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G = D102Dic6order 480 = 25·3·5

2nd semidirect product of D10 and Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D102Dic6, Dic3.10D20, (C6×D5)⋊2Q8, C605C46C2, C6.34(Q8×D5), Dic3⋊C45D5, C6.14(C2×D20), (C2×C20).20D6, C10.14(S3×D4), C2.18(S3×D20), C30.45(C2×D4), C30.43(C2×Q8), C32(D102Q8), (C2×C12).19D10, (C2×C60).9C22, C1515(C22⋊Q8), (C5×Dic3).7D4, C2.16(D5×Dic6), D10⋊C4.3S3, C6.Dic1018C2, (C2×Dic5).35D6, C10.16(C2×Dic6), (C22×D5).50D6, C30.120(C4○D4), C6.72(D42D5), (C2×C30).112C23, C51(Dic3.D4), C10.71(D42S3), (C2×Dic3).105D10, D10⋊Dic3.12C2, (C6×Dic5).66C22, C2.17(C30.C23), (C10×Dic3).70C22, (C2×Dic15).90C22, (C2×C15⋊Q8)⋊8C2, (C2×C4).48(S3×D5), (C2×D5×Dic3).6C2, (C5×Dic3⋊C4)⋊5C2, (D5×C2×C6).20C22, C22.178(C2×S3×D5), (C3×D10⋊C4).3C2, (C2×C6).124(C22×D5), (C2×C10).124(C22×S3), SmallGroup(480,498)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D102Dic6
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — D102Dic6
C15C2×C30 — D102Dic6
C1C22C2×C4

Generators and relations for D102Dic6
 G = < a,b,c,d | a10=b2=c12=1, d2=c6, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 748 in 148 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×2], Dic3 [×3], C12 [×2], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×3], C20 [×4], D10 [×2], D10 [×2], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12, C2×C12, C22×C6, C3×D5 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C5×Dic3 [×2], C5×Dic3, C3×Dic5, Dic15 [×2], C60, C6×D5 [×2], C6×D5 [×2], C2×C30, C4⋊Dic5 [×2], D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, Dic3.D4, D5×Dic3 [×2], C15⋊Q8 [×2], C6×Dic5, C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, D102Q8, D10⋊Dic3, C6.Dic10, C3×D10⋊C4, C5×Dic3⋊C4, C605C4, C2×D5×Dic3, C2×C15⋊Q8, D102Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], Dic6 [×2], C22×S3, C22⋊Q8, D20 [×2], C22×D5, C2×Dic6, S3×D4, D42S3, S3×D5, C2×D20, D42D5, Q8×D5, Dic3.D4, C2×S3×D5, D102Q8, D5×Dic6, S3×D20, C30.C23, D102Dic6

Smallest permutation representation of D102Dic6
On 240 points
Generators in S240
(1 157 51 74 201 38 175 183 125 62)(2 63 126 184 176 39 202 75 52 158)(3 159 53 76 203 40 177 185 127 64)(4 65 128 186 178 41 204 77 54 160)(5 161 55 78 193 42 179 187 129 66)(6 67 130 188 180 43 194 79 56 162)(7 163 57 80 195 44 169 189 131 68)(8 69 132 190 170 45 196 81 58 164)(9 165 59 82 197 46 171 191 121 70)(10 71 122 192 172 47 198 83 60 166)(11 167 49 84 199 48 173 181 123 72)(12 61 124 182 174 37 200 73 50 168)(13 233 107 116 214 154 86 141 220 36)(14 25 221 142 87 155 215 117 108 234)(15 235 97 118 216 156 88 143 222 26)(16 27 223 144 89 145 205 119 98 236)(17 237 99 120 206 146 90 133 224 28)(18 29 225 134 91 147 207 109 100 238)(19 239 101 110 208 148 92 135 226 30)(20 31 227 136 93 149 209 111 102 240)(21 229 103 112 210 150 94 137 228 32)(22 33 217 138 95 151 211 113 104 230)(23 231 105 114 212 152 96 139 218 34)(24 35 219 140 85 153 213 115 106 232)
(1 68)(2 170)(3 70)(4 172)(5 72)(6 174)(7 62)(8 176)(9 64)(10 178)(11 66)(12 180)(13 208)(14 240)(15 210)(16 230)(17 212)(18 232)(19 214)(20 234)(21 216)(22 236)(23 206)(24 238)(25 102)(26 150)(27 104)(28 152)(29 106)(30 154)(31 108)(32 156)(33 98)(34 146)(35 100)(36 148)(37 162)(38 195)(39 164)(40 197)(41 166)(42 199)(43 168)(44 201)(45 158)(46 203)(47 160)(48 193)(49 187)(50 194)(51 189)(52 196)(53 191)(54 198)(55 181)(56 200)(57 183)(58 202)(59 185)(60 204)(61 188)(63 190)(65 192)(67 182)(69 184)(71 186)(73 79)(74 169)(75 81)(76 171)(77 83)(78 173)(80 175)(82 177)(84 179)(85 147)(86 226)(87 149)(88 228)(89 151)(90 218)(91 153)(92 220)(93 155)(94 222)(95 145)(96 224)(97 103)(99 105)(101 107)(109 219)(110 233)(111 221)(112 235)(113 223)(114 237)(115 225)(116 239)(117 227)(118 229)(119 217)(120 231)(121 159)(122 128)(123 161)(124 130)(125 163)(126 132)(127 165)(129 167)(131 157)(133 139)(134 213)(135 141)(136 215)(137 143)(138 205)(140 207)(142 209)(144 211)
(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 20 7 14)(2 19 8 13)(3 18 9 24)(4 17 10 23)(5 16 11 22)(6 15 12 21)(25 157 31 163)(26 168 32 162)(27 167 33 161)(28 166 34 160)(29 165 35 159)(30 164 36 158)(37 150 43 156)(38 149 44 155)(39 148 45 154)(40 147 46 153)(41 146 47 152)(42 145 48 151)(49 217 55 223)(50 228 56 222)(51 227 57 221)(52 226 58 220)(53 225 59 219)(54 224 60 218)(61 229 67 235)(62 240 68 234)(63 239 69 233)(64 238 70 232)(65 237 71 231)(66 236 72 230)(73 137 79 143)(74 136 80 142)(75 135 81 141)(76 134 82 140)(77 133 83 139)(78 144 84 138)(85 203 91 197)(86 202 92 196)(87 201 93 195)(88 200 94 194)(89 199 95 193)(90 198 96 204)(97 124 103 130)(98 123 104 129)(99 122 105 128)(100 121 106 127)(101 132 107 126)(102 131 108 125)(109 191 115 185)(110 190 116 184)(111 189 117 183)(112 188 118 182)(113 187 119 181)(114 186 120 192)(169 215 175 209)(170 214 176 208)(171 213 177 207)(172 212 178 206)(173 211 179 205)(174 210 180 216)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L30A···30F60A···60H
order122222344444444556666610···101212121215152020202020···2030···3060···60
size11111010246612203030602222220202···244202044444412···124···44···4

60 irreducible representations

dim11111111222222222222444444444
type++++++++++-++++++-++-+--+-+-
imageC1C2C2C2C2C2C2C2S3D4Q8D5D6D6D6C4○D4D10D10Dic6D20S3×D4D42S3S3×D5D42D5Q8×D5C2×S3×D5D5×Dic6S3×D20C30.C23
kernelD102Dic6D10⋊Dic3C6.Dic10C3×D10⋊C4C5×Dic3⋊C4C605C4C2×D5×Dic3C2×C15⋊Q8D10⋊C4C5×Dic3C6×D5Dic3⋊C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12D10Dic3C10C10C2×C4C6C6C22C2C2C2
# reps11111111122211124248112222444

Matrix representation of D102Dic6 in GL6(𝔽61)

17170000
4410000
001000
000100
000010
000001
,
17170000
1440000
0060000
0006000
000010
000001
,
29540000
59320000
00604100
0055100
0000601
0000600
,
100000
010000
00113700
0005000
00003846
00002323

G:=sub<GL(6,GF(61))| [17,44,0,0,0,0,17,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,1,0,0,0,0,17,44,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[29,59,0,0,0,0,54,32,0,0,0,0,0,0,60,55,0,0,0,0,41,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,37,50,0,0,0,0,0,0,38,23,0,0,0,0,46,23] >;

D102Dic6 in GAP, Magma, Sage, TeX

D_{10}\rtimes_2{\rm Dic}_6
% in TeX

G:=Group("D10:2Dic6");
// GroupNames label

G:=SmallGroup(480,498);
// by ID

G=gap.SmallGroup(480,498);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,142,1356,18822]);
// 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=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽