metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊2Dic10, Dic5.18D12, D6⋊C4.3D5, (S3×C10)⋊2Q8, C60⋊5C4⋊5C2, C6.18(D4×D5), C5⋊2(C4.D12), C2.20(D5×D12), (C2×C20).18D6, C30.42(C2×D4), C10.33(S3×Q8), C30.40(C2×Q8), (C2×C12).15D10, C10.D4⋊5S3, C10.18(C2×D12), (C2×C60).7C22, C15⋊12(C22⋊Q8), (C3×Dic5).9D4, D6⋊Dic5.12C2, C30.Q8⋊21C2, C6.15(C2×Dic10), C2.16(S3×Dic10), C30.118(C4○D4), C6.71(D4⋊2D5), (C2×C30).107C23, (C2×Dic3).33D10, (C2×Dic5).108D6, (C22×S3).39D10, C10.70(D4⋊2S3), C3⋊1(Dic5.14D4), (C6×Dic5).62C22, C2.16(C30.C23), (C10×Dic3).65C22, (C2×Dic15).87C22, (C2×C15⋊Q8)⋊6C2, (C5×D6⋊C4).3C2, (C2×C4).44(S3×D5), (C2×S3×Dic5).5C2, C22.174(C2×S3×D5), (C3×C10.D4)⋊5C2, (S3×C2×C10).19C22, (C2×C6).119(C22×D5), (C2×C10).119(C22×S3), SmallGroup(480,493)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊2Dic10
G = < a,b,c,d | a6=b2=c20=1, d2=c10, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >
Subgroups: 716 in 148 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C22⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4⋊Dic3, D6⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, S3×C10, S3×C10, C2×C30, C10.D4, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, C4.D12, S3×Dic5, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, Dic5.14D4, D6⋊Dic5, C30.Q8, C3×C10.D4, C5×D6⋊C4, C60⋊5C4, C2×S3×Dic5, C2×C15⋊Q8, D6⋊2Dic10
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, D12, C22×S3, C22⋊Q8, Dic10, C22×D5, C2×D12, D4⋊2S3, S3×Q8, S3×D5, C2×Dic10, D4×D5, D4⋊2D5, C4.D12, C2×S3×D5, Dic5.14D4, S3×Dic10, D5×D12, C30.C23, D6⋊2Dic10
►On 240 points
Generators in S240
(1 179 109 146 40 58)(2 59 21 147 110 180)(3 161 111 148 22 60)(4 41 23 149 112 162)(5 163 113 150 24 42)(6 43 25 151 114 164)(7 165 115 152 26 44)(8 45 27 153 116 166)(9 167 117 154 28 46)(10 47 29 155 118 168)(11 169 119 156 30 48)(12 49 31 157 120 170)(13 171 101 158 32 50)(14 51 33 159 102 172)(15 173 103 160 34 52)(16 53 35 141 104 174)(17 175 105 142 36 54)(18 55 37 143 106 176)(19 177 107 144 38 56)(20 57 39 145 108 178)(61 198 201 223 131 98)(62 99 132 224 202 199)(63 200 203 225 133 100)(64 81 134 226 204 181)(65 182 205 227 135 82)(66 83 136 228 206 183)(67 184 207 229 137 84)(68 85 138 230 208 185)(69 186 209 231 139 86)(70 87 140 232 210 187)(71 188 211 233 121 88)(72 89 122 234 212 189)(73 190 213 235 123 90)(74 91 124 236 214 191)(75 192 215 237 125 92)(76 93 126 238 216 193)(77 194 217 239 127 94)(78 95 128 240 218 195)(79 196 219 221 129 96)(80 97 130 222 220 197)
(1 58)(2 21)(3 60)(4 23)(5 42)(6 25)(7 44)(8 27)(9 46)(10 29)(11 48)(12 31)(13 50)(14 33)(15 52)(16 35)(17 54)(18 37)(19 56)(20 39)(22 161)(24 163)(26 165)(28 167)(30 169)(32 171)(34 173)(36 175)(38 177)(40 179)(61 223)(62 202)(63 225)(64 204)(65 227)(66 206)(67 229)(68 208)(69 231)(70 210)(71 233)(72 212)(73 235)(74 214)(75 237)(76 216)(77 239)(78 218)(79 221)(80 220)(81 226)(82 135)(83 228)(84 137)(85 230)(86 139)(87 232)(88 121)(89 234)(90 123)(91 236)(92 125)(93 238)(94 127)(95 240)(96 129)(97 222)(98 131)(99 224)(100 133)(101 158)(103 160)(105 142)(107 144)(109 146)(111 148)(113 150)(115 152)(117 154)(119 156)(141 174)(143 176)(145 178)(147 180)(149 162)(151 164)(153 166)(155 168)(157 170)(159 172)(182 205)(184 207)(186 209)(188 211)(190 213)(192 215)(194 217)(196 219)(198 201)(200 203)
(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 98 11 88)(2 97 12 87)(3 96 13 86)(4 95 14 85)(5 94 15 84)(6 93 16 83)(7 92 17 82)(8 91 18 81)(9 90 19 100)(10 89 20 99)(21 222 31 232)(22 221 32 231)(23 240 33 230)(24 239 34 229)(25 238 35 228)(26 237 36 227)(27 236 37 226)(28 235 38 225)(29 234 39 224)(30 233 40 223)(41 128 51 138)(42 127 52 137)(43 126 53 136)(44 125 54 135)(45 124 55 134)(46 123 56 133)(47 122 57 132)(48 121 58 131)(49 140 59 130)(50 139 60 129)(61 169 71 179)(62 168 72 178)(63 167 73 177)(64 166 74 176)(65 165 75 175)(66 164 76 174)(67 163 77 173)(68 162 78 172)(69 161 79 171)(70 180 80 170)(101 186 111 196)(102 185 112 195)(103 184 113 194)(104 183 114 193)(105 182 115 192)(106 181 116 191)(107 200 117 190)(108 199 118 189)(109 198 119 188)(110 197 120 187)(141 206 151 216)(142 205 152 215)(143 204 153 214)(144 203 154 213)(145 202 155 212)(146 201 156 211)(147 220 157 210)(148 219 158 209)(149 218 159 208)(150 217 160 207)
G:=sub<Sym(240)| (1,179,109,146,40,58)(2,59,21,147,110,180)(3,161,111,148,22,60)(4,41,23,149,112,162)(5,163,113,150,24,42)(6,43,25,151,114,164)(7,165,115,152,26,44)(8,45,27,153,116,166)(9,167,117,154,28,46)(10,47,29,155,118,168)(11,169,119,156,30,48)(12,49,31,157,120,170)(13,171,101,158,32,50)(14,51,33,159,102,172)(15,173,103,160,34,52)(16,53,35,141,104,174)(17,175,105,142,36,54)(18,55,37,143,106,176)(19,177,107,144,38,56)(20,57,39,145,108,178)(61,198,201,223,131,98)(62,99,132,224,202,199)(63,200,203,225,133,100)(64,81,134,226,204,181)(65,182,205,227,135,82)(66,83,136,228,206,183)(67,184,207,229,137,84)(68,85,138,230,208,185)(69,186,209,231,139,86)(70,87,140,232,210,187)(71,188,211,233,121,88)(72,89,122,234,212,189)(73,190,213,235,123,90)(74,91,124,236,214,191)(75,192,215,237,125,92)(76,93,126,238,216,193)(77,194,217,239,127,94)(78,95,128,240,218,195)(79,196,219,221,129,96)(80,97,130,222,220,197), (1,58)(2,21)(3,60)(4,23)(5,42)(6,25)(7,44)(8,27)(9,46)(10,29)(11,48)(12,31)(13,50)(14,33)(15,52)(16,35)(17,54)(18,37)(19,56)(20,39)(22,161)(24,163)(26,165)(28,167)(30,169)(32,171)(34,173)(36,175)(38,177)(40,179)(61,223)(62,202)(63,225)(64,204)(65,227)(66,206)(67,229)(68,208)(69,231)(70,210)(71,233)(72,212)(73,235)(74,214)(75,237)(76,216)(77,239)(78,218)(79,221)(80,220)(81,226)(82,135)(83,228)(84,137)(85,230)(86,139)(87,232)(88,121)(89,234)(90,123)(91,236)(92,125)(93,238)(94,127)(95,240)(96,129)(97,222)(98,131)(99,224)(100,133)(101,158)(103,160)(105,142)(107,144)(109,146)(111,148)(113,150)(115,152)(117,154)(119,156)(141,174)(143,176)(145,178)(147,180)(149,162)(151,164)(153,166)(155,168)(157,170)(159,172)(182,205)(184,207)(186,209)(188,211)(190,213)(192,215)(194,217)(196,219)(198,201)(200,203), (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), (1,98,11,88)(2,97,12,87)(3,96,13,86)(4,95,14,85)(5,94,15,84)(6,93,16,83)(7,92,17,82)(8,91,18,81)(9,90,19,100)(10,89,20,99)(21,222,31,232)(22,221,32,231)(23,240,33,230)(24,239,34,229)(25,238,35,228)(26,237,36,227)(27,236,37,226)(28,235,38,225)(29,234,39,224)(30,233,40,223)(41,128,51,138)(42,127,52,137)(43,126,53,136)(44,125,54,135)(45,124,55,134)(46,123,56,133)(47,122,57,132)(48,121,58,131)(49,140,59,130)(50,139,60,129)(61,169,71,179)(62,168,72,178)(63,167,73,177)(64,166,74,176)(65,165,75,175)(66,164,76,174)(67,163,77,173)(68,162,78,172)(69,161,79,171)(70,180,80,170)(101,186,111,196)(102,185,112,195)(103,184,113,194)(104,183,114,193)(105,182,115,192)(106,181,116,191)(107,200,117,190)(108,199,118,189)(109,198,119,188)(110,197,120,187)(141,206,151,216)(142,205,152,215)(143,204,153,214)(144,203,154,213)(145,202,155,212)(146,201,156,211)(147,220,157,210)(148,219,158,209)(149,218,159,208)(150,217,160,207)>;
G:=Group( (1,179,109,146,40,58)(2,59,21,147,110,180)(3,161,111,148,22,60)(4,41,23,149,112,162)(5,163,113,150,24,42)(6,43,25,151,114,164)(7,165,115,152,26,44)(8,45,27,153,116,166)(9,167,117,154,28,46)(10,47,29,155,118,168)(11,169,119,156,30,48)(12,49,31,157,120,170)(13,171,101,158,32,50)(14,51,33,159,102,172)(15,173,103,160,34,52)(16,53,35,141,104,174)(17,175,105,142,36,54)(18,55,37,143,106,176)(19,177,107,144,38,56)(20,57,39,145,108,178)(61,198,201,223,131,98)(62,99,132,224,202,199)(63,200,203,225,133,100)(64,81,134,226,204,181)(65,182,205,227,135,82)(66,83,136,228,206,183)(67,184,207,229,137,84)(68,85,138,230,208,185)(69,186,209,231,139,86)(70,87,140,232,210,187)(71,188,211,233,121,88)(72,89,122,234,212,189)(73,190,213,235,123,90)(74,91,124,236,214,191)(75,192,215,237,125,92)(76,93,126,238,216,193)(77,194,217,239,127,94)(78,95,128,240,218,195)(79,196,219,221,129,96)(80,97,130,222,220,197), (1,58)(2,21)(3,60)(4,23)(5,42)(6,25)(7,44)(8,27)(9,46)(10,29)(11,48)(12,31)(13,50)(14,33)(15,52)(16,35)(17,54)(18,37)(19,56)(20,39)(22,161)(24,163)(26,165)(28,167)(30,169)(32,171)(34,173)(36,175)(38,177)(40,179)(61,223)(62,202)(63,225)(64,204)(65,227)(66,206)(67,229)(68,208)(69,231)(70,210)(71,233)(72,212)(73,235)(74,214)(75,237)(76,216)(77,239)(78,218)(79,221)(80,220)(81,226)(82,135)(83,228)(84,137)(85,230)(86,139)(87,232)(88,121)(89,234)(90,123)(91,236)(92,125)(93,238)(94,127)(95,240)(96,129)(97,222)(98,131)(99,224)(100,133)(101,158)(103,160)(105,142)(107,144)(109,146)(111,148)(113,150)(115,152)(117,154)(119,156)(141,174)(143,176)(145,178)(147,180)(149,162)(151,164)(153,166)(155,168)(157,170)(159,172)(182,205)(184,207)(186,209)(188,211)(190,213)(192,215)(194,217)(196,219)(198,201)(200,203), (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), (1,98,11,88)(2,97,12,87)(3,96,13,86)(4,95,14,85)(5,94,15,84)(6,93,16,83)(7,92,17,82)(8,91,18,81)(9,90,19,100)(10,89,20,99)(21,222,31,232)(22,221,32,231)(23,240,33,230)(24,239,34,229)(25,238,35,228)(26,237,36,227)(27,236,37,226)(28,235,38,225)(29,234,39,224)(30,233,40,223)(41,128,51,138)(42,127,52,137)(43,126,53,136)(44,125,54,135)(45,124,55,134)(46,123,56,133)(47,122,57,132)(48,121,58,131)(49,140,59,130)(50,139,60,129)(61,169,71,179)(62,168,72,178)(63,167,73,177)(64,166,74,176)(65,165,75,175)(66,164,76,174)(67,163,77,173)(68,162,78,172)(69,161,79,171)(70,180,80,170)(101,186,111,196)(102,185,112,195)(103,184,113,194)(104,183,114,193)(105,182,115,192)(106,181,116,191)(107,200,117,190)(108,199,118,189)(109,198,119,188)(110,197,120,187)(141,206,151,216)(142,205,152,215)(143,204,153,214)(144,203,154,213)(145,202,155,212)(146,201,156,211)(147,220,157,210)(148,219,158,209)(149,218,159,208)(150,217,160,207) );
G=PermutationGroup([[(1,179,109,146,40,58),(2,59,21,147,110,180),(3,161,111,148,22,60),(4,41,23,149,112,162),(5,163,113,150,24,42),(6,43,25,151,114,164),(7,165,115,152,26,44),(8,45,27,153,116,166),(9,167,117,154,28,46),(10,47,29,155,118,168),(11,169,119,156,30,48),(12,49,31,157,120,170),(13,171,101,158,32,50),(14,51,33,159,102,172),(15,173,103,160,34,52),(16,53,35,141,104,174),(17,175,105,142,36,54),(18,55,37,143,106,176),(19,177,107,144,38,56),(20,57,39,145,108,178),(61,198,201,223,131,98),(62,99,132,224,202,199),(63,200,203,225,133,100),(64,81,134,226,204,181),(65,182,205,227,135,82),(66,83,136,228,206,183),(67,184,207,229,137,84),(68,85,138,230,208,185),(69,186,209,231,139,86),(70,87,140,232,210,187),(71,188,211,233,121,88),(72,89,122,234,212,189),(73,190,213,235,123,90),(74,91,124,236,214,191),(75,192,215,237,125,92),(76,93,126,238,216,193),(77,194,217,239,127,94),(78,95,128,240,218,195),(79,196,219,221,129,96),(80,97,130,222,220,197)], [(1,58),(2,21),(3,60),(4,23),(5,42),(6,25),(7,44),(8,27),(9,46),(10,29),(11,48),(12,31),(13,50),(14,33),(15,52),(16,35),(17,54),(18,37),(19,56),(20,39),(22,161),(24,163),(26,165),(28,167),(30,169),(32,171),(34,173),(36,175),(38,177),(40,179),(61,223),(62,202),(63,225),(64,204),(65,227),(66,206),(67,229),(68,208),(69,231),(70,210),(71,233),(72,212),(73,235),(74,214),(75,237),(76,216),(77,239),(78,218),(79,221),(80,220),(81,226),(82,135),(83,228),(84,137),(85,230),(86,139),(87,232),(88,121),(89,234),(90,123),(91,236),(92,125),(93,238),(94,127),(95,240),(96,129),(97,222),(98,131),(99,224),(100,133),(101,158),(103,160),(105,142),(107,144),(109,146),(111,148),(113,150),(115,152),(117,154),(119,156),(141,174),(143,176),(145,178),(147,180),(149,162),(151,164),(153,166),(155,168),(157,170),(159,172),(182,205),(184,207),(186,209),(188,211),(190,213),(192,215),(194,217),(196,219),(198,201),(200,203)], [(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)], [(1,98,11,88),(2,97,12,87),(3,96,13,86),(4,95,14,85),(5,94,15,84),(6,93,16,83),(7,92,17,82),(8,91,18,81),(9,90,19,100),(10,89,20,99),(21,222,31,232),(22,221,32,231),(23,240,33,230),(24,239,34,229),(25,238,35,228),(26,237,36,227),(27,236,37,226),(28,235,38,225),(29,234,39,224),(30,233,40,223),(41,128,51,138),(42,127,52,137),(43,126,53,136),(44,125,54,135),(45,124,55,134),(46,123,56,133),(47,122,57,132),(48,121,58,131),(49,140,59,130),(50,139,60,129),(61,169,71,179),(62,168,72,178),(63,167,73,177),(64,166,74,176),(65,165,75,175),(66,164,76,174),(67,163,77,173),(68,162,78,172),(69,161,79,171),(70,180,80,170),(101,186,111,196),(102,185,112,195),(103,184,113,194),(104,183,114,193),(105,182,115,192),(106,181,116,191),(107,200,117,190),(108,199,118,189),(109,198,119,188),(110,197,120,187),(141,206,151,216),(142,205,152,215),(143,204,153,214),(144,203,154,213),(145,202,155,212),(146,201,156,211),(147,220,157,210),(148,219,158,209),(149,218,159,208),(150,217,160,207)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 4 | 10 | 10 | 12 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | + | - | - | - | + | + | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D12 | Dic10 | D4⋊2S3 | S3×Q8 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | S3×Dic10 | D5×D12 | C30.C23 |
| kernel | D6⋊2Dic10 | D6⋊Dic5 | C30.Q8 | C3×C10.D4 | C5×D6⋊C4 | C60⋊5C4 | C2×S3×Dic5 | C2×C15⋊Q8 | C10.D4 | C3×Dic5 | S3×C10 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | Dic5 | D6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D6⋊2Dic10 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 57 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 4 | 36 | 0 | 0 | 0 | 0 |
| 25 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 41 | 0 | 0 |
| 0 | 0 | 16 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 17 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,57,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,60,60],[4,25,0,0,0,0,36,27,0,0,0,0,0,0,40,16,0,0,0,0,41,21,0,0,0,0,0,0,0,60,0,0,0,0,60,0],[0,11,0,0,0,0,11,0,0,0,0,0,0,0,50,17,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
D6⋊2Dic10 in GAP, Magma, Sage, TeX
D_6\rtimes_2{\rm Dic}_{10} % in TeX
G:=Group("D6:2Dic10"); // GroupNames label
G:=SmallGroup(480,493);
// by ID
G=gap.SmallGroup(480,493);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,254,219,142,1356,18822]);
// 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=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations