metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.13D10, C60.34C23, Dic10.26D6, Dic30.11C22, C5⋊Q16⋊1S3, C5⋊2C8.7D6, Q8⋊3S3.D5, Q8.9(S3×D5), C15⋊7Q16⋊4C2, C5⋊7(Q16⋊S3), (C5×Q8).36D6, (C3×Q8).2D10, (S3×Dic10)⋊3C2, (C4×S3).10D10, (S3×C10).13D4, C20.D6⋊6C2, C10.150(S3×D4), C30.196(C2×D4), D12.D5⋊6C2, D6.9(C5⋊D4), D6.Dic5⋊6C2, C3⋊3(D4.9D10), C15⋊17(C8.C22), C20.34(C22×S3), (C5×Dic3).39D4, C12.34(C22×D5), (Q8×C15).4C22, (S3×C20).12C22, C15⋊3C8.10C22, (C5×D12).12C22, Dic3.18(C5⋊D4), (C3×Dic10).10C22, C4.34(C2×S3×D5), (C3×C5⋊Q16)⋊2C2, C2.31(S3×C5⋊D4), C6.53(C2×C5⋊D4), (C3×C5⋊2C8).8C22, (C5×Q8⋊3S3).1C2, SmallGroup(480,586)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.26D6
G = < a,b,c,d | a20=d2=1, b2=c6=a10, bab-1=a-1, cac-1=dad=a11, cbc-1=dbd=a5b, dcd=a10c5 >
Subgroups: 556 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C3×Q8, C3×Q8, C5×S3, C30, C8.C22, C5⋊2C8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C5×D4, C5×Q8, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, C5×Dic3, C3×Dic5, Dic15, C60, C60, S3×C10, S3×C10, C4.Dic5, D4.D5, C5⋊Q16, C5⋊Q16, C2×Dic10, C5×C4○D4, Q16⋊S3, C3×C5⋊2C8, C15⋊3C8, S3×Dic5, C15⋊Q8, C3×Dic10, S3×C20, S3×C20, C5×D12, C5×D12, Dic30, Q8×C15, D4.9D10, D6.Dic5, C20.D6, D12.D5, C3×C5⋊Q16, C15⋊7Q16, S3×Dic10, C5×Q8⋊3S3, Dic10.26D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q16⋊S3, C2×S3×D5, D4.9D10, S3×C5⋊D4, Dic10.26D6
►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 50 11 60)(2 49 12 59)(3 48 13 58)(4 47 14 57)(5 46 15 56)(6 45 16 55)(7 44 17 54)(8 43 18 53)(9 42 19 52)(10 41 20 51)(21 223 31 233)(22 222 32 232)(23 221 33 231)(24 240 34 230)(25 239 35 229)(26 238 36 228)(27 237 37 227)(28 236 38 226)(29 235 39 225)(30 234 40 224)(61 212 71 202)(62 211 72 201)(63 210 73 220)(64 209 74 219)(65 208 75 218)(66 207 76 217)(67 206 77 216)(68 205 78 215)(69 204 79 214)(70 203 80 213)(81 161 91 171)(82 180 92 170)(83 179 93 169)(84 178 94 168)(85 177 95 167)(86 176 96 166)(87 175 97 165)(88 174 98 164)(89 173 99 163)(90 172 100 162)(101 158 111 148)(102 157 112 147)(103 156 113 146)(104 155 114 145)(105 154 115 144)(106 153 116 143)(107 152 117 142)(108 151 118 141)(109 150 119 160)(110 149 120 159)(121 187 131 197)(122 186 132 196)(123 185 133 195)(124 184 134 194)(125 183 135 193)(126 182 136 192)(127 181 137 191)(128 200 138 190)(129 199 139 189)(130 198 140 188)
(1 174 24 182 158 70 11 164 34 192 148 80)(2 165 25 193 159 61 12 175 35 183 149 71)(3 176 26 184 160 72 13 166 36 194 150 62)(4 167 27 195 141 63 14 177 37 185 151 73)(5 178 28 186 142 74 15 168 38 196 152 64)(6 169 29 197 143 65 16 179 39 187 153 75)(7 180 30 188 144 76 17 170 40 198 154 66)(8 171 31 199 145 67 18 161 21 189 155 77)(9 162 32 190 146 78 19 172 22 200 156 68)(10 173 33 181 147 69 20 163 23 191 157 79)(41 94 231 132 102 219 51 84 221 122 112 209)(42 85 232 123 103 210 52 95 222 133 113 220)(43 96 233 134 104 201 53 86 223 124 114 211)(44 87 234 125 105 212 54 97 224 135 115 202)(45 98 235 136 106 203 55 88 225 126 116 213)(46 89 236 127 107 214 56 99 226 137 117 204)(47 100 237 138 108 205 57 90 227 128 118 215)(48 91 238 129 109 216 58 81 228 139 119 206)(49 82 239 140 110 207 59 92 229 130 120 217)(50 93 240 131 111 218 60 83 230 121 101 208)
(1 70)(2 61)(3 72)(4 63)(5 74)(6 65)(7 76)(8 67)(9 78)(10 69)(11 80)(12 71)(13 62)(14 73)(15 64)(16 75)(17 66)(18 77)(19 68)(20 79)(21 189)(22 200)(23 191)(24 182)(25 193)(26 184)(27 195)(28 186)(29 197)(30 188)(31 199)(32 190)(33 181)(34 192)(35 183)(36 194)(37 185)(38 196)(39 187)(40 198)(41 219)(42 210)(43 201)(44 212)(45 203)(46 214)(47 205)(48 216)(49 207)(50 218)(51 209)(52 220)(53 211)(54 202)(55 213)(56 204)(57 215)(58 206)(59 217)(60 208)(81 119)(82 110)(83 101)(84 112)(85 103)(86 114)(87 105)(88 116)(89 107)(90 118)(91 109)(92 120)(93 111)(94 102)(95 113)(96 104)(97 115)(98 106)(99 117)(100 108)(121 230)(122 221)(123 232)(124 223)(125 234)(126 225)(127 236)(128 227)(129 238)(130 229)(131 240)(132 231)(133 222)(134 233)(135 224)(136 235)(137 226)(138 237)(139 228)(140 239)(141 167)(142 178)(143 169)(144 180)(145 171)(146 162)(147 173)(148 164)(149 175)(150 166)(151 177)(152 168)(153 179)(154 170)(155 161)(156 172)(157 163)(158 174)(159 165)(160 176)
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), (1,50,11,60)(2,49,12,59)(3,48,13,58)(4,47,14,57)(5,46,15,56)(6,45,16,55)(7,44,17,54)(8,43,18,53)(9,42,19,52)(10,41,20,51)(21,223,31,233)(22,222,32,232)(23,221,33,231)(24,240,34,230)(25,239,35,229)(26,238,36,228)(27,237,37,227)(28,236,38,226)(29,235,39,225)(30,234,40,224)(61,212,71,202)(62,211,72,201)(63,210,73,220)(64,209,74,219)(65,208,75,218)(66,207,76,217)(67,206,77,216)(68,205,78,215)(69,204,79,214)(70,203,80,213)(81,161,91,171)(82,180,92,170)(83,179,93,169)(84,178,94,168)(85,177,95,167)(86,176,96,166)(87,175,97,165)(88,174,98,164)(89,173,99,163)(90,172,100,162)(101,158,111,148)(102,157,112,147)(103,156,113,146)(104,155,114,145)(105,154,115,144)(106,153,116,143)(107,152,117,142)(108,151,118,141)(109,150,119,160)(110,149,120,159)(121,187,131,197)(122,186,132,196)(123,185,133,195)(124,184,134,194)(125,183,135,193)(126,182,136,192)(127,181,137,191)(128,200,138,190)(129,199,139,189)(130,198,140,188), (1,174,24,182,158,70,11,164,34,192,148,80)(2,165,25,193,159,61,12,175,35,183,149,71)(3,176,26,184,160,72,13,166,36,194,150,62)(4,167,27,195,141,63,14,177,37,185,151,73)(5,178,28,186,142,74,15,168,38,196,152,64)(6,169,29,197,143,65,16,179,39,187,153,75)(7,180,30,188,144,76,17,170,40,198,154,66)(8,171,31,199,145,67,18,161,21,189,155,77)(9,162,32,190,146,78,19,172,22,200,156,68)(10,173,33,181,147,69,20,163,23,191,157,79)(41,94,231,132,102,219,51,84,221,122,112,209)(42,85,232,123,103,210,52,95,222,133,113,220)(43,96,233,134,104,201,53,86,223,124,114,211)(44,87,234,125,105,212,54,97,224,135,115,202)(45,98,235,136,106,203,55,88,225,126,116,213)(46,89,236,127,107,214,56,99,226,137,117,204)(47,100,237,138,108,205,57,90,227,128,118,215)(48,91,238,129,109,216,58,81,228,139,119,206)(49,82,239,140,110,207,59,92,229,130,120,217)(50,93,240,131,111,218,60,83,230,121,101,208), (1,70)(2,61)(3,72)(4,63)(5,74)(6,65)(7,76)(8,67)(9,78)(10,69)(11,80)(12,71)(13,62)(14,73)(15,64)(16,75)(17,66)(18,77)(19,68)(20,79)(21,189)(22,200)(23,191)(24,182)(25,193)(26,184)(27,195)(28,186)(29,197)(30,188)(31,199)(32,190)(33,181)(34,192)(35,183)(36,194)(37,185)(38,196)(39,187)(40,198)(41,219)(42,210)(43,201)(44,212)(45,203)(46,214)(47,205)(48,216)(49,207)(50,218)(51,209)(52,220)(53,211)(54,202)(55,213)(56,204)(57,215)(58,206)(59,217)(60,208)(81,119)(82,110)(83,101)(84,112)(85,103)(86,114)(87,105)(88,116)(89,107)(90,118)(91,109)(92,120)(93,111)(94,102)(95,113)(96,104)(97,115)(98,106)(99,117)(100,108)(121,230)(122,221)(123,232)(124,223)(125,234)(126,225)(127,236)(128,227)(129,238)(130,229)(131,240)(132,231)(133,222)(134,233)(135,224)(136,235)(137,226)(138,237)(139,228)(140,239)(141,167)(142,178)(143,169)(144,180)(145,171)(146,162)(147,173)(148,164)(149,175)(150,166)(151,177)(152,168)(153,179)(154,170)(155,161)(156,172)(157,163)(158,174)(159,165)(160,176)>;
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), (1,50,11,60)(2,49,12,59)(3,48,13,58)(4,47,14,57)(5,46,15,56)(6,45,16,55)(7,44,17,54)(8,43,18,53)(9,42,19,52)(10,41,20,51)(21,223,31,233)(22,222,32,232)(23,221,33,231)(24,240,34,230)(25,239,35,229)(26,238,36,228)(27,237,37,227)(28,236,38,226)(29,235,39,225)(30,234,40,224)(61,212,71,202)(62,211,72,201)(63,210,73,220)(64,209,74,219)(65,208,75,218)(66,207,76,217)(67,206,77,216)(68,205,78,215)(69,204,79,214)(70,203,80,213)(81,161,91,171)(82,180,92,170)(83,179,93,169)(84,178,94,168)(85,177,95,167)(86,176,96,166)(87,175,97,165)(88,174,98,164)(89,173,99,163)(90,172,100,162)(101,158,111,148)(102,157,112,147)(103,156,113,146)(104,155,114,145)(105,154,115,144)(106,153,116,143)(107,152,117,142)(108,151,118,141)(109,150,119,160)(110,149,120,159)(121,187,131,197)(122,186,132,196)(123,185,133,195)(124,184,134,194)(125,183,135,193)(126,182,136,192)(127,181,137,191)(128,200,138,190)(129,199,139,189)(130,198,140,188), (1,174,24,182,158,70,11,164,34,192,148,80)(2,165,25,193,159,61,12,175,35,183,149,71)(3,176,26,184,160,72,13,166,36,194,150,62)(4,167,27,195,141,63,14,177,37,185,151,73)(5,178,28,186,142,74,15,168,38,196,152,64)(6,169,29,197,143,65,16,179,39,187,153,75)(7,180,30,188,144,76,17,170,40,198,154,66)(8,171,31,199,145,67,18,161,21,189,155,77)(9,162,32,190,146,78,19,172,22,200,156,68)(10,173,33,181,147,69,20,163,23,191,157,79)(41,94,231,132,102,219,51,84,221,122,112,209)(42,85,232,123,103,210,52,95,222,133,113,220)(43,96,233,134,104,201,53,86,223,124,114,211)(44,87,234,125,105,212,54,97,224,135,115,202)(45,98,235,136,106,203,55,88,225,126,116,213)(46,89,236,127,107,214,56,99,226,137,117,204)(47,100,237,138,108,205,57,90,227,128,118,215)(48,91,238,129,109,216,58,81,228,139,119,206)(49,82,239,140,110,207,59,92,229,130,120,217)(50,93,240,131,111,218,60,83,230,121,101,208), (1,70)(2,61)(3,72)(4,63)(5,74)(6,65)(7,76)(8,67)(9,78)(10,69)(11,80)(12,71)(13,62)(14,73)(15,64)(16,75)(17,66)(18,77)(19,68)(20,79)(21,189)(22,200)(23,191)(24,182)(25,193)(26,184)(27,195)(28,186)(29,197)(30,188)(31,199)(32,190)(33,181)(34,192)(35,183)(36,194)(37,185)(38,196)(39,187)(40,198)(41,219)(42,210)(43,201)(44,212)(45,203)(46,214)(47,205)(48,216)(49,207)(50,218)(51,209)(52,220)(53,211)(54,202)(55,213)(56,204)(57,215)(58,206)(59,217)(60,208)(81,119)(82,110)(83,101)(84,112)(85,103)(86,114)(87,105)(88,116)(89,107)(90,118)(91,109)(92,120)(93,111)(94,102)(95,113)(96,104)(97,115)(98,106)(99,117)(100,108)(121,230)(122,221)(123,232)(124,223)(125,234)(126,225)(127,236)(128,227)(129,238)(130,229)(131,240)(132,231)(133,222)(134,233)(135,224)(136,235)(137,226)(138,237)(139,228)(140,239)(141,167)(142,178)(143,169)(144,180)(145,171)(146,162)(147,173)(148,164)(149,175)(150,166)(151,177)(152,168)(153,179)(154,170)(155,161)(156,172)(157,163)(158,174)(159,165)(160,176) );
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)], [(1,50,11,60),(2,49,12,59),(3,48,13,58),(4,47,14,57),(5,46,15,56),(6,45,16,55),(7,44,17,54),(8,43,18,53),(9,42,19,52),(10,41,20,51),(21,223,31,233),(22,222,32,232),(23,221,33,231),(24,240,34,230),(25,239,35,229),(26,238,36,228),(27,237,37,227),(28,236,38,226),(29,235,39,225),(30,234,40,224),(61,212,71,202),(62,211,72,201),(63,210,73,220),(64,209,74,219),(65,208,75,218),(66,207,76,217),(67,206,77,216),(68,205,78,215),(69,204,79,214),(70,203,80,213),(81,161,91,171),(82,180,92,170),(83,179,93,169),(84,178,94,168),(85,177,95,167),(86,176,96,166),(87,175,97,165),(88,174,98,164),(89,173,99,163),(90,172,100,162),(101,158,111,148),(102,157,112,147),(103,156,113,146),(104,155,114,145),(105,154,115,144),(106,153,116,143),(107,152,117,142),(108,151,118,141),(109,150,119,160),(110,149,120,159),(121,187,131,197),(122,186,132,196),(123,185,133,195),(124,184,134,194),(125,183,135,193),(126,182,136,192),(127,181,137,191),(128,200,138,190),(129,199,139,189),(130,198,140,188)], [(1,174,24,182,158,70,11,164,34,192,148,80),(2,165,25,193,159,61,12,175,35,183,149,71),(3,176,26,184,160,72,13,166,36,194,150,62),(4,167,27,195,141,63,14,177,37,185,151,73),(5,178,28,186,142,74,15,168,38,196,152,64),(6,169,29,197,143,65,16,179,39,187,153,75),(7,180,30,188,144,76,17,170,40,198,154,66),(8,171,31,199,145,67,18,161,21,189,155,77),(9,162,32,190,146,78,19,172,22,200,156,68),(10,173,33,181,147,69,20,163,23,191,157,79),(41,94,231,132,102,219,51,84,221,122,112,209),(42,85,232,123,103,210,52,95,222,133,113,220),(43,96,233,134,104,201,53,86,223,124,114,211),(44,87,234,125,105,212,54,97,224,135,115,202),(45,98,235,136,106,203,55,88,225,126,116,213),(46,89,236,127,107,214,56,99,226,137,117,204),(47,100,237,138,108,205,57,90,227,128,118,215),(48,91,238,129,109,216,58,81,228,139,119,206),(49,82,239,140,110,207,59,92,229,130,120,217),(50,93,240,131,111,218,60,83,230,121,101,208)], [(1,70),(2,61),(3,72),(4,63),(5,74),(6,65),(7,76),(8,67),(9,78),(10,69),(11,80),(12,71),(13,62),(14,73),(15,64),(16,75),(17,66),(18,77),(19,68),(20,79),(21,189),(22,200),(23,191),(24,182),(25,193),(26,184),(27,195),(28,186),(29,197),(30,188),(31,199),(32,190),(33,181),(34,192),(35,183),(36,194),(37,185),(38,196),(39,187),(40,198),(41,219),(42,210),(43,201),(44,212),(45,203),(46,214),(47,205),(48,216),(49,207),(50,218),(51,209),(52,220),(53,211),(54,202),(55,213),(56,204),(57,215),(58,206),(59,217),(60,208),(81,119),(82,110),(83,101),(84,112),(85,103),(86,114),(87,105),(88,116),(89,107),(90,118),(91,109),(92,120),(93,111),(94,102),(95,113),(96,104),(97,115),(98,106),(99,117),(100,108),(121,230),(122,221),(123,232),(124,223),(125,234),(126,225),(127,236),(128,227),(129,238),(130,229),(131,240),(132,231),(133,222),(134,233),(135,224),(136,235),(137,226),(138,237),(139,228),(140,239),(141,167),(142,178),(143,169),(144,180),(145,171),(146,162),(147,173),(148,164),(149,175),(150,166),(151,177),(152,168),(153,179),(154,170),(155,161),(156,172),(157,163),(158,174),(159,165),(160,176)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6 | 8A | 8B | 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 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 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 | 2 | 2 | 4 | 6 | 20 | 60 | 2 | 2 | 2 | 20 | 60 | 2 | 2 | 12 | ··· | 12 | 4 | 8 | 40 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 20 | 20 | 4 | 4 | 8 | ··· | 8 |
48 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | S3×D4 | S3×D5 | Q16⋊S3 | C2×S3×D5 | D4.9D10 | S3×C5⋊D4 | Dic10.26D6 |
| kernel | Dic10.26D6 | D6.Dic5 | C20.D6 | D12.D5 | C3×C5⋊Q16 | C15⋊7Q16 | S3×Dic10 | C5×Q8⋊3S3 | C5⋊Q16 | C5×Dic3 | S3×C10 | Q8⋊3S3 | C5⋊2C8 | Dic10 | C5×Q8 | C4×S3 | D12 | C3×Q8 | Dic3 | D6 | C15 | 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 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of Dic10.26D6 ►in GL6(𝔽241)
| 189 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 64 | 16 | 0 | 0 | 0 | 0 |
| 211 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 163 | 0 | 120 | 0 |
| 0 | 0 | 0 | 163 | 0 | 120 |
| 0 | 0 | 120 | 0 | 78 | 0 |
| 0 | 0 | 0 | 120 | 0 | 78 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 93 | 148 | 46 | 195 |
| 0 | 0 | 93 | 186 | 46 | 92 |
| 0 | 0 | 46 | 195 | 148 | 93 |
| 0 | 0 | 46 | 92 | 148 | 55 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 148 | 93 | 195 | 46 |
| 0 | 0 | 186 | 93 | 92 | 46 |
| 0 | 0 | 195 | 46 | 93 | 148 |
| 0 | 0 | 92 | 46 | 55 | 148 |
G:=sub<GL(6,GF(241))| [189,240,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[64,211,0,0,0,0,16,177,0,0,0,0,0,0,163,0,120,0,0,0,0,163,0,120,0,0,120,0,78,0,0,0,0,120,0,78],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,93,93,46,46,0,0,148,186,195,92,0,0,46,46,148,148,0,0,195,92,93,55],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,148,186,195,92,0,0,93,93,46,46,0,0,195,92,93,55,0,0,46,46,148,148] >;
Dic10.26D6 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}._{26}D_6 % in TeX
G:=Group("Dic10.26D6"); // GroupNames label
G:=SmallGroup(480,586);
// by ID
G=gap.SmallGroup(480,586);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,100,675,185,80,1356,18822]);
// 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=a^10*c^5>;
// generators/relations