Dic10.D6 - GroupNames

G = Dic₁₀.D₆ order 480 = 2⁵·3·5

1^st non-split extension by Dic₁₀ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₅⋊₁Q₁₆, C₄₀.₁₇D₆, Dic₁₂⋊₄D₅, Dic₂₀⋊₄S₃, D₃₀.₂₂D₄, C₂₄.₁₇D₁₀, Dic₁₀.₁D₆, Dic₆.₁D₁₀, C₆₀.₁₄₃C₂³, C₁₂₀.₃₆C₂², Dic₁₅.₂₇D₄, C₅⋊₂(S₃×Q₁₆), C₃⋊₂(D₅×Q₁₆), C₁₅⋊₃(C₂×Q₁₆), C₈.₂₉(S₃×D₅), C₆.₃₂(D₄×D₅), (C₈×D₁₅).₂C₂, C₁₅⋊Q₁₆⋊₉C₂, C₁₀.₃₂(S₃×D₄), C₃₀.₁₈(C₂×D₄), D₁₅⋊Q₈.₂C₂, (C₅×Dic₁₂)⋊₆C₂, (C₃×Dic₂₀)⋊₇C₂, C₂₀.₇₅(C₂²×S₃), C₁₂.₇₄(C₂²×D₅), C₂.₁₀(C₂₀⋊D₆), C₁₅⋊₃C₈.₄₀C₂², (C₄×D₁₅).₅₅C₂², (C₅×Dic₆).₂₇C₂², (C₃×Dic₁₀).₂₇C₂², C₄.₁₁₆(C₂×S₃×D₅), SmallGroup(480,340)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆₀ — Dic₁₀.D₆

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆₀ — C₃×Dic₁₀ — D₁₅⋊Q₈ — Dic₁₀.D₆

Lower central

C₁₅ — C₃₀ — C₆₀ — Dic₁₀.D₆

Upper central

C₁ — C₂ — C₄ — C₈

Generators and relations for Dic₁₀.D₆
G = < a,b,c,d | a¹²=d²=1, b²=c¹⁰=a⁶, bab^-1=cac^-1=a^-1, dad=a⁵, cbc^-1=a³b, bd=db, dcd=c⁹ >

Subgroups: 636 in 120 conjugacy classes, 40 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₈, C₈, C₂×C₄, Q₈, D₅, C₁₀, Dic₃, C₁₂, C₁₂, D₆, C₁₅, C₂×C₈, Q₁₆, C₂×Q₈, Dic₅, C₂₀, C₂₀, D₁₀, C₃⋊C₈, C₂₄, Dic₆, Dic₆, C₄×S₃, C₃×Q₈, D₁₅, C₃₀, C₂×Q₁₆, C₅⋊₂C₈, C₄₀, Dic₁₀, Dic₁₀, C₄×D₅, C₅×Q₈, S₃×C₈, Dic₁₂, C₃⋊Q₁₆, C₃×Q₁₆, S₃×Q₈, C₅×Dic₃, C₃×Dic₅, Dic₁₅, C₆₀, D₃₀, C₈×D₅, Dic₂₀, C₅⋊Q₁₆, C₅×Q₁₆, Q₈×D₅, S₃×Q₁₆, C₁₅⋊₃C₈, C₁₂₀, D₃₀.C₂, C₁₅⋊Q₈, C₃×Dic₁₀, C₅×Dic₆, C₄×D₁₅, D₅×Q₁₆, C₁₅⋊Q₁₆, C₃×Dic₂₀, C₅×Dic₁₂, C₈×D₁₅, D₁₅⋊Q₈, Dic₁₀.D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, Q₁₆, C₂×D₄, D₁₀, C₂²×S₃, C₂×Q₁₆, C₂²×D₅, S₃×D₄, S₃×D₅, D₄×D₅, S₃×Q₁₆, C₂×S₃×D₅, D₅×Q₁₆, C₂₀⋊D₆, Dic₁₀.D₆

Smallest permutation representation of Dic₁₀.D₆
►On 240 points

Generators in S₂₄₀

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

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

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

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

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

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

51 conjugacy classes

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	5A	5B	6	8A	8B	8C	8D	10A	10B	12A	12B	12C	15A	15B	20A	20B	20C	20D	20E	20F	24A	24B	30A	30B	40A	40B	40C	40D	60A	60B	60C	60D	120A	···	120H
order	1	2	2	2	3	4	4	4	4	4	4	5	5	6	8	8	8	8	10	10	12	12	12	15	15	20	20	20	20	20	20	24	24	30	30	40	40	40	40	60	60	60	60	120	···	120
size	1	1	15	15	2	2	12	12	20	20	30	2	2	2	2	2	30	30	2	2	4	40	40	4	4	4	4	24	24	24	24	4	4	4	4	4	4	4	4	4	4	4	4	4	···	4

51 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	-	+	+	+	+	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₅	D₆	D₆	Q₁₆	D₁₀	D₁₀	S₃×D₄	S₃×D₅	D₄×D₅	S₃×Q₁₆	C₂×S₃×D₅	D₅×Q₁₆	C₂₀⋊D₆	Dic₁₀.D₆
kernel	Dic₁₀.D₆	C₁₅⋊Q₁₆	C₃×Dic₂₀	C₅×Dic₁₂	C₈×D₁₅	D₁₅⋊Q₈	Dic₂₀	Dic₁₅	D₃₀	Dic₁₂	C₄₀	Dic₁₀	D₁₅	C₂₄	Dic₆	C₁₀	C₈	C₆	C₅	C₄	C₃	C₂	C₁
# reps	1	2	1	1	1	2	1	1	1	2	1	2	4	2	4	1	2	2	2	2	4	4	8

Matrix representation of Dic₁₀.D₆ ►in GL₆(𝔽₂₄₁)

0	1	0	0	0	0
240	0	0	0	0	0
0	0	240	1	0	0
0	0	240	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

209	136	0	0	0	0
136	32	0	0	0	0
0	0	0	240	0	0
0	0	240	0	0	0
0	0	0	0	240	0
0	0	0	0	0	240

61	80	0	0	0	0
80	180	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	190
0	0	0	0	52	189

240	0	0	0	0	0
0	240	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	52	190
0	0	0	0	53	189

G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[209,136,0,0,0,0,136,32,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[61,80,0,0,0,0,80,180,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,52,0,0,0,0,190,189],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,52,53,0,0,0,0,190,189] >;

Dic₁₀.D₆ in GAP, Magma, Sage, TeX

{\rm Dic}_{10}.D_6

% in TeX

G:=Group("Dic10.D6");

// GroupNames label

G:=SmallGroup(480,340);

// by ID

G=gap.SmallGroup(480,340);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,135,142,675,346,80,1356,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^12=d^2=1,b^2=c^10=a^6,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^9>;

// generators/relations

G = Dic10.D6 order 480 = 25·3·5

1st non-split extension by Dic10 of D6 acting via D6/C3=C22

G = Dic₁₀.D₆ order 480 = 2⁵·3·5

1^st non-split extension by Dic₁₀ of D₆ acting via D₆/C₃=C₂²