metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊5Dic3, C6.88(C4×D4), C6.24(C4×Q8), C2.6(D4×Dic3), C2.3(Q8×Dic3), C22.24(S3×Q8), C2.6(D6⋊Q8), (C2×Dic3).18Q8, C22.108(S3×D4), (C22×C4).324D6, C6.45(C22⋊Q8), (C2×Dic3).176D4, C2.5(D6.D4), C2.7(Dic3.Q8), C6.19(C42.C2), C6.25(C42⋊2C2), C6.45(C42⋊C2), C22.57(C4○D12), C6.C42.27C2, (C22×C6).344C23, C23.303(C22×S3), C22.57(D4⋊2S3), (C22×C12).391C22, C3⋊7(C23.63C23), C22.25(Q8⋊3S3), C2.9(C23.26D6), C6.49(C22.D4), C22.41(C22×Dic3), (C22×Dic3).190C22, (C3×C4⋊C4)⋊8C4, (C6×C4⋊C4).14C2, (C2×C4⋊C4).17S3, (C2×C6).79(C2×Q8), (C2×C6).331(C2×D4), C2.6(C4⋊C4⋊S3), (C2×C4×Dic3).36C2, (C2×C12).185(C2×C4), (C2×C6).83(C4○D4), (C2×C4⋊Dic3).17C2, (C2×C4).17(C2×Dic3), (C2×C6).180(C22×C4), SmallGroup(192,539)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊5Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, bab-1=a-1, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 344 in 154 conjugacy classes, 75 normal (51 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Dic3, C4⋊Dic3, C3×C4⋊C4, C22×Dic3, C22×C12, C23.63C23, C6.C42, C2×C4×Dic3, C2×C4⋊Dic3, C6×C4⋊C4, C4⋊C4⋊5Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C22×C4, C2×D4, C2×Q8, C4○D4, C2×Dic3, C22×S3, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C22×Dic3, C23.63C23, Dic3.Q8, D6.D4, D6⋊Q8, C4⋊C4⋊S3, C23.26D6, D4×Dic3, Q8×Dic3, C4⋊C4⋊5Dic3
►Regular action on 192 points
Generators in S192
(1 49 23 41)(2 50 24 42)(3 51 19 37)(4 52 20 38)(5 53 21 39)(6 54 22 40)(7 45 29 32)(8 46 30 33)(9 47 25 34)(10 48 26 35)(11 43 27 36)(12 44 28 31)(13 160 181 173)(14 161 182 174)(15 162 183 169)(16 157 184 170)(17 158 185 171)(18 159 186 172)(55 86 68 99)(56 87 69 100)(57 88 70 101)(58 89 71 102)(59 90 72 97)(60 85 67 98)(61 84 75 91)(62 79 76 92)(63 80 77 93)(64 81 78 94)(65 82 73 95)(66 83 74 96)(103 144 117 130)(104 139 118 131)(105 140 119 132)(106 141 120 127)(107 142 115 128)(108 143 116 129)(109 149 122 136)(110 150 123 137)(111 145 124 138)(112 146 125 133)(113 147 126 134)(114 148 121 135)(151 178 165 192)(152 179 166 187)(153 180 167 188)(154 175 168 189)(155 176 163 190)(156 177 164 191)
(1 65 12 55)(2 66 7 56)(3 61 8 57)(4 62 9 58)(5 63 10 59)(6 64 11 60)(13 149 190 139)(14 150 191 140)(15 145 192 141)(16 146 187 142)(17 147 188 143)(18 148 189 144)(19 75 30 70)(20 76 25 71)(21 77 26 72)(22 78 27 67)(23 73 28 68)(24 74 29 69)(31 86 41 82)(32 87 42 83)(33 88 37 84)(34 89 38 79)(35 90 39 80)(36 85 40 81)(43 98 54 94)(44 99 49 95)(45 100 50 96)(46 101 51 91)(47 102 52 92)(48 97 53 93)(103 159 114 154)(104 160 109 155)(105 161 110 156)(106 162 111 151)(107 157 112 152)(108 158 113 153)(115 170 125 166)(116 171 126 167)(117 172 121 168)(118 173 122 163)(119 174 123 164)(120 169 124 165)(127 183 138 178)(128 184 133 179)(129 185 134 180)(130 186 135 175)(131 181 136 176)(132 182 137 177)
(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)
(1 104 4 107)(2 103 5 106)(3 108 6 105)(7 114 10 111)(8 113 11 110)(9 112 12 109)(13 102 16 99)(14 101 17 98)(15 100 18 97)(19 116 22 119)(20 115 23 118)(21 120 24 117)(25 125 28 122)(26 124 29 121)(27 123 30 126)(31 131 34 128)(32 130 35 127)(33 129 36 132)(37 134 40 137)(38 133 41 136)(39 138 42 135)(43 140 46 143)(44 139 47 142)(45 144 48 141)(49 149 52 146)(50 148 53 145)(51 147 54 150)(55 155 58 152)(56 154 59 151)(57 153 60 156)(61 158 64 161)(62 157 65 160)(63 162 66 159)(67 164 70 167)(68 163 71 166)(69 168 72 165)(73 173 76 170)(74 172 77 169)(75 171 78 174)(79 179 82 176)(80 178 83 175)(81 177 84 180)(85 182 88 185)(86 181 89 184)(87 186 90 183)(91 188 94 191)(92 187 95 190)(93 192 96 189)
G:=sub<Sym(192)| (1,49,23,41)(2,50,24,42)(3,51,19,37)(4,52,20,38)(5,53,21,39)(6,54,22,40)(7,45,29,32)(8,46,30,33)(9,47,25,34)(10,48,26,35)(11,43,27,36)(12,44,28,31)(13,160,181,173)(14,161,182,174)(15,162,183,169)(16,157,184,170)(17,158,185,171)(18,159,186,172)(55,86,68,99)(56,87,69,100)(57,88,70,101)(58,89,71,102)(59,90,72,97)(60,85,67,98)(61,84,75,91)(62,79,76,92)(63,80,77,93)(64,81,78,94)(65,82,73,95)(66,83,74,96)(103,144,117,130)(104,139,118,131)(105,140,119,132)(106,141,120,127)(107,142,115,128)(108,143,116,129)(109,149,122,136)(110,150,123,137)(111,145,124,138)(112,146,125,133)(113,147,126,134)(114,148,121,135)(151,178,165,192)(152,179,166,187)(153,180,167,188)(154,175,168,189)(155,176,163,190)(156,177,164,191), (1,65,12,55)(2,66,7,56)(3,61,8,57)(4,62,9,58)(5,63,10,59)(6,64,11,60)(13,149,190,139)(14,150,191,140)(15,145,192,141)(16,146,187,142)(17,147,188,143)(18,148,189,144)(19,75,30,70)(20,76,25,71)(21,77,26,72)(22,78,27,67)(23,73,28,68)(24,74,29,69)(31,86,41,82)(32,87,42,83)(33,88,37,84)(34,89,38,79)(35,90,39,80)(36,85,40,81)(43,98,54,94)(44,99,49,95)(45,100,50,96)(46,101,51,91)(47,102,52,92)(48,97,53,93)(103,159,114,154)(104,160,109,155)(105,161,110,156)(106,162,111,151)(107,157,112,152)(108,158,113,153)(115,170,125,166)(116,171,126,167)(117,172,121,168)(118,173,122,163)(119,174,123,164)(120,169,124,165)(127,183,138,178)(128,184,133,179)(129,185,134,180)(130,186,135,175)(131,181,136,176)(132,182,137,177), (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), (1,104,4,107)(2,103,5,106)(3,108,6,105)(7,114,10,111)(8,113,11,110)(9,112,12,109)(13,102,16,99)(14,101,17,98)(15,100,18,97)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,125,28,122)(26,124,29,121)(27,123,30,126)(31,131,34,128)(32,130,35,127)(33,129,36,132)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,155,58,152)(56,154,59,151)(57,153,60,156)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,179,82,176)(80,178,83,175)(81,177,84,180)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189)>;
G:=Group( (1,49,23,41)(2,50,24,42)(3,51,19,37)(4,52,20,38)(5,53,21,39)(6,54,22,40)(7,45,29,32)(8,46,30,33)(9,47,25,34)(10,48,26,35)(11,43,27,36)(12,44,28,31)(13,160,181,173)(14,161,182,174)(15,162,183,169)(16,157,184,170)(17,158,185,171)(18,159,186,172)(55,86,68,99)(56,87,69,100)(57,88,70,101)(58,89,71,102)(59,90,72,97)(60,85,67,98)(61,84,75,91)(62,79,76,92)(63,80,77,93)(64,81,78,94)(65,82,73,95)(66,83,74,96)(103,144,117,130)(104,139,118,131)(105,140,119,132)(106,141,120,127)(107,142,115,128)(108,143,116,129)(109,149,122,136)(110,150,123,137)(111,145,124,138)(112,146,125,133)(113,147,126,134)(114,148,121,135)(151,178,165,192)(152,179,166,187)(153,180,167,188)(154,175,168,189)(155,176,163,190)(156,177,164,191), (1,65,12,55)(2,66,7,56)(3,61,8,57)(4,62,9,58)(5,63,10,59)(6,64,11,60)(13,149,190,139)(14,150,191,140)(15,145,192,141)(16,146,187,142)(17,147,188,143)(18,148,189,144)(19,75,30,70)(20,76,25,71)(21,77,26,72)(22,78,27,67)(23,73,28,68)(24,74,29,69)(31,86,41,82)(32,87,42,83)(33,88,37,84)(34,89,38,79)(35,90,39,80)(36,85,40,81)(43,98,54,94)(44,99,49,95)(45,100,50,96)(46,101,51,91)(47,102,52,92)(48,97,53,93)(103,159,114,154)(104,160,109,155)(105,161,110,156)(106,162,111,151)(107,157,112,152)(108,158,113,153)(115,170,125,166)(116,171,126,167)(117,172,121,168)(118,173,122,163)(119,174,123,164)(120,169,124,165)(127,183,138,178)(128,184,133,179)(129,185,134,180)(130,186,135,175)(131,181,136,176)(132,182,137,177), (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), (1,104,4,107)(2,103,5,106)(3,108,6,105)(7,114,10,111)(8,113,11,110)(9,112,12,109)(13,102,16,99)(14,101,17,98)(15,100,18,97)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,125,28,122)(26,124,29,121)(27,123,30,126)(31,131,34,128)(32,130,35,127)(33,129,36,132)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,155,58,152)(56,154,59,151)(57,153,60,156)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,179,82,176)(80,178,83,175)(81,177,84,180)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189) );
G=PermutationGroup([[(1,49,23,41),(2,50,24,42),(3,51,19,37),(4,52,20,38),(5,53,21,39),(6,54,22,40),(7,45,29,32),(8,46,30,33),(9,47,25,34),(10,48,26,35),(11,43,27,36),(12,44,28,31),(13,160,181,173),(14,161,182,174),(15,162,183,169),(16,157,184,170),(17,158,185,171),(18,159,186,172),(55,86,68,99),(56,87,69,100),(57,88,70,101),(58,89,71,102),(59,90,72,97),(60,85,67,98),(61,84,75,91),(62,79,76,92),(63,80,77,93),(64,81,78,94),(65,82,73,95),(66,83,74,96),(103,144,117,130),(104,139,118,131),(105,140,119,132),(106,141,120,127),(107,142,115,128),(108,143,116,129),(109,149,122,136),(110,150,123,137),(111,145,124,138),(112,146,125,133),(113,147,126,134),(114,148,121,135),(151,178,165,192),(152,179,166,187),(153,180,167,188),(154,175,168,189),(155,176,163,190),(156,177,164,191)], [(1,65,12,55),(2,66,7,56),(3,61,8,57),(4,62,9,58),(5,63,10,59),(6,64,11,60),(13,149,190,139),(14,150,191,140),(15,145,192,141),(16,146,187,142),(17,147,188,143),(18,148,189,144),(19,75,30,70),(20,76,25,71),(21,77,26,72),(22,78,27,67),(23,73,28,68),(24,74,29,69),(31,86,41,82),(32,87,42,83),(33,88,37,84),(34,89,38,79),(35,90,39,80),(36,85,40,81),(43,98,54,94),(44,99,49,95),(45,100,50,96),(46,101,51,91),(47,102,52,92),(48,97,53,93),(103,159,114,154),(104,160,109,155),(105,161,110,156),(106,162,111,151),(107,157,112,152),(108,158,113,153),(115,170,125,166),(116,171,126,167),(117,172,121,168),(118,173,122,163),(119,174,123,164),(120,169,124,165),(127,183,138,178),(128,184,133,179),(129,185,134,180),(130,186,135,175),(131,181,136,176),(132,182,137,177)], [(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)], [(1,104,4,107),(2,103,5,106),(3,108,6,105),(7,114,10,111),(8,113,11,110),(9,112,12,109),(13,102,16,99),(14,101,17,98),(15,100,18,97),(19,116,22,119),(20,115,23,118),(21,120,24,117),(25,125,28,122),(26,124,29,121),(27,123,30,126),(31,131,34,128),(32,130,35,127),(33,129,36,132),(37,134,40,137),(38,133,41,136),(39,138,42,135),(43,140,46,143),(44,139,47,142),(45,144,48,141),(49,149,52,146),(50,148,53,145),(51,147,54,150),(55,155,58,152),(56,154,59,151),(57,153,60,156),(61,158,64,161),(62,157,65,160),(63,162,66,159),(67,164,70,167),(68,163,71,166),(69,168,72,165),(73,173,76,170),(74,172,77,169),(75,171,78,174),(79,179,82,176),(80,178,83,175),(81,177,84,180),(85,182,88,185),(86,181,89,184),(87,186,90,183),(91,188,94,191),(92,187,95,190),(93,192,96,189)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | C4○D4 | C4○D12 | S3×D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 |
| kernel | C4⋊C4⋊5Dic3 | C6.C42 | C2×C4×Dic3 | C2×C4⋊Dic3 | C6×C4⋊C4 | C3×C4⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×Dic3 | C4⋊C4 | C22×C4 | C2×C6 | C22 | C22 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 4 | 3 | 8 | 8 | 1 | 1 | 1 | 1 |
Matrix representation of C4⋊C4⋊5Dic3 ►in GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 10 | 11 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,2,5],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,10,0,0,0,0,1,11],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,10,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C4⋊C4⋊5Dic3 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_5{\rm Dic}_3 % in TeX
G:=Group("C4:C4:5Dic3"); // GroupNames label
G:=SmallGroup(192,539);
// by ID
G=gap.SmallGroup(192,539);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,422,387,184,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations