metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.15D10, C60.43C23, Dic10.27D6, D60.15C22, C5⋊D24⋊7C2, C5⋊Q16⋊7S3, C15⋊22(C4○D8), Q8⋊3S3⋊3D5, C5⋊2C8.20D6, Q8.12(S3×D5), (C3×Q8).7D10, (C5×Q8).39D6, D60⋊C2⋊3C2, C5⋊5(D24⋊C2), Q8⋊2D15⋊7C2, (C4×S3).26D10, (S3×C10).15D4, C30.205(C2×D4), C10.153(S3×D4), C20.D6⋊8C2, D6.4(C5⋊D4), C3⋊5(D4.8D10), C20.43(C22×S3), (C5×Dic3).41D4, C12.43(C22×D5), (S3×C20).15C22, C15⋊3C8.17C22, (C5×D12).16C22, (Q8×C15).13C22, Dic3.23(C5⋊D4), (C3×Dic10).15C22, C4.43(C2×S3×D5), (S3×C5⋊2C8)⋊7C2, (C3×C5⋊Q16)⋊5C2, C2.34(S3×C5⋊D4), C6.56(C2×C5⋊D4), (C5×Q8⋊3S3)⋊3C2, (C3×C5⋊2C8).13C22, SmallGroup(480,595)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.27D6
G = < a,b,c,d | a20=d2=1, b2=c6=a10, bab-1=a-1, cac-1=dad=a11, cbc-1=dbd=a5b, dcd=c5 >
Subgroups: 652 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, C12, D6, D6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C3×Q8, C3×Q8, C5×S3, D15, C30, C4○D8, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, S3×C8, D24, Q8⋊2S3, C3×Q16, Q8⋊3S3, Q8⋊3S3, C5×Dic3, C3×Dic5, C60, C60, S3×C10, S3×C10, D30, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, D24⋊C2, C3×C5⋊2C8, C15⋊3C8, D30.C2, C5⋊D12, C3×Dic10, S3×C20, S3×C20, C5×D12, C5×D12, D60, Q8×C15, D4.8D10, S3×C5⋊2C8, C5⋊D24, C20.D6, C3×C5⋊Q16, Q8⋊2D15, D60⋊C2, C5×Q8⋊3S3, Dic10.27D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D24⋊C2, C2×S3×D5, D4.8D10, S3×C5⋊D4, Dic10.27D6
►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)
(1 182 11 192)(2 181 12 191)(3 200 13 190)(4 199 14 189)(5 198 15 188)(6 197 16 187)(7 196 17 186)(8 195 18 185)(9 194 19 184)(10 193 20 183)(21 162 31 172)(22 161 32 171)(23 180 33 170)(24 179 34 169)(25 178 35 168)(26 177 36 167)(27 176 37 166)(28 175 38 165)(29 174 39 164)(30 173 40 163)(41 155 51 145)(42 154 52 144)(43 153 53 143)(44 152 54 142)(45 151 55 141)(46 150 56 160)(47 149 57 159)(48 148 58 158)(49 147 59 157)(50 146 60 156)(61 205 71 215)(62 204 72 214)(63 203 73 213)(64 202 74 212)(65 201 75 211)(66 220 76 210)(67 219 77 209)(68 218 78 208)(69 217 79 207)(70 216 80 206)(81 136 91 126)(82 135 92 125)(83 134 93 124)(84 133 94 123)(85 132 95 122)(86 131 96 121)(87 130 97 140)(88 129 98 139)(89 128 99 138)(90 127 100 137)(101 225 111 235)(102 224 112 234)(103 223 113 233)(104 222 114 232)(105 221 115 231)(106 240 116 230)(107 239 117 229)(108 238 118 228)(109 237 119 227)(110 236 120 226)
(1 39 154 102 75 129 11 29 144 112 65 139)(2 30 155 113 76 140 12 40 145 103 66 130)(3 21 156 104 77 131 13 31 146 114 67 121)(4 32 157 115 78 122 14 22 147 105 68 132)(5 23 158 106 79 133 15 33 148 116 69 123)(6 34 159 117 80 124 16 24 149 107 70 134)(7 25 160 108 61 135 17 35 150 118 71 125)(8 36 141 119 62 126 18 26 151 109 72 136)(9 27 142 110 63 137 19 37 152 120 73 127)(10 38 143 101 64 128 20 28 153 111 74 138)(41 238 220 92 181 168 51 228 210 82 191 178)(42 229 201 83 182 179 52 239 211 93 192 169)(43 240 202 94 183 170 53 230 212 84 193 180)(44 231 203 85 184 161 54 221 213 95 194 171)(45 222 204 96 185 172 55 232 214 86 195 162)(46 233 205 87 186 163 56 223 215 97 196 173)(47 224 206 98 187 174 57 234 216 88 197 164)(48 235 207 89 188 165 58 225 217 99 198 175)(49 226 208 100 189 176 59 236 218 90 199 166)(50 237 209 91 190 167 60 227 219 81 200 177)
(1 134)(2 125)(3 136)(4 127)(5 138)(6 129)(7 140)(8 131)(9 122)(10 133)(11 124)(12 135)(13 126)(14 137)(15 128)(16 139)(17 130)(18 121)(19 132)(20 123)(21 62)(22 73)(23 64)(24 75)(25 66)(26 77)(27 68)(28 79)(29 70)(30 61)(31 72)(32 63)(33 74)(34 65)(35 76)(36 67)(37 78)(38 69)(39 80)(40 71)(41 233)(42 224)(43 235)(44 226)(45 237)(46 228)(47 239)(48 230)(49 221)(50 232)(51 223)(52 234)(53 225)(54 236)(55 227)(56 238)(57 229)(58 240)(59 231)(60 222)(81 185)(82 196)(83 187)(84 198)(85 189)(86 200)(87 191)(88 182)(89 193)(90 184)(91 195)(92 186)(93 197)(94 188)(95 199)(96 190)(97 181)(98 192)(99 183)(100 194)(101 148)(102 159)(103 150)(104 141)(105 152)(106 143)(107 154)(108 145)(109 156)(110 147)(111 158)(112 149)(113 160)(114 151)(115 142)(116 153)(117 144)(118 155)(119 146)(120 157)(161 208)(162 219)(163 210)(164 201)(165 212)(166 203)(167 214)(168 205)(169 216)(170 207)(171 218)(172 209)(173 220)(174 211)(175 202)(176 213)(177 204)(178 215)(179 206)(180 217)
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), (1,182,11,192)(2,181,12,191)(3,200,13,190)(4,199,14,189)(5,198,15,188)(6,197,16,187)(7,196,17,186)(8,195,18,185)(9,194,19,184)(10,193,20,183)(21,162,31,172)(22,161,32,171)(23,180,33,170)(24,179,34,169)(25,178,35,168)(26,177,36,167)(27,176,37,166)(28,175,38,165)(29,174,39,164)(30,173,40,163)(41,155,51,145)(42,154,52,144)(43,153,53,143)(44,152,54,142)(45,151,55,141)(46,150,56,160)(47,149,57,159)(48,148,58,158)(49,147,59,157)(50,146,60,156)(61,205,71,215)(62,204,72,214)(63,203,73,213)(64,202,74,212)(65,201,75,211)(66,220,76,210)(67,219,77,209)(68,218,78,208)(69,217,79,207)(70,216,80,206)(81,136,91,126)(82,135,92,125)(83,134,93,124)(84,133,94,123)(85,132,95,122)(86,131,96,121)(87,130,97,140)(88,129,98,139)(89,128,99,138)(90,127,100,137)(101,225,111,235)(102,224,112,234)(103,223,113,233)(104,222,114,232)(105,221,115,231)(106,240,116,230)(107,239,117,229)(108,238,118,228)(109,237,119,227)(110,236,120,226), (1,39,154,102,75,129,11,29,144,112,65,139)(2,30,155,113,76,140,12,40,145,103,66,130)(3,21,156,104,77,131,13,31,146,114,67,121)(4,32,157,115,78,122,14,22,147,105,68,132)(5,23,158,106,79,133,15,33,148,116,69,123)(6,34,159,117,80,124,16,24,149,107,70,134)(7,25,160,108,61,135,17,35,150,118,71,125)(8,36,141,119,62,126,18,26,151,109,72,136)(9,27,142,110,63,137,19,37,152,120,73,127)(10,38,143,101,64,128,20,28,153,111,74,138)(41,238,220,92,181,168,51,228,210,82,191,178)(42,229,201,83,182,179,52,239,211,93,192,169)(43,240,202,94,183,170,53,230,212,84,193,180)(44,231,203,85,184,161,54,221,213,95,194,171)(45,222,204,96,185,172,55,232,214,86,195,162)(46,233,205,87,186,163,56,223,215,97,196,173)(47,224,206,98,187,174,57,234,216,88,197,164)(48,235,207,89,188,165,58,225,217,99,198,175)(49,226,208,100,189,176,59,236,218,90,199,166)(50,237,209,91,190,167,60,227,219,81,200,177), (1,134)(2,125)(3,136)(4,127)(5,138)(6,129)(7,140)(8,131)(9,122)(10,133)(11,124)(12,135)(13,126)(14,137)(15,128)(16,139)(17,130)(18,121)(19,132)(20,123)(21,62)(22,73)(23,64)(24,75)(25,66)(26,77)(27,68)(28,79)(29,70)(30,61)(31,72)(32,63)(33,74)(34,65)(35,76)(36,67)(37,78)(38,69)(39,80)(40,71)(41,233)(42,224)(43,235)(44,226)(45,237)(46,228)(47,239)(48,230)(49,221)(50,232)(51,223)(52,234)(53,225)(54,236)(55,227)(56,238)(57,229)(58,240)(59,231)(60,222)(81,185)(82,196)(83,187)(84,198)(85,189)(86,200)(87,191)(88,182)(89,193)(90,184)(91,195)(92,186)(93,197)(94,188)(95,199)(96,190)(97,181)(98,192)(99,183)(100,194)(101,148)(102,159)(103,150)(104,141)(105,152)(106,143)(107,154)(108,145)(109,156)(110,147)(111,158)(112,149)(113,160)(114,151)(115,142)(116,153)(117,144)(118,155)(119,146)(120,157)(161,208)(162,219)(163,210)(164,201)(165,212)(166,203)(167,214)(168,205)(169,216)(170,207)(171,218)(172,209)(173,220)(174,211)(175,202)(176,213)(177,204)(178,215)(179,206)(180,217)>;
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), (1,182,11,192)(2,181,12,191)(3,200,13,190)(4,199,14,189)(5,198,15,188)(6,197,16,187)(7,196,17,186)(8,195,18,185)(9,194,19,184)(10,193,20,183)(21,162,31,172)(22,161,32,171)(23,180,33,170)(24,179,34,169)(25,178,35,168)(26,177,36,167)(27,176,37,166)(28,175,38,165)(29,174,39,164)(30,173,40,163)(41,155,51,145)(42,154,52,144)(43,153,53,143)(44,152,54,142)(45,151,55,141)(46,150,56,160)(47,149,57,159)(48,148,58,158)(49,147,59,157)(50,146,60,156)(61,205,71,215)(62,204,72,214)(63,203,73,213)(64,202,74,212)(65,201,75,211)(66,220,76,210)(67,219,77,209)(68,218,78,208)(69,217,79,207)(70,216,80,206)(81,136,91,126)(82,135,92,125)(83,134,93,124)(84,133,94,123)(85,132,95,122)(86,131,96,121)(87,130,97,140)(88,129,98,139)(89,128,99,138)(90,127,100,137)(101,225,111,235)(102,224,112,234)(103,223,113,233)(104,222,114,232)(105,221,115,231)(106,240,116,230)(107,239,117,229)(108,238,118,228)(109,237,119,227)(110,236,120,226), (1,39,154,102,75,129,11,29,144,112,65,139)(2,30,155,113,76,140,12,40,145,103,66,130)(3,21,156,104,77,131,13,31,146,114,67,121)(4,32,157,115,78,122,14,22,147,105,68,132)(5,23,158,106,79,133,15,33,148,116,69,123)(6,34,159,117,80,124,16,24,149,107,70,134)(7,25,160,108,61,135,17,35,150,118,71,125)(8,36,141,119,62,126,18,26,151,109,72,136)(9,27,142,110,63,137,19,37,152,120,73,127)(10,38,143,101,64,128,20,28,153,111,74,138)(41,238,220,92,181,168,51,228,210,82,191,178)(42,229,201,83,182,179,52,239,211,93,192,169)(43,240,202,94,183,170,53,230,212,84,193,180)(44,231,203,85,184,161,54,221,213,95,194,171)(45,222,204,96,185,172,55,232,214,86,195,162)(46,233,205,87,186,163,56,223,215,97,196,173)(47,224,206,98,187,174,57,234,216,88,197,164)(48,235,207,89,188,165,58,225,217,99,198,175)(49,226,208,100,189,176,59,236,218,90,199,166)(50,237,209,91,190,167,60,227,219,81,200,177), (1,134)(2,125)(3,136)(4,127)(5,138)(6,129)(7,140)(8,131)(9,122)(10,133)(11,124)(12,135)(13,126)(14,137)(15,128)(16,139)(17,130)(18,121)(19,132)(20,123)(21,62)(22,73)(23,64)(24,75)(25,66)(26,77)(27,68)(28,79)(29,70)(30,61)(31,72)(32,63)(33,74)(34,65)(35,76)(36,67)(37,78)(38,69)(39,80)(40,71)(41,233)(42,224)(43,235)(44,226)(45,237)(46,228)(47,239)(48,230)(49,221)(50,232)(51,223)(52,234)(53,225)(54,236)(55,227)(56,238)(57,229)(58,240)(59,231)(60,222)(81,185)(82,196)(83,187)(84,198)(85,189)(86,200)(87,191)(88,182)(89,193)(90,184)(91,195)(92,186)(93,197)(94,188)(95,199)(96,190)(97,181)(98,192)(99,183)(100,194)(101,148)(102,159)(103,150)(104,141)(105,152)(106,143)(107,154)(108,145)(109,156)(110,147)(111,158)(112,149)(113,160)(114,151)(115,142)(116,153)(117,144)(118,155)(119,146)(120,157)(161,208)(162,219)(163,210)(164,201)(165,212)(166,203)(167,214)(168,205)(169,216)(170,207)(171,218)(172,209)(173,220)(174,211)(175,202)(176,213)(177,204)(178,215)(179,206)(180,217) );
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)], [(1,182,11,192),(2,181,12,191),(3,200,13,190),(4,199,14,189),(5,198,15,188),(6,197,16,187),(7,196,17,186),(8,195,18,185),(9,194,19,184),(10,193,20,183),(21,162,31,172),(22,161,32,171),(23,180,33,170),(24,179,34,169),(25,178,35,168),(26,177,36,167),(27,176,37,166),(28,175,38,165),(29,174,39,164),(30,173,40,163),(41,155,51,145),(42,154,52,144),(43,153,53,143),(44,152,54,142),(45,151,55,141),(46,150,56,160),(47,149,57,159),(48,148,58,158),(49,147,59,157),(50,146,60,156),(61,205,71,215),(62,204,72,214),(63,203,73,213),(64,202,74,212),(65,201,75,211),(66,220,76,210),(67,219,77,209),(68,218,78,208),(69,217,79,207),(70,216,80,206),(81,136,91,126),(82,135,92,125),(83,134,93,124),(84,133,94,123),(85,132,95,122),(86,131,96,121),(87,130,97,140),(88,129,98,139),(89,128,99,138),(90,127,100,137),(101,225,111,235),(102,224,112,234),(103,223,113,233),(104,222,114,232),(105,221,115,231),(106,240,116,230),(107,239,117,229),(108,238,118,228),(109,237,119,227),(110,236,120,226)], [(1,39,154,102,75,129,11,29,144,112,65,139),(2,30,155,113,76,140,12,40,145,103,66,130),(3,21,156,104,77,131,13,31,146,114,67,121),(4,32,157,115,78,122,14,22,147,105,68,132),(5,23,158,106,79,133,15,33,148,116,69,123),(6,34,159,117,80,124,16,24,149,107,70,134),(7,25,160,108,61,135,17,35,150,118,71,125),(8,36,141,119,62,126,18,26,151,109,72,136),(9,27,142,110,63,137,19,37,152,120,73,127),(10,38,143,101,64,128,20,28,153,111,74,138),(41,238,220,92,181,168,51,228,210,82,191,178),(42,229,201,83,182,179,52,239,211,93,192,169),(43,240,202,94,183,170,53,230,212,84,193,180),(44,231,203,85,184,161,54,221,213,95,194,171),(45,222,204,96,185,172,55,232,214,86,195,162),(46,233,205,87,186,163,56,223,215,97,196,173),(47,224,206,98,187,174,57,234,216,88,197,164),(48,235,207,89,188,165,58,225,217,99,198,175),(49,226,208,100,189,176,59,236,218,90,199,166),(50,237,209,91,190,167,60,227,219,81,200,177)], [(1,134),(2,125),(3,136),(4,127),(5,138),(6,129),(7,140),(8,131),(9,122),(10,133),(11,124),(12,135),(13,126),(14,137),(15,128),(16,139),(17,130),(18,121),(19,132),(20,123),(21,62),(22,73),(23,64),(24,75),(25,66),(26,77),(27,68),(28,79),(29,70),(30,61),(31,72),(32,63),(33,74),(34,65),(35,76),(36,67),(37,78),(38,69),(39,80),(40,71),(41,233),(42,224),(43,235),(44,226),(45,237),(46,228),(47,239),(48,230),(49,221),(50,232),(51,223),(52,234),(53,225),(54,236),(55,227),(56,238),(57,229),(58,240),(59,231),(60,222),(81,185),(82,196),(83,187),(84,198),(85,189),(86,200),(87,191),(88,182),(89,193),(90,184),(91,195),(92,186),(93,197),(94,188),(95,199),(96,190),(97,181),(98,192),(99,183),(100,194),(101,148),(102,159),(103,150),(104,141),(105,152),(106,143),(107,154),(108,145),(109,156),(110,147),(111,158),(112,149),(113,160),(114,151),(115,142),(116,153),(117,144),(118,155),(119,146),(120,157),(161,208),(162,219),(163,210),(164,201),(165,212),(166,203),(167,214),(168,205),(169,216),(170,207),(171,218),(172,209),(173,220),(174,211),(175,202),(176,213),(177,204),(178,215),(179,206),(180,217)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 15A | 15B | 20A | ··· | 20F | 20G | 20H | 20I | 20J | 24A | 24B | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 12 | 60 | 2 | 2 | 3 | 3 | 4 | 20 | 2 | 2 | 2 | 10 | 10 | 30 | 30 | 2 | 2 | 12 | ··· | 12 | 4 | 8 | 40 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 20 | 20 | 4 | 4 | 8 | ··· | 8 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | S3×D4 | S3×D5 | D24⋊C2 | C2×S3×D5 | D4.8D10 | S3×C5⋊D4 | Dic10.27D6 |
| kernel | Dic10.27D6 | S3×C5⋊2C8 | C5⋊D24 | C20.D6 | C3×C5⋊Q16 | Q8⋊2D15 | D60⋊C2 | C5×Q8⋊3S3 | C5⋊Q16 | C5×Dic3 | S3×C10 | Q8⋊3S3 | C5⋊2C8 | Dic10 | C5×Q8 | C4×S3 | D12 | C3×Q8 | C15 | Dic3 | D6 | C10 | Q8 | 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 | 4 | 4 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of Dic10.27D6 ►in GL6(𝔽241)
| 1 | 240 | 0 | 0 | 0 | 0 |
| 53 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 123 | 177 |
| 89 | 225 | 0 | 0 | 0 | 0 |
| 13 | 152 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 233 | 24 |
| 0 | 0 | 0 | 0 | 168 | 8 |
| 214 | 192 | 0 | 0 | 0 | 0 |
| 187 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 3 |
| 0 | 0 | 0 | 0 | 160 | 1 |
| 27 | 49 | 0 | 0 | 0 | 0 |
| 54 | 214 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 49 |
| 0 | 0 | 0 | 0 | 123 | 177 |
G:=sub<GL(6,GF(241))| [1,53,0,0,0,0,240,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,123,0,0,0,0,0,177],[89,13,0,0,0,0,225,152,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,233,168,0,0,0,0,24,8],[214,187,0,0,0,0,192,27,0,0,0,0,0,0,1,240,0,0,0,0,1,0,0,0,0,0,0,0,240,160,0,0,0,0,3,1],[27,54,0,0,0,0,49,214,0,0,0,0,0,0,1,0,0,0,0,0,1,240,0,0,0,0,0,0,64,123,0,0,0,0,49,177] >;
Dic10.27D6 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}._{27}D_6 % in TeX
G:=Group("Dic10.27D6"); // GroupNames label
G:=SmallGroup(480,595);
// by ID
G=gap.SmallGroup(480,595);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,100,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=d^2=1,b^2=c^6=a^10,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^5>;
// generators/relations