metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.3D6, C40.32D6, D30.23D4, D12.3D10, C24.32D10, Dic6.2D10, Dic10.2D6, C120.50C22, C60.144C23, Dic15.28D4, C40⋊C2⋊6S3, C24⋊C2⋊6D5, C15⋊7(C4○D8), C8.32(S3×D5), C6.33(D4×D5), (C8×D15)⋊11C2, C15⋊D8⋊12C2, C30.26(C2×D4), C10.33(S3×D4), D12⋊D5⋊9C2, D20⋊S3⋊9C2, C15⋊Q16⋊10C2, C5⋊2(Q8.7D6), C3⋊2(SD16⋊3D5), C20.79(C22×S3), C12.79(C22×D5), C2.11(C20⋊D6), C15⋊3C8.41C22, (C5×D12).27C22, (C3×D20).27C22, (C4×D15).56C22, (C5×Dic6).28C22, (C3×Dic10).28C22, C4.117(C2×S3×D5), (C5×C24⋊C2)⋊8C2, (C3×C40⋊C2)⋊10C2, SmallGroup(480,352)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.D10
G = < a,b,c,d | a20=c6=1, b2=d2=a10, bab-1=cac-1=a-1, dad-1=a9, cbc-1=a15b, bd=db, dcd-1=c-1 >
Subgroups: 716 in 124 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C3×D5, D15, C30, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8⋊2D5, Q8.7D6, C15⋊3C8, C120, D5×Dic3, S3×Dic5, C3⋊D20, C5⋊D12, C3×Dic10, C3×D20, C5×Dic6, C5×D12, C4×D15, SD16⋊3D5, C15⋊D8, C15⋊Q16, C3×C40⋊C2, C5×C24⋊C2, C8×D15, D20⋊S3, D12⋊D5, Dic6.D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, C22×D5, S3×D4, S3×D5, D4×D5, Q8.7D6, C2×S3×D5, SD16⋊3D5, C20⋊D6, Dic6.D10
►On 240 points
Generators in S240
(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 87 11 97)(2 86 12 96)(3 85 13 95)(4 84 14 94)(5 83 15 93)(6 82 16 92)(7 81 17 91)(8 100 18 90)(9 99 19 89)(10 98 20 88)(21 201 31 211)(22 220 32 210)(23 219 33 209)(24 218 34 208)(25 217 35 207)(26 216 36 206)(27 215 37 205)(28 214 38 204)(29 213 39 203)(30 212 40 202)(41 191 51 181)(42 190 52 200)(43 189 53 199)(44 188 54 198)(45 187 55 197)(46 186 56 196)(47 185 57 195)(48 184 58 194)(49 183 59 193)(50 182 60 192)(61 161 71 171)(62 180 72 170)(63 179 73 169)(64 178 74 168)(65 177 75 167)(66 176 76 166)(67 175 77 165)(68 174 78 164)(69 173 79 163)(70 172 80 162)(101 222 111 232)(102 221 112 231)(103 240 113 230)(104 239 114 229)(105 238 115 228)(106 237 116 227)(107 236 117 226)(108 235 118 225)(109 234 119 224)(110 233 120 223)(121 159 131 149)(122 158 132 148)(123 157 133 147)(124 156 134 146)(125 155 135 145)(126 154 136 144)(127 153 137 143)(128 152 138 142)(129 151 139 141)(130 150 140 160)
(1 161 121 223 36 58)(2 180 122 222 37 57)(3 179 123 221 38 56)(4 178 124 240 39 55)(5 177 125 239 40 54)(6 176 126 238 21 53)(7 175 127 237 22 52)(8 174 128 236 23 51)(9 173 129 235 24 50)(10 172 130 234 25 49)(11 171 131 233 26 48)(12 170 132 232 27 47)(13 169 133 231 28 46)(14 168 134 230 29 45)(15 167 135 229 30 44)(16 166 136 228 31 43)(17 165 137 227 32 42)(18 164 138 226 33 41)(19 163 139 225 34 60)(20 162 140 224 35 59)(61 154 120 201 184 82)(62 153 101 220 185 81)(63 152 102 219 186 100)(64 151 103 218 187 99)(65 150 104 217 188 98)(66 149 105 216 189 97)(67 148 106 215 190 96)(68 147 107 214 191 95)(69 146 108 213 192 94)(70 145 109 212 193 93)(71 144 110 211 194 92)(72 143 111 210 195 91)(73 142 112 209 196 90)(74 141 113 208 197 89)(75 160 114 207 198 88)(76 159 115 206 199 87)(77 158 116 205 200 86)(78 157 117 204 181 85)(79 156 118 203 182 84)(80 155 119 202 183 83)
(1 136 11 126)(2 125 12 135)(3 134 13 124)(4 123 14 133)(5 132 15 122)(6 121 16 131)(7 130 17 140)(8 139 18 129)(9 128 19 138)(10 137 20 127)(21 36 31 26)(22 25 32 35)(23 34 33 24)(27 30 37 40)(28 39 38 29)(41 235 51 225)(42 224 52 234)(43 233 53 223)(44 222 54 232)(45 231 55 221)(46 240 56 230)(47 229 57 239)(48 238 58 228)(49 227 59 237)(50 236 60 226)(61 76 71 66)(62 65 72 75)(63 74 73 64)(67 70 77 80)(68 79 78 69)(81 150 91 160)(82 159 92 149)(83 148 93 158)(84 157 94 147)(85 146 95 156)(86 155 96 145)(87 144 97 154)(88 153 98 143)(89 142 99 152)(90 151 100 141)(101 188 111 198)(102 197 112 187)(103 186 113 196)(104 195 114 185)(105 184 115 194)(106 193 116 183)(107 182 117 192)(108 191 118 181)(109 200 119 190)(110 189 120 199)(161 166 171 176)(162 175 172 165)(163 164 173 174)(167 180 177 170)(168 169 178 179)(201 206 211 216)(202 215 212 205)(203 204 213 214)(207 220 217 210)(208 209 218 219)
G:=sub<Sym(240)| (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), (1,87,11,97)(2,86,12,96)(3,85,13,95)(4,84,14,94)(5,83,15,93)(6,82,16,92)(7,81,17,91)(8,100,18,90)(9,99,19,89)(10,98,20,88)(21,201,31,211)(22,220,32,210)(23,219,33,209)(24,218,34,208)(25,217,35,207)(26,216,36,206)(27,215,37,205)(28,214,38,204)(29,213,39,203)(30,212,40,202)(41,191,51,181)(42,190,52,200)(43,189,53,199)(44,188,54,198)(45,187,55,197)(46,186,56,196)(47,185,57,195)(48,184,58,194)(49,183,59,193)(50,182,60,192)(61,161,71,171)(62,180,72,170)(63,179,73,169)(64,178,74,168)(65,177,75,167)(66,176,76,166)(67,175,77,165)(68,174,78,164)(69,173,79,163)(70,172,80,162)(101,222,111,232)(102,221,112,231)(103,240,113,230)(104,239,114,229)(105,238,115,228)(106,237,116,227)(107,236,117,226)(108,235,118,225)(109,234,119,224)(110,233,120,223)(121,159,131,149)(122,158,132,148)(123,157,133,147)(124,156,134,146)(125,155,135,145)(126,154,136,144)(127,153,137,143)(128,152,138,142)(129,151,139,141)(130,150,140,160), (1,161,121,223,36,58)(2,180,122,222,37,57)(3,179,123,221,38,56)(4,178,124,240,39,55)(5,177,125,239,40,54)(6,176,126,238,21,53)(7,175,127,237,22,52)(8,174,128,236,23,51)(9,173,129,235,24,50)(10,172,130,234,25,49)(11,171,131,233,26,48)(12,170,132,232,27,47)(13,169,133,231,28,46)(14,168,134,230,29,45)(15,167,135,229,30,44)(16,166,136,228,31,43)(17,165,137,227,32,42)(18,164,138,226,33,41)(19,163,139,225,34,60)(20,162,140,224,35,59)(61,154,120,201,184,82)(62,153,101,220,185,81)(63,152,102,219,186,100)(64,151,103,218,187,99)(65,150,104,217,188,98)(66,149,105,216,189,97)(67,148,106,215,190,96)(68,147,107,214,191,95)(69,146,108,213,192,94)(70,145,109,212,193,93)(71,144,110,211,194,92)(72,143,111,210,195,91)(73,142,112,209,196,90)(74,141,113,208,197,89)(75,160,114,207,198,88)(76,159,115,206,199,87)(77,158,116,205,200,86)(78,157,117,204,181,85)(79,156,118,203,182,84)(80,155,119,202,183,83), (1,136,11,126)(2,125,12,135)(3,134,13,124)(4,123,14,133)(5,132,15,122)(6,121,16,131)(7,130,17,140)(8,139,18,129)(9,128,19,138)(10,137,20,127)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,235,51,225)(42,224,52,234)(43,233,53,223)(44,222,54,232)(45,231,55,221)(46,240,56,230)(47,229,57,239)(48,238,58,228)(49,227,59,237)(50,236,60,226)(61,76,71,66)(62,65,72,75)(63,74,73,64)(67,70,77,80)(68,79,78,69)(81,150,91,160)(82,159,92,149)(83,148,93,158)(84,157,94,147)(85,146,95,156)(86,155,96,145)(87,144,97,154)(88,153,98,143)(89,142,99,152)(90,151,100,141)(101,188,111,198)(102,197,112,187)(103,186,113,196)(104,195,114,185)(105,184,115,194)(106,193,116,183)(107,182,117,192)(108,191,118,181)(109,200,119,190)(110,189,120,199)(161,166,171,176)(162,175,172,165)(163,164,173,174)(167,180,177,170)(168,169,178,179)(201,206,211,216)(202,215,212,205)(203,204,213,214)(207,220,217,210)(208,209,218,219)>;
G:=Group( (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), (1,87,11,97)(2,86,12,96)(3,85,13,95)(4,84,14,94)(5,83,15,93)(6,82,16,92)(7,81,17,91)(8,100,18,90)(9,99,19,89)(10,98,20,88)(21,201,31,211)(22,220,32,210)(23,219,33,209)(24,218,34,208)(25,217,35,207)(26,216,36,206)(27,215,37,205)(28,214,38,204)(29,213,39,203)(30,212,40,202)(41,191,51,181)(42,190,52,200)(43,189,53,199)(44,188,54,198)(45,187,55,197)(46,186,56,196)(47,185,57,195)(48,184,58,194)(49,183,59,193)(50,182,60,192)(61,161,71,171)(62,180,72,170)(63,179,73,169)(64,178,74,168)(65,177,75,167)(66,176,76,166)(67,175,77,165)(68,174,78,164)(69,173,79,163)(70,172,80,162)(101,222,111,232)(102,221,112,231)(103,240,113,230)(104,239,114,229)(105,238,115,228)(106,237,116,227)(107,236,117,226)(108,235,118,225)(109,234,119,224)(110,233,120,223)(121,159,131,149)(122,158,132,148)(123,157,133,147)(124,156,134,146)(125,155,135,145)(126,154,136,144)(127,153,137,143)(128,152,138,142)(129,151,139,141)(130,150,140,160), (1,161,121,223,36,58)(2,180,122,222,37,57)(3,179,123,221,38,56)(4,178,124,240,39,55)(5,177,125,239,40,54)(6,176,126,238,21,53)(7,175,127,237,22,52)(8,174,128,236,23,51)(9,173,129,235,24,50)(10,172,130,234,25,49)(11,171,131,233,26,48)(12,170,132,232,27,47)(13,169,133,231,28,46)(14,168,134,230,29,45)(15,167,135,229,30,44)(16,166,136,228,31,43)(17,165,137,227,32,42)(18,164,138,226,33,41)(19,163,139,225,34,60)(20,162,140,224,35,59)(61,154,120,201,184,82)(62,153,101,220,185,81)(63,152,102,219,186,100)(64,151,103,218,187,99)(65,150,104,217,188,98)(66,149,105,216,189,97)(67,148,106,215,190,96)(68,147,107,214,191,95)(69,146,108,213,192,94)(70,145,109,212,193,93)(71,144,110,211,194,92)(72,143,111,210,195,91)(73,142,112,209,196,90)(74,141,113,208,197,89)(75,160,114,207,198,88)(76,159,115,206,199,87)(77,158,116,205,200,86)(78,157,117,204,181,85)(79,156,118,203,182,84)(80,155,119,202,183,83), (1,136,11,126)(2,125,12,135)(3,134,13,124)(4,123,14,133)(5,132,15,122)(6,121,16,131)(7,130,17,140)(8,139,18,129)(9,128,19,138)(10,137,20,127)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,235,51,225)(42,224,52,234)(43,233,53,223)(44,222,54,232)(45,231,55,221)(46,240,56,230)(47,229,57,239)(48,238,58,228)(49,227,59,237)(50,236,60,226)(61,76,71,66)(62,65,72,75)(63,74,73,64)(67,70,77,80)(68,79,78,69)(81,150,91,160)(82,159,92,149)(83,148,93,158)(84,157,94,147)(85,146,95,156)(86,155,96,145)(87,144,97,154)(88,153,98,143)(89,142,99,152)(90,151,100,141)(101,188,111,198)(102,197,112,187)(103,186,113,196)(104,195,114,185)(105,184,115,194)(106,193,116,183)(107,182,117,192)(108,191,118,181)(109,200,119,190)(110,189,120,199)(161,166,171,176)(162,175,172,165)(163,164,173,174)(167,180,177,170)(168,169,178,179)(201,206,211,216)(202,215,212,205)(203,204,213,214)(207,220,217,210)(208,209,218,219) );
G=PermutationGroup([[(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)], [(1,87,11,97),(2,86,12,96),(3,85,13,95),(4,84,14,94),(5,83,15,93),(6,82,16,92),(7,81,17,91),(8,100,18,90),(9,99,19,89),(10,98,20,88),(21,201,31,211),(22,220,32,210),(23,219,33,209),(24,218,34,208),(25,217,35,207),(26,216,36,206),(27,215,37,205),(28,214,38,204),(29,213,39,203),(30,212,40,202),(41,191,51,181),(42,190,52,200),(43,189,53,199),(44,188,54,198),(45,187,55,197),(46,186,56,196),(47,185,57,195),(48,184,58,194),(49,183,59,193),(50,182,60,192),(61,161,71,171),(62,180,72,170),(63,179,73,169),(64,178,74,168),(65,177,75,167),(66,176,76,166),(67,175,77,165),(68,174,78,164),(69,173,79,163),(70,172,80,162),(101,222,111,232),(102,221,112,231),(103,240,113,230),(104,239,114,229),(105,238,115,228),(106,237,116,227),(107,236,117,226),(108,235,118,225),(109,234,119,224),(110,233,120,223),(121,159,131,149),(122,158,132,148),(123,157,133,147),(124,156,134,146),(125,155,135,145),(126,154,136,144),(127,153,137,143),(128,152,138,142),(129,151,139,141),(130,150,140,160)], [(1,161,121,223,36,58),(2,180,122,222,37,57),(3,179,123,221,38,56),(4,178,124,240,39,55),(5,177,125,239,40,54),(6,176,126,238,21,53),(7,175,127,237,22,52),(8,174,128,236,23,51),(9,173,129,235,24,50),(10,172,130,234,25,49),(11,171,131,233,26,48),(12,170,132,232,27,47),(13,169,133,231,28,46),(14,168,134,230,29,45),(15,167,135,229,30,44),(16,166,136,228,31,43),(17,165,137,227,32,42),(18,164,138,226,33,41),(19,163,139,225,34,60),(20,162,140,224,35,59),(61,154,120,201,184,82),(62,153,101,220,185,81),(63,152,102,219,186,100),(64,151,103,218,187,99),(65,150,104,217,188,98),(66,149,105,216,189,97),(67,148,106,215,190,96),(68,147,107,214,191,95),(69,146,108,213,192,94),(70,145,109,212,193,93),(71,144,110,211,194,92),(72,143,111,210,195,91),(73,142,112,209,196,90),(74,141,113,208,197,89),(75,160,114,207,198,88),(76,159,115,206,199,87),(77,158,116,205,200,86),(78,157,117,204,181,85),(79,156,118,203,182,84),(80,155,119,202,183,83)], [(1,136,11,126),(2,125,12,135),(3,134,13,124),(4,123,14,133),(5,132,15,122),(6,121,16,131),(7,130,17,140),(8,139,18,129),(9,128,19,138),(10,137,20,127),(21,36,31,26),(22,25,32,35),(23,34,33,24),(27,30,37,40),(28,39,38,29),(41,235,51,225),(42,224,52,234),(43,233,53,223),(44,222,54,232),(45,231,55,221),(46,240,56,230),(47,229,57,239),(48,238,58,228),(49,227,59,237),(50,236,60,226),(61,76,71,66),(62,65,72,75),(63,74,73,64),(67,70,77,80),(68,79,78,69),(81,150,91,160),(82,159,92,149),(83,148,93,158),(84,157,94,147),(85,146,95,156),(86,155,96,145),(87,144,97,154),(88,153,98,143),(89,142,99,152),(90,151,100,141),(101,188,111,198),(102,197,112,187),(103,186,113,196),(104,195,114,185),(105,184,115,194),(106,193,116,183),(107,182,117,192),(108,191,118,181),(109,200,119,190),(110,189,120,199),(161,166,171,176),(162,175,172,165),(163,164,173,174),(167,180,177,170),(168,169,178,179),(201,206,211,216),(202,215,212,205),(203,204,213,214),(207,220,217,210),(208,209,218,219)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 12 | 20 | 30 | 2 | 2 | 12 | 15 | 15 | 20 | 2 | 2 | 2 | 40 | 2 | 2 | 30 | 30 | 2 | 2 | 24 | 24 | 4 | 40 | 4 | 4 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 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 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4○D8 | S3×D4 | S3×D5 | D4×D5 | Q8.7D6 | C2×S3×D5 | SD16⋊3D5 | C20⋊D6 | Dic6.D10 |
| kernel | Dic6.D10 | C15⋊D8 | C15⋊Q16 | C3×C40⋊C2 | C5×C24⋊C2 | C8×D15 | D20⋊S3 | D12⋊D5 | C40⋊C2 | Dic15 | D30 | C24⋊C2 | C40 | Dic10 | D20 | C24 | Dic6 | D12 | C15 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of Dic6.D10 ►in GL6(𝔽241)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 16 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 77 | 107 | 0 | 0 |
| 0 | 0 | 109 | 164 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 52 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 141 | 77 | 0 | 0 |
| 0 | 0 | 180 | 100 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 49 |
| 0 | 0 | 0 | 0 | 177 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 192 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,189,0,0,0,0,0,0,64,16,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,189,0,0,0,0,0,240,0,0,0,0,0,0,77,109,0,0,0,0,107,164,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,52,0,0,0,0,0,1,0,0,0,0,0,0,141,180,0,0,0,0,77,100,0,0,0,0,0,0,240,177,0,0,0,0,49,2],[1,189,0,0,0,0,0,240,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,192,240] >;
Dic6.D10 in GAP, Magma, Sage, TeX
{\rm Dic}_6.D_{10} % in TeX
G:=Group("Dic6.D10"); // GroupNames label
G:=SmallGroup(480,352);
// by ID
G=gap.SmallGroup(480,352);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,58,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^6=1,b^2=d^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,d*a*d^-1=a^9,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations