metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.2M4(2), Dic5.22D12, D6⋊(C5⋊C8), C5⋊2(D6⋊C8), (S3×C10)⋊1C8, C10.8(S3×C8), C30.6(C2×C8), C15⋊1(C22⋊C8), C2.2(D6⋊F5), C2.2(D6.F5), C10.4(C8⋊S3), C10.10(D6⋊C4), (C22×S3).2F5, C22.13(S3×F5), (C3×Dic5).30D4, (C2×Dic15).5C4, C6.10(C22⋊F5), C3⋊1(C23.2F5), C30.10(C22⋊C4), C6.2(C22.F5), (C2×Dic5).143D6, Dic5.28(C3⋊D4), (C6×Dic5).136C22, (C6×C5⋊C8)⋊1C2, (C2×C5⋊C8)⋊1S3, C6.4(C2×C5⋊C8), C2.4(S3×C5⋊C8), (S3×C2×C10).2C4, (C2×C15⋊C8)⋊1C2, (C2×C30).4(C2×C4), (C2×C10).6(C4×S3), (C2×C6).14(C2×F5), (C2×S3×Dic5).10C2, SmallGroup(480,246)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.22D12
G = < a,b,c,d | a10=1, b2=c12=a5, d2=b, bab-1=a-1, cac-1=dad-1=a3, bc=cb, bd=db, dcd-1=a5bc11 >
Subgroups: 468 in 100 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C8, C2×C4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, C22×C4, Dic5, Dic5, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C22⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, C2×C3⋊C8, C2×C24, S3×C2×C4, C3×Dic5, Dic15, S3×C10, S3×C10, C2×C30, C2×C5⋊C8, C2×C5⋊C8, C22×Dic5, D6⋊C8, C3×C5⋊C8, C15⋊C8, S3×Dic5, C6×Dic5, C2×Dic15, S3×C2×C10, C23.2F5, C6×C5⋊C8, C2×C15⋊C8, C2×S3×Dic5, Dic5.22D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), F5, C4×S3, D12, C3⋊D4, C22⋊C8, C5⋊C8, C2×F5, S3×C8, C8⋊S3, D6⋊C4, C2×C5⋊C8, C22.F5, C22⋊F5, D6⋊C8, S3×F5, C23.2F5, D6⋊F5, S3×C5⋊C8, D6.F5, Dic5.22D12
►On 240 points
Generators in S240
(1 177 198 123 74 13 189 210 135 86)(2 124 190 87 199 14 136 178 75 211)(3 88 137 212 191 15 76 125 200 179)(4 213 77 180 138 16 201 89 192 126)(5 181 202 127 78 17 169 214 139 90)(6 128 170 91 203 18 140 182 79 215)(7 92 141 216 171 19 80 129 204 183)(8 193 81 184 142 20 205 93 172 130)(9 185 206 131 82 21 173 194 143 94)(10 132 174 95 207 22 144 186 83 195)(11 96 121 196 175 23 84 133 208 187)(12 197 85 188 122 24 209 73 176 134)(25 236 63 115 160 37 224 51 103 148)(26 116 225 149 64 38 104 237 161 52)(27 150 105 53 226 39 162 117 65 238)(28 54 163 239 106 40 66 151 227 118)(29 240 67 119 164 41 228 55 107 152)(30 120 229 153 68 42 108 217 165 56)(31 154 109 57 230 43 166 97 69 218)(32 58 167 219 110 44 70 155 231 98)(33 220 71 99 168 45 232 59 111 156)(34 100 233 157 72 46 112 221 145 60)(35 158 113 61 234 47 146 101 49 222)(36 62 147 223 114 48 50 159 235 102)
(1 71 13 59)(2 72 14 60)(3 49 15 61)(4 50 16 62)(5 51 17 63)(6 52 18 64)(7 53 19 65)(8 54 20 66)(9 55 21 67)(10 56 22 68)(11 57 23 69)(12 58 24 70)(25 202 37 214)(26 203 38 215)(27 204 39 216)(28 205 40 193)(29 206 41 194)(30 207 42 195)(31 208 43 196)(32 209 44 197)(33 210 45 198)(34 211 46 199)(35 212 47 200)(36 213 48 201)(73 110 85 98)(74 111 86 99)(75 112 87 100)(76 113 88 101)(77 114 89 102)(78 115 90 103)(79 116 91 104)(80 117 92 105)(81 118 93 106)(82 119 94 107)(83 120 95 108)(84 97 96 109)(121 154 133 166)(122 155 134 167)(123 156 135 168)(124 157 136 145)(125 158 137 146)(126 159 138 147)(127 160 139 148)(128 161 140 149)(129 162 141 150)(130 163 142 151)(131 164 143 152)(132 165 144 153)(169 236 181 224)(170 237 182 225)(171 238 183 226)(172 239 184 227)(173 240 185 228)(174 217 186 229)(175 218 187 230)(176 219 188 231)(177 220 189 232)(178 221 190 233)(179 222 191 234)(180 223 192 235)
(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 70 71 12 13 58 59 24)(2 11 72 57 14 23 60 69)(3 56 49 22 15 68 61 10)(4 21 50 67 16 9 62 55)(5 66 51 8 17 54 63 20)(6 7 52 53 18 19 64 65)(25 193 202 28 37 205 214 40)(26 27 203 204 38 39 215 216)(29 213 206 48 41 201 194 36)(30 47 207 200 42 35 195 212)(31 199 208 34 43 211 196 46)(32 33 209 210 44 45 197 198)(73 74 110 111 85 86 98 99)(75 84 112 97 87 96 100 109)(76 120 113 95 88 108 101 83)(77 94 114 107 89 82 102 119)(78 106 115 81 90 118 103 93)(79 80 116 117 91 92 104 105)(121 145 154 124 133 157 166 136)(122 123 155 156 134 135 167 168)(125 165 158 144 137 153 146 132)(126 143 159 152 138 131 147 164)(127 151 160 130 139 163 148 142)(128 129 161 162 140 141 149 150)(169 227 236 172 181 239 224 184)(170 171 237 238 182 183 225 226)(173 223 240 192 185 235 228 180)(174 191 217 234 186 179 229 222)(175 233 218 178 187 221 230 190)(176 177 219 220 188 189 231 232)
G:=sub<Sym(240)| (1,177,198,123,74,13,189,210,135,86)(2,124,190,87,199,14,136,178,75,211)(3,88,137,212,191,15,76,125,200,179)(4,213,77,180,138,16,201,89,192,126)(5,181,202,127,78,17,169,214,139,90)(6,128,170,91,203,18,140,182,79,215)(7,92,141,216,171,19,80,129,204,183)(8,193,81,184,142,20,205,93,172,130)(9,185,206,131,82,21,173,194,143,94)(10,132,174,95,207,22,144,186,83,195)(11,96,121,196,175,23,84,133,208,187)(12,197,85,188,122,24,209,73,176,134)(25,236,63,115,160,37,224,51,103,148)(26,116,225,149,64,38,104,237,161,52)(27,150,105,53,226,39,162,117,65,238)(28,54,163,239,106,40,66,151,227,118)(29,240,67,119,164,41,228,55,107,152)(30,120,229,153,68,42,108,217,165,56)(31,154,109,57,230,43,166,97,69,218)(32,58,167,219,110,44,70,155,231,98)(33,220,71,99,168,45,232,59,111,156)(34,100,233,157,72,46,112,221,145,60)(35,158,113,61,234,47,146,101,49,222)(36,62,147,223,114,48,50,159,235,102), (1,71,13,59)(2,72,14,60)(3,49,15,61)(4,50,16,62)(5,51,17,63)(6,52,18,64)(7,53,19,65)(8,54,20,66)(9,55,21,67)(10,56,22,68)(11,57,23,69)(12,58,24,70)(25,202,37,214)(26,203,38,215)(27,204,39,216)(28,205,40,193)(29,206,41,194)(30,207,42,195)(31,208,43,196)(32,209,44,197)(33,210,45,198)(34,211,46,199)(35,212,47,200)(36,213,48,201)(73,110,85,98)(74,111,86,99)(75,112,87,100)(76,113,88,101)(77,114,89,102)(78,115,90,103)(79,116,91,104)(80,117,92,105)(81,118,93,106)(82,119,94,107)(83,120,95,108)(84,97,96,109)(121,154,133,166)(122,155,134,167)(123,156,135,168)(124,157,136,145)(125,158,137,146)(126,159,138,147)(127,160,139,148)(128,161,140,149)(129,162,141,150)(130,163,142,151)(131,164,143,152)(132,165,144,153)(169,236,181,224)(170,237,182,225)(171,238,183,226)(172,239,184,227)(173,240,185,228)(174,217,186,229)(175,218,187,230)(176,219,188,231)(177,220,189,232)(178,221,190,233)(179,222,191,234)(180,223,192,235), (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), (1,70,71,12,13,58,59,24)(2,11,72,57,14,23,60,69)(3,56,49,22,15,68,61,10)(4,21,50,67,16,9,62,55)(5,66,51,8,17,54,63,20)(6,7,52,53,18,19,64,65)(25,193,202,28,37,205,214,40)(26,27,203,204,38,39,215,216)(29,213,206,48,41,201,194,36)(30,47,207,200,42,35,195,212)(31,199,208,34,43,211,196,46)(32,33,209,210,44,45,197,198)(73,74,110,111,85,86,98,99)(75,84,112,97,87,96,100,109)(76,120,113,95,88,108,101,83)(77,94,114,107,89,82,102,119)(78,106,115,81,90,118,103,93)(79,80,116,117,91,92,104,105)(121,145,154,124,133,157,166,136)(122,123,155,156,134,135,167,168)(125,165,158,144,137,153,146,132)(126,143,159,152,138,131,147,164)(127,151,160,130,139,163,148,142)(128,129,161,162,140,141,149,150)(169,227,236,172,181,239,224,184)(170,171,237,238,182,183,225,226)(173,223,240,192,185,235,228,180)(174,191,217,234,186,179,229,222)(175,233,218,178,187,221,230,190)(176,177,219,220,188,189,231,232)>;
G:=Group( (1,177,198,123,74,13,189,210,135,86)(2,124,190,87,199,14,136,178,75,211)(3,88,137,212,191,15,76,125,200,179)(4,213,77,180,138,16,201,89,192,126)(5,181,202,127,78,17,169,214,139,90)(6,128,170,91,203,18,140,182,79,215)(7,92,141,216,171,19,80,129,204,183)(8,193,81,184,142,20,205,93,172,130)(9,185,206,131,82,21,173,194,143,94)(10,132,174,95,207,22,144,186,83,195)(11,96,121,196,175,23,84,133,208,187)(12,197,85,188,122,24,209,73,176,134)(25,236,63,115,160,37,224,51,103,148)(26,116,225,149,64,38,104,237,161,52)(27,150,105,53,226,39,162,117,65,238)(28,54,163,239,106,40,66,151,227,118)(29,240,67,119,164,41,228,55,107,152)(30,120,229,153,68,42,108,217,165,56)(31,154,109,57,230,43,166,97,69,218)(32,58,167,219,110,44,70,155,231,98)(33,220,71,99,168,45,232,59,111,156)(34,100,233,157,72,46,112,221,145,60)(35,158,113,61,234,47,146,101,49,222)(36,62,147,223,114,48,50,159,235,102), (1,71,13,59)(2,72,14,60)(3,49,15,61)(4,50,16,62)(5,51,17,63)(6,52,18,64)(7,53,19,65)(8,54,20,66)(9,55,21,67)(10,56,22,68)(11,57,23,69)(12,58,24,70)(25,202,37,214)(26,203,38,215)(27,204,39,216)(28,205,40,193)(29,206,41,194)(30,207,42,195)(31,208,43,196)(32,209,44,197)(33,210,45,198)(34,211,46,199)(35,212,47,200)(36,213,48,201)(73,110,85,98)(74,111,86,99)(75,112,87,100)(76,113,88,101)(77,114,89,102)(78,115,90,103)(79,116,91,104)(80,117,92,105)(81,118,93,106)(82,119,94,107)(83,120,95,108)(84,97,96,109)(121,154,133,166)(122,155,134,167)(123,156,135,168)(124,157,136,145)(125,158,137,146)(126,159,138,147)(127,160,139,148)(128,161,140,149)(129,162,141,150)(130,163,142,151)(131,164,143,152)(132,165,144,153)(169,236,181,224)(170,237,182,225)(171,238,183,226)(172,239,184,227)(173,240,185,228)(174,217,186,229)(175,218,187,230)(176,219,188,231)(177,220,189,232)(178,221,190,233)(179,222,191,234)(180,223,192,235), (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), (1,70,71,12,13,58,59,24)(2,11,72,57,14,23,60,69)(3,56,49,22,15,68,61,10)(4,21,50,67,16,9,62,55)(5,66,51,8,17,54,63,20)(6,7,52,53,18,19,64,65)(25,193,202,28,37,205,214,40)(26,27,203,204,38,39,215,216)(29,213,206,48,41,201,194,36)(30,47,207,200,42,35,195,212)(31,199,208,34,43,211,196,46)(32,33,209,210,44,45,197,198)(73,74,110,111,85,86,98,99)(75,84,112,97,87,96,100,109)(76,120,113,95,88,108,101,83)(77,94,114,107,89,82,102,119)(78,106,115,81,90,118,103,93)(79,80,116,117,91,92,104,105)(121,145,154,124,133,157,166,136)(122,123,155,156,134,135,167,168)(125,165,158,144,137,153,146,132)(126,143,159,152,138,131,147,164)(127,151,160,130,139,163,148,142)(128,129,161,162,140,141,149,150)(169,227,236,172,181,239,224,184)(170,171,237,238,182,183,225,226)(173,223,240,192,185,235,228,180)(174,191,217,234,186,179,229,222)(175,233,218,178,187,221,230,190)(176,177,219,220,188,189,231,232) );
G=PermutationGroup([[(1,177,198,123,74,13,189,210,135,86),(2,124,190,87,199,14,136,178,75,211),(3,88,137,212,191,15,76,125,200,179),(4,213,77,180,138,16,201,89,192,126),(5,181,202,127,78,17,169,214,139,90),(6,128,170,91,203,18,140,182,79,215),(7,92,141,216,171,19,80,129,204,183),(8,193,81,184,142,20,205,93,172,130),(9,185,206,131,82,21,173,194,143,94),(10,132,174,95,207,22,144,186,83,195),(11,96,121,196,175,23,84,133,208,187),(12,197,85,188,122,24,209,73,176,134),(25,236,63,115,160,37,224,51,103,148),(26,116,225,149,64,38,104,237,161,52),(27,150,105,53,226,39,162,117,65,238),(28,54,163,239,106,40,66,151,227,118),(29,240,67,119,164,41,228,55,107,152),(30,120,229,153,68,42,108,217,165,56),(31,154,109,57,230,43,166,97,69,218),(32,58,167,219,110,44,70,155,231,98),(33,220,71,99,168,45,232,59,111,156),(34,100,233,157,72,46,112,221,145,60),(35,158,113,61,234,47,146,101,49,222),(36,62,147,223,114,48,50,159,235,102)], [(1,71,13,59),(2,72,14,60),(3,49,15,61),(4,50,16,62),(5,51,17,63),(6,52,18,64),(7,53,19,65),(8,54,20,66),(9,55,21,67),(10,56,22,68),(11,57,23,69),(12,58,24,70),(25,202,37,214),(26,203,38,215),(27,204,39,216),(28,205,40,193),(29,206,41,194),(30,207,42,195),(31,208,43,196),(32,209,44,197),(33,210,45,198),(34,211,46,199),(35,212,47,200),(36,213,48,201),(73,110,85,98),(74,111,86,99),(75,112,87,100),(76,113,88,101),(77,114,89,102),(78,115,90,103),(79,116,91,104),(80,117,92,105),(81,118,93,106),(82,119,94,107),(83,120,95,108),(84,97,96,109),(121,154,133,166),(122,155,134,167),(123,156,135,168),(124,157,136,145),(125,158,137,146),(126,159,138,147),(127,160,139,148),(128,161,140,149),(129,162,141,150),(130,163,142,151),(131,164,143,152),(132,165,144,153),(169,236,181,224),(170,237,182,225),(171,238,183,226),(172,239,184,227),(173,240,185,228),(174,217,186,229),(175,218,187,230),(176,219,188,231),(177,220,189,232),(178,221,190,233),(179,222,191,234),(180,223,192,235)], [(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)], [(1,70,71,12,13,58,59,24),(2,11,72,57,14,23,60,69),(3,56,49,22,15,68,61,10),(4,21,50,67,16,9,62,55),(5,66,51,8,17,54,63,20),(6,7,52,53,18,19,64,65),(25,193,202,28,37,205,214,40),(26,27,203,204,38,39,215,216),(29,213,206,48,41,201,194,36),(30,47,207,200,42,35,195,212),(31,199,208,34,43,211,196,46),(32,33,209,210,44,45,197,198),(73,74,110,111,85,86,98,99),(75,84,112,97,87,96,100,109),(76,120,113,95,88,108,101,83),(77,94,114,107,89,82,102,119),(78,106,115,81,90,118,103,93),(79,80,116,117,91,92,104,105),(121,145,154,124,133,157,166,136),(122,123,155,156,134,135,167,168),(125,165,158,144,137,153,146,132),(126,143,159,152,138,131,147,164),(127,151,160,130,139,163,148,142),(128,129,161,162,140,141,149,150),(169,227,236,172,181,239,224,184),(170,171,237,238,182,183,225,226),(173,223,240,192,185,235,228,180),(174,191,217,234,186,179,229,222),(175,233,218,178,187,221,230,190),(176,177,219,220,188,189,231,232)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 12A | 12B | 12C | 12D | 15 | 24A | ··· | 24H | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 24 | ··· | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 5 | 5 | 5 | 5 | 30 | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | 10 | 10 | 10 | 8 | 10 | ··· | 10 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | + | + | + | - | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | M4(2) | D12 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | F5 | C5⋊C8 | C2×F5 | C22.F5 | C22⋊F5 | S3×F5 | D6⋊F5 | S3×C5⋊C8 | D6.F5 |
| kernel | Dic5.22D12 | C6×C5⋊C8 | C2×C15⋊C8 | C2×S3×Dic5 | C2×Dic15 | S3×C2×C10 | S3×C10 | C2×C5⋊C8 | C3×Dic5 | C2×Dic5 | C30 | Dic5 | Dic5 | C2×C10 | C10 | C10 | C22×S3 | D6 | C2×C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 |
Matrix representation of Dic5.22D12 ►in GL8(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 237 | 0 | 52 | 189 |
| 0 | 0 | 0 | 0 | 180 | 0 | 52 | 240 |
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 212 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 221 | 0 | 0 |
| 0 | 0 | 0 | 0 | 101 | 167 | 20 | 212 |
| 0 | 0 | 0 | 0 | 36 | 140 | 47 | 221 |
| 0 | 160 | 0 | 0 | 0 | 0 | 0 | 0 |
| 96 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 213 | 237 | 239 | 0 |
| 0 | 0 | 0 | 0 | 55 | 219 | 0 | 239 |
| 0 | 0 | 0 | 0 | 51 | 206 | 28 | 4 |
| 0 | 0 | 0 | 0 | 215 | 122 | 186 | 22 |
| 0 | 160 | 0 | 0 | 0 | 0 | 0 | 0 |
| 145 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 213 | 237 | 239 | 0 |
| 0 | 0 | 0 | 0 | 55 | 219 | 0 | 239 |
| 0 | 0 | 0 | 0 | 31 | 235 | 28 | 4 |
| 0 | 0 | 0 | 0 | 168 | 142 | 186 | 22 |
G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,189,240,237,180,0,0,0,0,1,0,0,0,0,0,0,0,0,0,52,52,0,0,0,0,0,0,189,240],[64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,20,47,101,36,0,0,0,0,212,221,167,140,0,0,0,0,0,0,20,47,0,0,0,0,0,0,212,221],[0,96,0,0,0,0,0,0,160,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,213,55,51,215,0,0,0,0,237,219,206,122,0,0,0,0,239,0,28,186,0,0,0,0,0,239,4,22],[0,145,0,0,0,0,0,0,160,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,213,55,31,168,0,0,0,0,237,219,235,142,0,0,0,0,239,0,28,186,0,0,0,0,0,239,4,22] >;
Dic5.22D12 in GAP, Magma, Sage, TeX
{\rm Dic}_5._{22}D_{12} % in TeX
G:=Group("Dic5.22D12"); // GroupNames label
G:=SmallGroup(480,246);
// by ID
G=gap.SmallGroup(480,246);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=1,b^2=c^12=a^5,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^3,b*c=c*b,b*d=d*b,d*c*d^-1=a^5*b*c^11>;
// generators/relations