metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).16D14, C7⋊C8.33D4, C8.C22.D7, (C7×D4).16D4, C4.181(D4×D7), (C7×Q8).16D4, Q8.Dic7.C2, C4○D4.27D14, C28.200(C2×D4), C7⋊5(D4.5D4), D4.7(C7⋊D4), (C2×Q8).70D14, Q8.7(C7⋊D4), (C2×C28).19C23, C4.12D28⋊11C2, C28.53D4⋊12C2, C28.10D4⋊10C2, D4.9D14.2C2, (Q8×C14).97C22, C14.127(C4⋊D4), C4.Dic7.28C22, C2.33(Dic7⋊D4), C22.16(D4⋊2D7), (C7×M4(2)).26C22, (C2×Dic14).136C22, C4.56(C2×C7⋊D4), (C2×C7⋊Q16)⋊22C2, (C2×C7⋊C8).173C22, (C7×C8.C22).1C2, (C2×C4).19(C22×D7), (C2×C14).39(C4○D4), (C7×C4○D4).17C22, SmallGroup(448,738)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).16D14
G = < a,b,c,d | a8=b2=c14=1, d2=a6, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >
Subgroups: 396 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C14, C14, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C8.C22, C7⋊C8, C7⋊C8, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, D4.5D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4.D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×Dic14, Q8×C14, C7×C4○D4, C28.53D4, C4.12D28, C28.10D4, C2×C7⋊Q16, Q8.Dic7, D4.9D14, C7×C8.C22, M4(2).16D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4.5D4, D4×D7, D4⋊2D7, C2×C7⋊D4, Dic7⋊D4, M4(2).16D14
►On 224 points
Generators in S224
(1 174 159 60 20 123 194 76)(2 61 195 175 21 77 160 124)(3 176 161 62 22 125 196 78)(4 63 183 177 23 79 162 126)(5 178 163 64 24 113 184 80)(6 65 185 179 25 81 164 114)(7 180 165 66 26 115 186 82)(8 67 187 181 27 83 166 116)(9 182 167 68 28 117 188 84)(10 69 189 169 15 71 168 118)(11 170 155 70 16 119 190 72)(12 57 191 171 17 73 156 120)(13 172 157 58 18 121 192 74)(14 59 193 173 19 75 158 122)(29 93 52 129 111 151 211 204)(30 130 212 94 112 205 53 152)(31 95 54 131 99 153 213 206)(32 132 214 96 100 207 55 154)(33 97 56 133 101 141 215 208)(34 134 216 98 102 209 43 142)(35 85 44 135 103 143 217 210)(36 136 218 86 104 197 45 144)(37 87 46 137 105 145 219 198)(38 138 220 88 106 199 47 146)(39 89 48 139 107 147 221 200)(40 140 222 90 108 201 49 148)(41 91 50 127 109 149 223 202)(42 128 224 92 110 203 51 150)
(2 21)(4 23)(6 25)(8 27)(10 15)(12 17)(14 19)(29 111)(31 99)(33 101)(35 103)(37 105)(39 107)(41 109)(44 217)(46 219)(48 221)(50 223)(52 211)(54 213)(56 215)(58 74)(60 76)(62 78)(64 80)(66 82)(68 84)(70 72)(86 144)(88 146)(90 148)(92 150)(94 152)(96 154)(98 142)(113 178)(115 180)(117 182)(119 170)(121 172)(123 174)(125 176)(128 203)(130 205)(132 207)(134 209)(136 197)(138 199)(140 201)(156 191)(158 193)(160 195)(162 183)(164 185)(166 187)(168 189)
(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)
(1 132 194 154 20 207 159 96)(2 153 160 131 21 95 195 206)(3 130 196 152 22 205 161 94)(4 151 162 129 23 93 183 204)(5 128 184 150 24 203 163 92)(6 149 164 127 25 91 185 202)(7 140 186 148 26 201 165 90)(8 147 166 139 27 89 187 200)(9 138 188 146 28 199 167 88)(10 145 168 137 15 87 189 198)(11 136 190 144 16 197 155 86)(12 143 156 135 17 85 191 210)(13 134 192 142 18 209 157 98)(14 141 158 133 19 97 193 208)(29 177 211 63 111 126 52 79)(30 62 53 176 112 78 212 125)(31 175 213 61 99 124 54 77)(32 60 55 174 100 76 214 123)(33 173 215 59 101 122 56 75)(34 58 43 172 102 74 216 121)(35 171 217 57 103 120 44 73)(36 70 45 170 104 72 218 119)(37 169 219 69 105 118 46 71)(38 68 47 182 106 84 220 117)(39 181 221 67 107 116 48 83)(40 66 49 180 108 82 222 115)(41 179 223 65 109 114 50 81)(42 64 51 178 110 80 224 113)
G:=sub<Sym(224)| (1,174,159,60,20,123,194,76)(2,61,195,175,21,77,160,124)(3,176,161,62,22,125,196,78)(4,63,183,177,23,79,162,126)(5,178,163,64,24,113,184,80)(6,65,185,179,25,81,164,114)(7,180,165,66,26,115,186,82)(8,67,187,181,27,83,166,116)(9,182,167,68,28,117,188,84)(10,69,189,169,15,71,168,118)(11,170,155,70,16,119,190,72)(12,57,191,171,17,73,156,120)(13,172,157,58,18,121,192,74)(14,59,193,173,19,75,158,122)(29,93,52,129,111,151,211,204)(30,130,212,94,112,205,53,152)(31,95,54,131,99,153,213,206)(32,132,214,96,100,207,55,154)(33,97,56,133,101,141,215,208)(34,134,216,98,102,209,43,142)(35,85,44,135,103,143,217,210)(36,136,218,86,104,197,45,144)(37,87,46,137,105,145,219,198)(38,138,220,88,106,199,47,146)(39,89,48,139,107,147,221,200)(40,140,222,90,108,201,49,148)(41,91,50,127,109,149,223,202)(42,128,224,92,110,203,51,150), (2,21)(4,23)(6,25)(8,27)(10,15)(12,17)(14,19)(29,111)(31,99)(33,101)(35,103)(37,105)(39,107)(41,109)(44,217)(46,219)(48,221)(50,223)(52,211)(54,213)(56,215)(58,74)(60,76)(62,78)(64,80)(66,82)(68,84)(70,72)(86,144)(88,146)(90,148)(92,150)(94,152)(96,154)(98,142)(113,178)(115,180)(117,182)(119,170)(121,172)(123,174)(125,176)(128,203)(130,205)(132,207)(134,209)(136,197)(138,199)(140,201)(156,191)(158,193)(160,195)(162,183)(164,185)(166,187)(168,189), (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), (1,132,194,154,20,207,159,96)(2,153,160,131,21,95,195,206)(3,130,196,152,22,205,161,94)(4,151,162,129,23,93,183,204)(5,128,184,150,24,203,163,92)(6,149,164,127,25,91,185,202)(7,140,186,148,26,201,165,90)(8,147,166,139,27,89,187,200)(9,138,188,146,28,199,167,88)(10,145,168,137,15,87,189,198)(11,136,190,144,16,197,155,86)(12,143,156,135,17,85,191,210)(13,134,192,142,18,209,157,98)(14,141,158,133,19,97,193,208)(29,177,211,63,111,126,52,79)(30,62,53,176,112,78,212,125)(31,175,213,61,99,124,54,77)(32,60,55,174,100,76,214,123)(33,173,215,59,101,122,56,75)(34,58,43,172,102,74,216,121)(35,171,217,57,103,120,44,73)(36,70,45,170,104,72,218,119)(37,169,219,69,105,118,46,71)(38,68,47,182,106,84,220,117)(39,181,221,67,107,116,48,83)(40,66,49,180,108,82,222,115)(41,179,223,65,109,114,50,81)(42,64,51,178,110,80,224,113)>;
G:=Group( (1,174,159,60,20,123,194,76)(2,61,195,175,21,77,160,124)(3,176,161,62,22,125,196,78)(4,63,183,177,23,79,162,126)(5,178,163,64,24,113,184,80)(6,65,185,179,25,81,164,114)(7,180,165,66,26,115,186,82)(8,67,187,181,27,83,166,116)(9,182,167,68,28,117,188,84)(10,69,189,169,15,71,168,118)(11,170,155,70,16,119,190,72)(12,57,191,171,17,73,156,120)(13,172,157,58,18,121,192,74)(14,59,193,173,19,75,158,122)(29,93,52,129,111,151,211,204)(30,130,212,94,112,205,53,152)(31,95,54,131,99,153,213,206)(32,132,214,96,100,207,55,154)(33,97,56,133,101,141,215,208)(34,134,216,98,102,209,43,142)(35,85,44,135,103,143,217,210)(36,136,218,86,104,197,45,144)(37,87,46,137,105,145,219,198)(38,138,220,88,106,199,47,146)(39,89,48,139,107,147,221,200)(40,140,222,90,108,201,49,148)(41,91,50,127,109,149,223,202)(42,128,224,92,110,203,51,150), (2,21)(4,23)(6,25)(8,27)(10,15)(12,17)(14,19)(29,111)(31,99)(33,101)(35,103)(37,105)(39,107)(41,109)(44,217)(46,219)(48,221)(50,223)(52,211)(54,213)(56,215)(58,74)(60,76)(62,78)(64,80)(66,82)(68,84)(70,72)(86,144)(88,146)(90,148)(92,150)(94,152)(96,154)(98,142)(113,178)(115,180)(117,182)(119,170)(121,172)(123,174)(125,176)(128,203)(130,205)(132,207)(134,209)(136,197)(138,199)(140,201)(156,191)(158,193)(160,195)(162,183)(164,185)(166,187)(168,189), (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), (1,132,194,154,20,207,159,96)(2,153,160,131,21,95,195,206)(3,130,196,152,22,205,161,94)(4,151,162,129,23,93,183,204)(5,128,184,150,24,203,163,92)(6,149,164,127,25,91,185,202)(7,140,186,148,26,201,165,90)(8,147,166,139,27,89,187,200)(9,138,188,146,28,199,167,88)(10,145,168,137,15,87,189,198)(11,136,190,144,16,197,155,86)(12,143,156,135,17,85,191,210)(13,134,192,142,18,209,157,98)(14,141,158,133,19,97,193,208)(29,177,211,63,111,126,52,79)(30,62,53,176,112,78,212,125)(31,175,213,61,99,124,54,77)(32,60,55,174,100,76,214,123)(33,173,215,59,101,122,56,75)(34,58,43,172,102,74,216,121)(35,171,217,57,103,120,44,73)(36,70,45,170,104,72,218,119)(37,169,219,69,105,118,46,71)(38,68,47,182,106,84,220,117)(39,181,221,67,107,116,48,83)(40,66,49,180,108,82,222,115)(41,179,223,65,109,114,50,81)(42,64,51,178,110,80,224,113) );
G=PermutationGroup([[(1,174,159,60,20,123,194,76),(2,61,195,175,21,77,160,124),(3,176,161,62,22,125,196,78),(4,63,183,177,23,79,162,126),(5,178,163,64,24,113,184,80),(6,65,185,179,25,81,164,114),(7,180,165,66,26,115,186,82),(8,67,187,181,27,83,166,116),(9,182,167,68,28,117,188,84),(10,69,189,169,15,71,168,118),(11,170,155,70,16,119,190,72),(12,57,191,171,17,73,156,120),(13,172,157,58,18,121,192,74),(14,59,193,173,19,75,158,122),(29,93,52,129,111,151,211,204),(30,130,212,94,112,205,53,152),(31,95,54,131,99,153,213,206),(32,132,214,96,100,207,55,154),(33,97,56,133,101,141,215,208),(34,134,216,98,102,209,43,142),(35,85,44,135,103,143,217,210),(36,136,218,86,104,197,45,144),(37,87,46,137,105,145,219,198),(38,138,220,88,106,199,47,146),(39,89,48,139,107,147,221,200),(40,140,222,90,108,201,49,148),(41,91,50,127,109,149,223,202),(42,128,224,92,110,203,51,150)], [(2,21),(4,23),(6,25),(8,27),(10,15),(12,17),(14,19),(29,111),(31,99),(33,101),(35,103),(37,105),(39,107),(41,109),(44,217),(46,219),(48,221),(50,223),(52,211),(54,213),(56,215),(58,74),(60,76),(62,78),(64,80),(66,82),(68,84),(70,72),(86,144),(88,146),(90,148),(92,150),(94,152),(96,154),(98,142),(113,178),(115,180),(117,182),(119,170),(121,172),(123,174),(125,176),(128,203),(130,205),(132,207),(134,209),(136,197),(138,199),(140,201),(156,191),(158,193),(160,195),(162,183),(164,185),(166,187),(168,189)], [(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)], [(1,132,194,154,20,207,159,96),(2,153,160,131,21,95,195,206),(3,130,196,152,22,205,161,94),(4,151,162,129,23,93,183,204),(5,128,184,150,24,203,163,92),(6,149,164,127,25,91,185,202),(7,140,186,148,26,201,165,90),(8,147,166,139,27,89,187,200),(9,138,188,146,28,199,167,88),(10,145,168,137,15,87,189,198),(11,136,190,144,16,197,155,86),(12,143,156,135,17,85,191,210),(13,134,192,142,18,209,157,98),(14,141,158,133,19,97,193,208),(29,177,211,63,111,126,52,79),(30,62,53,176,112,78,212,125),(31,175,213,61,99,124,54,77),(32,60,55,174,100,76,214,123),(33,173,215,59,101,122,56,75),(34,58,43,172,102,74,216,121),(35,171,217,57,103,120,44,73),(36,70,45,170,104,72,218,119),(37,169,219,69,105,118,46,71),(38,68,47,182,106,84,220,117),(39,181,221,67,107,116,48,83),(40,66,49,180,108,82,222,115),(41,179,223,65,109,114,50,81),(42,64,51,178,110,80,224,113)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 56 | 2 | 2 | 2 | 8 | 14 | 14 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | D4.5D4 | D4×D7 | D4⋊2D7 | M4(2).16D14 |
| kernel | M4(2).16D14 | C28.53D4 | C4.12D28 | C28.10D4 | C2×C7⋊Q16 | Q8.Dic7 | D4.9D14 | C7×C8.C22 | C7⋊C8 | C7×D4 | C7×Q8 | C8.C22 | C2×C14 | M4(2) | C2×Q8 | C4○D4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 3 |
Matrix representation of M4(2).16D14 ►in GL6(𝔽113)
| 0 | 112 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 110 | 59 | 16 | 23 |
| 0 | 0 | 74 | 73 | 42 | 22 |
| 0 | 0 | 102 | 82 | 67 | 106 |
| 0 | 0 | 68 | 14 | 59 | 89 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 79 | 112 | 0 |
| 0 | 0 | 65 | 48 | 0 | 112 |
| 101 | 16 | 0 | 0 | 0 | 0 |
| 16 | 101 | 0 | 0 | 0 | 0 |
| 0 | 0 | 93 | 21 | 81 | 0 |
| 0 | 0 | 45 | 69 | 81 | 111 |
| 0 | 0 | 42 | 71 | 112 | 34 |
| 0 | 0 | 35 | 78 | 0 | 65 |
| 105 | 96 | 0 | 0 | 0 | 0 |
| 17 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 99 | 18 | 98 | 31 |
| 0 | 0 | 54 | 24 | 103 | 39 |
| 0 | 0 | 49 | 101 | 24 | 38 |
| 0 | 0 | 48 | 105 | 7 | 79 |
G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,112,0,0,0,0,0,0,0,110,74,102,68,0,0,59,73,82,14,0,0,16,42,67,59,0,0,23,22,106,89],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,27,65,0,0,0,1,79,48,0,0,0,0,112,0,0,0,0,0,0,112],[101,16,0,0,0,0,16,101,0,0,0,0,0,0,93,45,42,35,0,0,21,69,71,78,0,0,81,81,112,0,0,0,0,111,34,65],[105,17,0,0,0,0,96,8,0,0,0,0,0,0,99,54,49,48,0,0,18,24,101,105,0,0,98,103,24,7,0,0,31,39,38,79] >;
M4(2).16D14 in GAP, Magma, Sage, TeX
M_4(2)._{16}D_{14} % in TeX
G:=Group("M4(2).16D14"); // GroupNames label
G:=SmallGroup(448,738);
// by ID
G=gap.SmallGroup(448,738);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,219,184,1123,297,136,1684,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;
// generators/relations