direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C30, C60.199D4, C120⋊49C22, C60.293C23, C8⋊3(C2×C30), (C2×C8)⋊5C30, C40⋊13(C2×C6), (C2×C40)⋊13C6, Q8⋊2(C2×C30), (C2×Q8)⋊5C30, C4.7(D4×C15), (C2×C120)⋊29C2, C24⋊13(C2×C10), (C2×C24)⋊13C10, (Q8×C10)⋊14C6, (Q8×C30)⋊24C2, (C6×Q8)⋊10C10, (C2×D4).6C30, D4.1(C2×C30), C20.42(C3×D4), C10.75(C6×D4), C6.75(D4×C10), C2.12(D4×C30), C12.42(C5×D4), (C6×D4).13C10, (D4×C30).27C2, (D4×C10).13C6, C30.458(C2×D4), (C2×C30).196D4, C4.2(C22×C30), C20.45(C22×C6), (Q8×C15)⋊36C22, C22.15(D4×C15), C12.45(C22×C10), (C2×C60).583C22, (D4×C15).50C22, (C5×Q8)⋊11(C2×C6), (C3×Q8)⋊9(C2×C10), (C2×C6).53(C5×D4), (C2×C4).27(C2×C30), (C5×D4).11(C2×C6), (C2×C10).53(C3×D4), (C2×C20).129(C2×C6), (C3×D4).11(C2×C10), (C2×C12).130(C2×C10), SmallGroup(480,938)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C30
G = < a,b,c | a30=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Subgroups: 216 in 136 conjugacy classes, 88 normal (40 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C30, C30, C30, C2×SD16, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C2×C24, C3×SD16, C6×D4, C6×Q8, C60, C60, C2×C30, C2×C30, C2×C40, C5×SD16, D4×C10, Q8×C10, C6×SD16, C120, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, Q8×C15, C22×C30, C10×SD16, C2×C120, C15×SD16, D4×C30, Q8×C30, SD16×C30
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, SD16, C2×D4, C2×C10, C3×D4, C22×C6, C30, C2×SD16, C5×D4, C22×C10, C3×SD16, C6×D4, C2×C30, C5×SD16, D4×C10, C6×SD16, D4×C15, C22×C30, C10×SD16, C15×SD16, D4×C30, SD16×C30
►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)
(1 114 40 180 131 89 240 193)(2 115 41 151 132 90 211 194)(3 116 42 152 133 61 212 195)(4 117 43 153 134 62 213 196)(5 118 44 154 135 63 214 197)(6 119 45 155 136 64 215 198)(7 120 46 156 137 65 216 199)(8 91 47 157 138 66 217 200)(9 92 48 158 139 67 218 201)(10 93 49 159 140 68 219 202)(11 94 50 160 141 69 220 203)(12 95 51 161 142 70 221 204)(13 96 52 162 143 71 222 205)(14 97 53 163 144 72 223 206)(15 98 54 164 145 73 224 207)(16 99 55 165 146 74 225 208)(17 100 56 166 147 75 226 209)(18 101 57 167 148 76 227 210)(19 102 58 168 149 77 228 181)(20 103 59 169 150 78 229 182)(21 104 60 170 121 79 230 183)(22 105 31 171 122 80 231 184)(23 106 32 172 123 81 232 185)(24 107 33 173 124 82 233 186)(25 108 34 174 125 83 234 187)(26 109 35 175 126 84 235 188)(27 110 36 176 127 85 236 189)(28 111 37 177 128 86 237 190)(29 112 38 178 129 87 238 191)(30 113 39 179 130 88 239 192)
(31 231)(32 232)(33 233)(34 234)(35 235)(36 236)(37 237)(38 238)(39 239)(40 240)(41 211)(42 212)(43 213)(44 214)(45 215)(46 216)(47 217)(48 218)(49 219)(50 220)(51 221)(52 222)(53 223)(54 224)(55 225)(56 226)(57 227)(58 228)(59 229)(60 230)(61 195)(62 196)(63 197)(64 198)(65 199)(66 200)(67 201)(68 202)(69 203)(70 204)(71 205)(72 206)(73 207)(74 208)(75 209)(76 210)(77 181)(78 182)(79 183)(80 184)(81 185)(82 186)(83 187)(84 188)(85 189)(86 190)(87 191)(88 192)(89 193)(90 194)(91 157)(92 158)(93 159)(94 160)(95 161)(96 162)(97 163)(98 164)(99 165)(100 166)(101 167)(102 168)(103 169)(104 170)(105 171)(106 172)(107 173)(108 174)(109 175)(110 176)(111 177)(112 178)(113 179)(114 180)(115 151)(116 152)(117 153)(118 154)(119 155)(120 156)
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), (1,114,40,180,131,89,240,193)(2,115,41,151,132,90,211,194)(3,116,42,152,133,61,212,195)(4,117,43,153,134,62,213,196)(5,118,44,154,135,63,214,197)(6,119,45,155,136,64,215,198)(7,120,46,156,137,65,216,199)(8,91,47,157,138,66,217,200)(9,92,48,158,139,67,218,201)(10,93,49,159,140,68,219,202)(11,94,50,160,141,69,220,203)(12,95,51,161,142,70,221,204)(13,96,52,162,143,71,222,205)(14,97,53,163,144,72,223,206)(15,98,54,164,145,73,224,207)(16,99,55,165,146,74,225,208)(17,100,56,166,147,75,226,209)(18,101,57,167,148,76,227,210)(19,102,58,168,149,77,228,181)(20,103,59,169,150,78,229,182)(21,104,60,170,121,79,230,183)(22,105,31,171,122,80,231,184)(23,106,32,172,123,81,232,185)(24,107,33,173,124,82,233,186)(25,108,34,174,125,83,234,187)(26,109,35,175,126,84,235,188)(27,110,36,176,127,85,236,189)(28,111,37,177,128,86,237,190)(29,112,38,178,129,87,238,191)(30,113,39,179,130,88,239,192), (31,231)(32,232)(33,233)(34,234)(35,235)(36,236)(37,237)(38,238)(39,239)(40,240)(41,211)(42,212)(43,213)(44,214)(45,215)(46,216)(47,217)(48,218)(49,219)(50,220)(51,221)(52,222)(53,223)(54,224)(55,225)(56,226)(57,227)(58,228)(59,229)(60,230)(61,195)(62,196)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,205)(72,206)(73,207)(74,208)(75,209)(76,210)(77,181)(78,182)(79,183)(80,184)(81,185)(82,186)(83,187)(84,188)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,175)(110,176)(111,177)(112,178)(113,179)(114,180)(115,151)(116,152)(117,153)(118,154)(119,155)(120,156)>;
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), (1,114,40,180,131,89,240,193)(2,115,41,151,132,90,211,194)(3,116,42,152,133,61,212,195)(4,117,43,153,134,62,213,196)(5,118,44,154,135,63,214,197)(6,119,45,155,136,64,215,198)(7,120,46,156,137,65,216,199)(8,91,47,157,138,66,217,200)(9,92,48,158,139,67,218,201)(10,93,49,159,140,68,219,202)(11,94,50,160,141,69,220,203)(12,95,51,161,142,70,221,204)(13,96,52,162,143,71,222,205)(14,97,53,163,144,72,223,206)(15,98,54,164,145,73,224,207)(16,99,55,165,146,74,225,208)(17,100,56,166,147,75,226,209)(18,101,57,167,148,76,227,210)(19,102,58,168,149,77,228,181)(20,103,59,169,150,78,229,182)(21,104,60,170,121,79,230,183)(22,105,31,171,122,80,231,184)(23,106,32,172,123,81,232,185)(24,107,33,173,124,82,233,186)(25,108,34,174,125,83,234,187)(26,109,35,175,126,84,235,188)(27,110,36,176,127,85,236,189)(28,111,37,177,128,86,237,190)(29,112,38,178,129,87,238,191)(30,113,39,179,130,88,239,192), (31,231)(32,232)(33,233)(34,234)(35,235)(36,236)(37,237)(38,238)(39,239)(40,240)(41,211)(42,212)(43,213)(44,214)(45,215)(46,216)(47,217)(48,218)(49,219)(50,220)(51,221)(52,222)(53,223)(54,224)(55,225)(56,226)(57,227)(58,228)(59,229)(60,230)(61,195)(62,196)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,205)(72,206)(73,207)(74,208)(75,209)(76,210)(77,181)(78,182)(79,183)(80,184)(81,185)(82,186)(83,187)(84,188)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,175)(110,176)(111,177)(112,178)(113,179)(114,180)(115,151)(116,152)(117,153)(118,154)(119,155)(120,156) );
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)], [(1,114,40,180,131,89,240,193),(2,115,41,151,132,90,211,194),(3,116,42,152,133,61,212,195),(4,117,43,153,134,62,213,196),(5,118,44,154,135,63,214,197),(6,119,45,155,136,64,215,198),(7,120,46,156,137,65,216,199),(8,91,47,157,138,66,217,200),(9,92,48,158,139,67,218,201),(10,93,49,159,140,68,219,202),(11,94,50,160,141,69,220,203),(12,95,51,161,142,70,221,204),(13,96,52,162,143,71,222,205),(14,97,53,163,144,72,223,206),(15,98,54,164,145,73,224,207),(16,99,55,165,146,74,225,208),(17,100,56,166,147,75,226,209),(18,101,57,167,148,76,227,210),(19,102,58,168,149,77,228,181),(20,103,59,169,150,78,229,182),(21,104,60,170,121,79,230,183),(22,105,31,171,122,80,231,184),(23,106,32,172,123,81,232,185),(24,107,33,173,124,82,233,186),(25,108,34,174,125,83,234,187),(26,109,35,175,126,84,235,188),(27,110,36,176,127,85,236,189),(28,111,37,177,128,86,237,190),(29,112,38,178,129,87,238,191),(30,113,39,179,130,88,239,192)], [(31,231),(32,232),(33,233),(34,234),(35,235),(36,236),(37,237),(38,238),(39,239),(40,240),(41,211),(42,212),(43,213),(44,214),(45,215),(46,216),(47,217),(48,218),(49,219),(50,220),(51,221),(52,222),(53,223),(54,224),(55,225),(56,226),(57,227),(58,228),(59,229),(60,230),(61,195),(62,196),(63,197),(64,198),(65,199),(66,200),(67,201),(68,202),(69,203),(70,204),(71,205),(72,206),(73,207),(74,208),(75,209),(76,210),(77,181),(78,182),(79,183),(80,184),(81,185),(82,186),(83,187),(84,188),(85,189),(86,190),(87,191),(88,192),(89,193),(90,194),(91,157),(92,158),(93,159),(94,160),(95,161),(96,162),(97,163),(98,164),(99,165),(100,166),(101,167),(102,168),(103,169),(104,170),(105,171),(106,172),(107,173),(108,174),(109,175),(110,176),(111,177),(112,178),(113,179),(114,180),(115,151),(116,152),(117,153),(118,154),(119,155),(120,156)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20P | 24A | ··· | 24H | 30A | ··· | 30X | 30Y | ··· | 30AN | 40A | ··· | 40P | 60A | ··· | 60P | 60Q | ··· | 60AF | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | C30 | D4 | D4 | SD16 | C3×D4 | C3×D4 | C5×D4 | C5×D4 | C3×SD16 | C5×SD16 | D4×C15 | D4×C15 | C15×SD16 |
| kernel | SD16×C30 | C2×C120 | C15×SD16 | D4×C30 | Q8×C30 | C10×SD16 | C6×SD16 | C2×C40 | C5×SD16 | D4×C10 | Q8×C10 | C2×C24 | C3×SD16 | C6×D4 | C6×Q8 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C60 | C2×C30 | C30 | C20 | C2×C10 | C12 | C2×C6 | C10 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 2 | 4 | 16 | 4 | 4 | 8 | 8 | 32 | 8 | 8 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 8 | 16 | 8 | 8 | 32 |
Matrix representation of SD16×C30 ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 141 | 0 |
| 0 | 0 | 141 |
| 240 | 0 | 0 |
| 0 | 203 | 222 |
| 0 | 38 | 0 |
| 240 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 240 |
G:=sub<GL(3,GF(241))| [240,0,0,0,141,0,0,0,141],[240,0,0,0,203,38,0,222,0],[240,0,0,0,1,0,0,1,240] >;
SD16×C30 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{30} % in TeX
G:=Group("SD16xC30"); // GroupNames label
G:=SmallGroup(480,938);
// by ID
G=gap.SmallGroup(480,938);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1680,1709,15125,7572,124]);
// Polycyclic
G:=Group<a,b,c|a^30=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations