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G = D61Dic10order 480 = 25·3·5

1st semidirect product of D6 and Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D61Dic10, D6⋊C4.8D5, (S3×C10)⋊1Q8, C53(D63Q8), (C2×C20).15D6, C6.131(D4×D5), C30.37(C2×Q8), C10.32(S3×Q8), (C2×Dic10)⋊5S3, C30.131(C2×D4), C1510(C22⋊Q8), (C3×Dic5).7D4, (C6×Dic10)⋊15C2, C30.57(C4○D4), (C2×C12).224D10, D6⋊Dic5.11C2, C6.Dic1016C2, C30.Q816C2, C30.4Q815C2, C6.14(C2×Dic10), C2.15(S3×Dic10), C6.12(D42D5), (C2×C30).100C23, (C2×C60).255C22, (C2×Dic5).106D6, (C2×Dic3).30D10, (C22×S3).36D10, C2.15(D12⋊D5), C10.32(Q83S3), Dic5.20(C3⋊D4), C36(Dic5.14D4), (C6×Dic5).57C22, (C2×Dic15).80C22, (C10×Dic3).60C22, (C5×D6⋊C4).8C2, (C2×C4).39(S3×D5), C2.15(D5×C3⋊D4), (C2×S3×Dic5).4C2, C10.33(C2×C3⋊D4), C22.169(C2×S3×D5), (S3×C2×C10).15C22, (C2×C6).112(C22×D5), (C2×C10).112(C22×S3), SmallGroup(480,486)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D61Dic10
C1C5C15C30C2×C30C6×Dic5C2×S3×Dic5 — D61Dic10
C15C2×C30 — D61Dic10
C1C22C2×C4

Generators and relations for D61Dic10
 G = < a,b,c,d | a6=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 668 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 [×3], C12 [×4], D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×4], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×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], C4⋊Dic3, D6⋊C4, D6⋊C4, S3×C2×C4, C6×Q8, C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×2], C60, S3×C10 [×2], S3×C10 [×2], C2×C30, C10.D4 [×2], C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, D63Q8, S3×Dic5 [×2], C3×Dic10 [×2], C6×Dic5 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, Dic5.14D4, D6⋊Dic5, C30.Q8, C6.Dic10, C5×D6⋊C4, C30.4Q8, C2×S3×Dic5, C6×Dic10, D61Dic10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C22⋊Q8, Dic10 [×2], C22×D5, S3×Q8, Q83S3, C2×C3⋊D4, S3×D5, C2×Dic10, D4×D5, D42D5, D63Q8, C2×S3×D5, Dic5.14D4, S3×Dic10, D12⋊D5, D5×C3⋊D4, D61Dic10

Smallest permutation representation of D61Dic10
On 240 points
Generators in S240
(1 107 213 23 100 129)(2 108 214 24 81 130)(3 109 215 25 82 131)(4 110 216 26 83 132)(5 111 217 27 84 133)(6 112 218 28 85 134)(7 113 219 29 86 135)(8 114 220 30 87 136)(9 115 201 31 88 137)(10 116 202 32 89 138)(11 117 203 33 90 139)(12 118 204 34 91 140)(13 119 205 35 92 121)(14 120 206 36 93 122)(15 101 207 37 94 123)(16 102 208 38 95 124)(17 103 209 39 96 125)(18 104 210 40 97 126)(19 105 211 21 98 127)(20 106 212 22 99 128)(41 194 223 144 162 73)(42 195 224 145 163 74)(43 196 225 146 164 75)(44 197 226 147 165 76)(45 198 227 148 166 77)(46 199 228 149 167 78)(47 200 229 150 168 79)(48 181 230 151 169 80)(49 182 231 152 170 61)(50 183 232 153 171 62)(51 184 233 154 172 63)(52 185 234 155 173 64)(53 186 235 156 174 65)(54 187 236 157 175 66)(55 188 237 158 176 67)(56 189 238 159 177 68)(57 190 239 160 178 69)(58 191 240 141 179 70)(59 192 221 142 180 71)(60 193 222 143 161 72)
(1 129)(2 214)(3 131)(4 216)(5 133)(6 218)(7 135)(8 220)(9 137)(10 202)(11 139)(12 204)(13 121)(14 206)(15 123)(16 208)(17 125)(18 210)(19 127)(20 212)(21 211)(22 128)(23 213)(24 130)(25 215)(26 132)(27 217)(28 134)(29 219)(30 136)(31 201)(32 138)(33 203)(34 140)(35 205)(36 122)(37 207)(38 124)(39 209)(40 126)(42 145)(44 147)(46 149)(48 151)(50 153)(52 155)(54 157)(56 159)(58 141)(60 143)(61 182)(62 171)(63 184)(64 173)(65 186)(66 175)(67 188)(68 177)(69 190)(70 179)(71 192)(72 161)(73 194)(74 163)(75 196)(76 165)(77 198)(78 167)(79 200)(80 169)(82 109)(84 111)(86 113)(88 115)(90 117)(92 119)(94 101)(96 103)(98 105)(100 107)(162 223)(164 225)(166 227)(168 229)(170 231)(172 233)(174 235)(176 237)(178 239)(180 221)(181 230)(183 232)(185 234)(187 236)(189 238)(191 240)(193 222)(195 224)(197 226)(199 228)
(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)
(1 238 11 228)(2 237 12 227)(3 236 13 226)(4 235 14 225)(5 234 15 224)(6 233 16 223)(7 232 17 222)(8 231 18 221)(9 230 19 240)(10 229 20 239)(21 70 31 80)(22 69 32 79)(23 68 33 78)(24 67 34 77)(25 66 35 76)(26 65 36 75)(27 64 37 74)(28 63 38 73)(29 62 39 72)(30 61 40 71)(41 85 51 95)(42 84 52 94)(43 83 53 93)(44 82 54 92)(45 81 55 91)(46 100 56 90)(47 99 57 89)(48 98 58 88)(49 97 59 87)(50 96 60 86)(101 145 111 155)(102 144 112 154)(103 143 113 153)(104 142 114 152)(105 141 115 151)(106 160 116 150)(107 159 117 149)(108 158 118 148)(109 157 119 147)(110 156 120 146)(121 197 131 187)(122 196 132 186)(123 195 133 185)(124 194 134 184)(125 193 135 183)(126 192 136 182)(127 191 137 181)(128 190 138 200)(129 189 139 199)(130 188 140 198)(161 219 171 209)(162 218 172 208)(163 217 173 207)(164 216 174 206)(165 215 175 205)(166 214 176 204)(167 213 177 203)(168 212 178 202)(169 211 179 201)(170 210 180 220)

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

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

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

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○D4D10D10D10C3⋊D4Dic10S3×Q8Q83S3S3×D5D4×D5D42D5C2×S3×D5S3×Dic10D12⋊D5D5×C3⋊D4
kernelD61Dic10D6⋊Dic5C30.Q8C6.Dic10C5×D6⋊C4C30.4Q8C2×S3×Dic5C6×Dic10C2×Dic10C3×Dic5S3×C10D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3Dic5D6C10C10C2×C4C6C6C22C2C2C2
# reps11111111122221222248112222444

Matrix representation of D61Dic10 in GL4(𝔽61) generated by

1100
60000
0010
0001
,
1100
06000
0010
0001
,
91800
435200
005429
003259
,
1000
0100
003629
003125
G:=sub<GL(4,GF(61))| [1,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[9,43,0,0,18,52,0,0,0,0,54,32,0,0,29,59],[1,0,0,0,0,1,0,0,0,0,36,31,0,0,29,25] >;

D61Dic10 in GAP, Magma, Sage, TeX

D_6\rtimes_1{\rm Dic}_{10}
% in TeX

G:=Group("D6:1Dic10");
// GroupNames label

G:=SmallGroup(480,486);
// by ID

G=gap.SmallGroup(480,486);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,254,219,58,1356,18822]);
// Polycyclic

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

׿
×
𝔽