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G = D6⋊Dic10order 480 = 25·3·5

1st semidirect product of D6 and Dic10 acting via Dic10/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D65Dic10, (S3×C10)⋊5Q8, C57(D6⋊Q8), C152(C22⋊Q8), C30.20(C2×Q8), C10.26(S3×Q8), C10.D42S3, (C2×C20).257D6, C30.105(C2×D4), C10.127(S3×D4), D6⋊Dic5.4C2, C6.Dic102C2, (C2×Dic5).8D6, C6.8(C2×Dic10), C30.19(C4○D4), C6.19(C4○D20), (C2×C12).176D10, C31(C20.48D4), (C2×C30).42C23, C30.4Q828C2, (C5×Dic3).33D4, C2.10(S3×Dic10), C10.22(C4○D12), (C2×C60).401C22, (C22×S3).65D10, (C2×Dic3).135D10, Dic3.14(C5⋊D4), (C6×Dic5).24C22, C2.12(D6.D10), (C2×Dic15).47C22, (C10×Dic3).159C22, (C2×C15⋊Q8)⋊3C2, (S3×C2×C4).9D5, (S3×C2×C20).19C2, (C2×C4).69(S3×D5), C2.10(S3×C5⋊D4), C6.28(C2×C5⋊D4), C22.131(C2×S3×D5), (S3×C2×C10).77C22, (C2×C6).54(C22×D5), (C3×C10.D4)⋊29C2, (C2×C10).54(C22×S3), SmallGroup(480,428)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D6⋊Dic10
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — D6⋊Dic10
C15C2×C30 — D6⋊Dic10
C1C22C2×C4

Generators and relations for D6⋊Dic10
 G = < a,b,c,d | a6=b2=c20=1, d2=c10, bab=cac-1=dad-1=a-1, cbc-1=a4b, dbd-1=ab, dcd-1=c-1 >

Subgroups: 652 in 148 conjugacy classes, 52 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 [×2], Dic3 [×2], C12 [×3], D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×4], C20 [×3], 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 [×2], C2×C20, C2×C20 [×3], C22×C10, Dic3⋊C4 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60, S3×C10 [×2], S3×C10 [×2], C2×C30, C10.D4, C10.D4, C4⋊Dic5, C23.D5 [×2], C2×Dic10, C22×C20, D6⋊Q8, C15⋊Q8 [×2], C6×Dic5 [×2], S3×C20 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C20.48D4, D6⋊Dic5 [×2], C6.Dic10, C3×C10.D4, C30.4Q8, C2×C15⋊Q8, S3×C2×C20, D6⋊Dic10
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], C5⋊D4 [×2], C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, C2×Dic10, C4○D20, C2×C5⋊D4, D6⋊Q8, C2×S3×D5, C20.48D4, S3×Dic10, D6.D10, S3×C5⋊D4, D6⋊Dic10

Smallest permutation representation of D6⋊Dic10
On 240 points
Generators in S240
(1 151 110 180 193 217)(2 218 194 161 111 152)(3 153 112 162 195 219)(4 220 196 163 113 154)(5 155 114 164 197 201)(6 202 198 165 115 156)(7 157 116 166 199 203)(8 204 200 167 117 158)(9 159 118 168 181 205)(10 206 182 169 119 160)(11 141 120 170 183 207)(12 208 184 171 101 142)(13 143 102 172 185 209)(14 210 186 173 103 144)(15 145 104 174 187 211)(16 212 188 175 105 146)(17 147 106 176 189 213)(18 214 190 177 107 148)(19 149 108 178 191 215)(20 216 192 179 109 150)(21 122 85 54 66 222)(22 223 67 55 86 123)(23 124 87 56 68 224)(24 225 69 57 88 125)(25 126 89 58 70 226)(26 227 71 59 90 127)(27 128 91 60 72 228)(28 229 73 41 92 129)(29 130 93 42 74 230)(30 231 75 43 94 131)(31 132 95 44 76 232)(32 233 77 45 96 133)(33 134 97 46 78 234)(34 235 79 47 98 135)(35 136 99 48 80 236)(36 237 61 49 100 137)(37 138 81 50 62 238)(38 239 63 51 82 139)(39 140 83 52 64 240)(40 221 65 53 84 121)
(1 120)(2 184)(3 102)(4 186)(5 104)(6 188)(7 106)(8 190)(9 108)(10 192)(11 110)(12 194)(13 112)(14 196)(15 114)(16 198)(17 116)(18 200)(19 118)(20 182)(21 232)(22 133)(23 234)(24 135)(25 236)(26 137)(27 238)(28 139)(29 240)(30 121)(31 222)(32 123)(33 224)(34 125)(35 226)(36 127)(37 228)(38 129)(39 230)(40 131)(41 63)(42 83)(43 65)(44 85)(45 67)(46 87)(47 69)(48 89)(49 71)(50 91)(51 73)(52 93)(53 75)(54 95)(55 77)(56 97)(57 79)(58 99)(59 61)(60 81)(62 128)(64 130)(66 132)(68 134)(70 136)(72 138)(74 140)(76 122)(78 124)(80 126)(82 229)(84 231)(86 233)(88 235)(90 237)(92 239)(94 221)(96 223)(98 225)(100 227)(101 111)(103 113)(105 115)(107 117)(109 119)(141 151)(142 161)(143 153)(144 163)(145 155)(146 165)(147 157)(148 167)(149 159)(150 169)(152 171)(154 173)(156 175)(158 177)(160 179)(162 209)(164 211)(166 213)(168 215)(170 217)(172 219)(174 201)(176 203)(178 205)(180 207)(181 191)(183 193)(185 195)(187 197)(189 199)(202 212)(204 214)(206 216)(208 218)(210 220)
(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 21 11 31)(2 40 12 30)(3 39 13 29)(4 38 14 28)(5 37 15 27)(6 36 16 26)(7 35 17 25)(8 34 18 24)(9 33 19 23)(10 32 20 22)(41 163 51 173)(42 162 52 172)(43 161 53 171)(44 180 54 170)(45 179 55 169)(46 178 56 168)(47 177 57 167)(48 176 58 166)(49 175 59 165)(50 174 60 164)(61 105 71 115)(62 104 72 114)(63 103 73 113)(64 102 74 112)(65 101 75 111)(66 120 76 110)(67 119 77 109)(68 118 78 108)(69 117 79 107)(70 116 80 106)(81 187 91 197)(82 186 92 196)(83 185 93 195)(84 184 94 194)(85 183 95 193)(86 182 96 192)(87 181 97 191)(88 200 98 190)(89 199 99 189)(90 198 100 188)(121 208 131 218)(122 207 132 217)(123 206 133 216)(124 205 134 215)(125 204 135 214)(126 203 136 213)(127 202 137 212)(128 201 138 211)(129 220 139 210)(130 219 140 209)(141 232 151 222)(142 231 152 221)(143 230 153 240)(144 229 154 239)(145 228 155 238)(146 227 156 237)(147 226 157 236)(148 225 158 235)(149 224 159 234)(150 223 160 233)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E12F15A15B20A···20H20I···20P30A···30F60A···60H
order1222223444444445566610···1010···10121212121212151520···2020···2030···3060···60
size1111662226620206060222222···26···64420202020442···26···64···44···4

72 irreducible representations

dim1111111222222222222224444444
type+++++++++-++++++-+-++-
imageC1C2C2C2C2C2C2S3D4Q8D5D6D6C4○D4D10D10D10C5⋊D4Dic10C4○D12C4○D20S3×D4S3×Q8S3×D5C2×S3×D5S3×Dic10D6.D10S3×C5⋊D4
kernelD6⋊Dic10D6⋊Dic5C6.Dic10C3×C10.D4C30.4Q8C2×C15⋊Q8S3×C2×C20C10.D4C5×Dic3S3×C10S3×C2×C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3Dic3D6C10C6C10C10C2×C4C22C2C2C2
# reps1211111122221222288481122444

Matrix representation of D6⋊Dic10 in GL4(𝔽61) generated by

1100
60000
0010
0001
,
1100
06000
00600
00060
,
50000
111100
003632
00234
,
524300
52900
00729
001354
G:=sub<GL(4,GF(61))| [1,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,60,0,0,0,0,60,0,0,0,0,60],[50,11,0,0,0,11,0,0,0,0,36,2,0,0,32,34],[52,52,0,0,43,9,0,0,0,0,7,13,0,0,29,54] >;

D6⋊Dic10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,428);
// by ID

G=gap.SmallGroup(480,428);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,142,1356,18822]);
// Polycyclic

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

׿
×
𝔽