Dic3, Dic30 [×2], C2×Dic15 [×2], C2×C60, S3×C2×C10, Dic5.14D4, D6⋊Dic5, Dic155C4, C6.Dic10, C3×C10.D4, C5×D6⋊C4, C2×S3×Dic5, C2×Dic30, D63Dic10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], C22×S3, C22⋊Q8, Dic10 [×2], C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, C2×Dic10, D4×D5, D42D5, D6⋊Q8, C2×S3×D5, Dic5.14D4, S3×Dic10, D125D5, D10⋊D6, D63Dic10

Smallest permutation representation of D63Dic10
On 240 points
Generators in S240
(1 65 236 60 156 135)(2 66 237 41 157 136)(3 67 238 42 158 137)(4 68 239 43 159 138)(5 69 240 44 160 139)(6 70 221 45 141 140)(7 71 222 46 142 121)(8 72 223 47 143 122)(9 73 224 48 144 123)(10 74 225 49 145 124)(11 75 226 50 146 125)(12 76 227 51 147 126)(13 77 228 52 148 127)(14 78 229 53 149 128)(15 79 230 54 150 129)(16 80 231 55 151 130)(17 61 232 56 152 131)(18 62 233 57 153 132)(19 63 234 58 154 133)(20 64 235 59 155 134)(21 91 218 119 164 192)(22 92 219 120 165 193)(23 93 220 101 166 194)(24 94 201 102 167 195)(25 95 202 103 168 196)(26 96 203 104 169 197)(27 97 204 105 170 198)(28 98 205 106 171 199)(29 99 206 107 172 200)(30 100 207 108 173 181)(31 81 208 109 174 182)(32 82 209 110 175 183)(33 83 210 111 176 184)(34 84 211 112 177 185)(35 85 212 113 178 186)(36 86 213 114 179 187)(37 87 214 115 180 188)(38 88 215 116 161 189)(39 89 216 117 162 190)(40 90 217 118 163 191)
(1 125)(2 227)(3 127)(4 229)(5 129)(6 231)(7 131)(8 233)(9 133)(10 235)(11 135)(12 237)(13 137)(14 239)(15 139)(16 221)(17 121)(18 223)(19 123)(20 225)(21 182)(22 209)(23 184)(24 211)(25 186)(26 213)(27 188)(28 215)(29 190)(30 217)(31 192)(32 219)(33 194)(34 201)(35 196)(36 203)(37 198)(38 205)(39 200)(40 207)(41 126)(42 228)(43 128)(44 230)(45 130)(46 232)(47 132)(48 234)(49 134)(50 236)(51 136)(52 238)(53 138)(54 240)(55 140)(56 222)(57 122)(58 224)(59 124)(60 226)(61 142)(62 72)(63 144)(64 74)(65 146)(66 76)(67 148)(68 78)(69 150)(70 80)(71 152)(73 154)(75 156)(77 158)(79 160)(81 164)(82 92)(83 166)(84 94)(85 168)(86 96)(87 170)(88 98)(89 172)(90 100)(91 174)(93 176)(95 178)(97 180)(99 162)(101 210)(102 185)(103 212)(104 187)(105 214)(106 189)(107 216)(108 191)(109 218)(110 193)(111 220)(112 195)(113 202)(114 197)(115 204)(116 199)(117 206)(118 181)(119 208)(120 183)(141 151)(143 153)(145 155)(147 157)(149 159)(161 171)(163 173)(165 175)(167 177)(169 179)
(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)
(1 105 11 115)(2 104 12 114)(3 103 13 113)(4 102 14 112)(5 101 15 111)(6 120 16 110)(7 119 17 109)(8 118 18 108)(9 117 19 107)(10 116 20 106)(21 56 31 46)(22 55 32 45)(23 54 33 44)(24 53 34 43)(25 52 35 42)(26 51 36 41)(27 50 37 60)(28 49 38 59)(29 48 39 58)(30 47 40 57)(61 208 71 218)(62 207 72 217)(63 206 73 216)(64 205 74 215)(65 204 75 214)(66 203 76 213)(67 202 77 212)(68 201 78 211)(69 220 79 210)(70 219 80 209)(81 222 91 232)(82 221 92 231)(83 240 93 230)(84 239 94 229)(85 238 95 228)(86 237 96 227)(87 236 97 226)(88 235 98 225)(89 234 99 224)(90 233 100 223)(121 164 131 174)(122 163 132 173)(123 162 133 172)(124 161 134 171)(125 180 135 170)(126 179 136 169)(127 178 137 168)(128 177 138 167)(129 176 139 166)(130 175 140 165)(141 193 151 183)(142 192 152 182)(143 191 153 181)(144 190 154 200)(145 189 155 199)(146 188 156 198)(147 187 157 197)(148 186 158 196)(149 185 159 195)(150 184 160 194)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222223444444445566610···10101010101212121212121515202020202020202030···3060···60
size1111662410101220303060222222···2121212124420202020444444121212124···44···4

60 irreducible representations

dim11111111222222222222444444444
type++++++++++-++++++-+-++-+--+
imageC1C2C2C2C2C2C2C2S3D4Q8D5D6D6C4○D4D10D10D10Dic10C4○D12S3×D4S3×Q8S3×D5D4×D5D42D5C2×S3×D5S3×Dic10D125D5D10⋊D6
kernelD63Dic10D6⋊Dic5Dic155C4C6.Dic10C3×C10.D4C5×D6⋊C4C2×S3×Dic5C2×Dic30C10.D4Dic15S3×C10D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3D6C10C10C10C2×C4C6C6C22C2C2C2
# reps11111111122221222284112222444

Matrix representation of D63Dic10 in GL4(𝔽61) generated by

1000
0100
0011
00600
,
60000
06000
006060
0001
,
33000
383700
003815
004623
,
25800
225900
00110
005050
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,60,1],[33,38,0,0,0,37,0,0,0,0,38,46,0,0,15,23],[2,22,0,0,58,59,0,0,0,0,11,50,0,0,0,50] >;

D63Dic10 in GAP, Magma, Sage, TeX

D_6\rtimes_3{\rm Dic}_{10}
% in TeX

G:=Group("D6:3Dic10");
// GroupNames label

G:=SmallGroup(480,508);
// by ID

G=gap.SmallGroup(480,508);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,590,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=c^10,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽