direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Q8×D7, C12.41D14, Dic14⋊10C6, C84.41C22, C42.44C23, C7⋊5(C6×Q8), C21⋊9(C2×Q8), C4.6(C6×D7), (Q8×C21)⋊5C2, (C7×Q8)⋊12C6, (C4×D7).4C6, C28.27(C2×C6), (C12×D7).4C2, D14.10(C2×C6), Dic7.9(C2×C6), C6.44(C22×D7), (C3×Dic14)⋊10C2, C14.21(C22×C6), (C6×D7).16C22, (C3×Dic7).16C22, C2.8(C2×C6×D7), SmallGroup(336,180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8×D7
G = < a,b,c,d,e | a3=b4=d7=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 232 in 76 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C7, C2×C4, Q8, Q8, C12, C12, C2×C6, D7, C14, C2×Q8, C21, C2×C12, C3×Q8, C3×Q8, Dic7, C28, D14, C3×D7, C42, C6×Q8, Dic14, C4×D7, C7×Q8, C3×Dic7, C84, C6×D7, Q8×D7, C3×Dic14, C12×D7, Q8×C21, C3×Q8×D7
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, D7, C2×Q8, C3×Q8, C22×C6, D14, C3×D7, C6×Q8, C22×D7, C6×D7, Q8×D7, C2×C6×D7, C3×Q8×D7
►On 168 points
Generators in S168
(1 34 20)(2 35 21)(3 29 15)(4 30 16)(5 31 17)(6 32 18)(7 33 19)(8 36 22)(9 37 23)(10 38 24)(11 39 25)(12 40 26)(13 41 27)(14 42 28)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)(85 113 99)(86 114 100)(87 115 101)(88 116 102)(89 117 103)(90 118 104)(91 119 105)(92 120 106)(93 121 107)(94 122 108)(95 123 109)(96 124 110)(97 125 111)(98 126 112)(127 155 141)(128 156 142)(129 157 143)(130 158 144)(131 159 145)(132 160 146)(133 161 147)(134 162 148)(135 163 149)(136 164 150)(137 165 151)(138 166 152)(139 167 153)(140 168 154)
(1 55 13 48)(2 56 14 49)(3 50 8 43)(4 51 9 44)(5 52 10 45)(6 53 11 46)(7 54 12 47)(15 64 22 57)(16 65 23 58)(17 66 24 59)(18 67 25 60)(19 68 26 61)(20 69 27 62)(21 70 28 63)(29 78 36 71)(30 79 37 72)(31 80 38 73)(32 81 39 74)(33 82 40 75)(34 83 41 76)(35 84 42 77)(85 127 92 134)(86 128 93 135)(87 129 94 136)(88 130 95 137)(89 131 96 138)(90 132 97 139)(91 133 98 140)(99 141 106 148)(100 142 107 149)(101 143 108 150)(102 144 109 151)(103 145 110 152)(104 146 111 153)(105 147 112 154)(113 155 120 162)(114 156 121 163)(115 157 122 164)(116 158 123 165)(117 159 124 166)(118 160 125 167)(119 161 126 168)
(1 97 13 90)(2 98 14 91)(3 92 8 85)(4 93 9 86)(5 94 10 87)(6 95 11 88)(7 96 12 89)(15 106 22 99)(16 107 23 100)(17 108 24 101)(18 109 25 102)(19 110 26 103)(20 111 27 104)(21 112 28 105)(29 120 36 113)(30 121 37 114)(31 122 38 115)(32 123 39 116)(33 124 40 117)(34 125 41 118)(35 126 42 119)(43 134 50 127)(44 135 51 128)(45 136 52 129)(46 137 53 130)(47 138 54 131)(48 139 55 132)(49 140 56 133)(57 148 64 141)(58 149 65 142)(59 150 66 143)(60 151 67 144)(61 152 68 145)(62 153 69 146)(63 154 70 147)(71 162 78 155)(72 163 79 156)(73 164 80 157)(74 165 81 158)(75 166 82 159)(76 167 83 160)(77 168 84 161)
(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)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)(113 122)(114 121)(115 120)(116 126)(117 125)(118 124)(119 123)(127 136)(128 135)(129 134)(130 140)(131 139)(132 138)(133 137)(141 150)(142 149)(143 148)(144 154)(145 153)(146 152)(147 151)(155 164)(156 163)(157 162)(158 168)(159 167)(160 166)(161 165)
G:=sub<Sym(168)| (1,34,20)(2,35,21)(3,29,15)(4,30,16)(5,31,17)(6,32,18)(7,33,19)(8,36,22)(9,37,23)(10,38,24)(11,39,25)(12,40,26)(13,41,27)(14,42,28)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70)(85,113,99)(86,114,100)(87,115,101)(88,116,102)(89,117,103)(90,118,104)(91,119,105)(92,120,106)(93,121,107)(94,122,108)(95,123,109)(96,124,110)(97,125,111)(98,126,112)(127,155,141)(128,156,142)(129,157,143)(130,158,144)(131,159,145)(132,160,146)(133,161,147)(134,162,148)(135,163,149)(136,164,150)(137,165,151)(138,166,152)(139,167,153)(140,168,154), (1,55,13,48)(2,56,14,49)(3,50,8,43)(4,51,9,44)(5,52,10,45)(6,53,11,46)(7,54,12,47)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,78,36,71)(30,79,37,72)(31,80,38,73)(32,81,39,74)(33,82,40,75)(34,83,41,76)(35,84,42,77)(85,127,92,134)(86,128,93,135)(87,129,94,136)(88,130,95,137)(89,131,96,138)(90,132,97,139)(91,133,98,140)(99,141,106,148)(100,142,107,149)(101,143,108,150)(102,144,109,151)(103,145,110,152)(104,146,111,153)(105,147,112,154)(113,155,120,162)(114,156,121,163)(115,157,122,164)(116,158,123,165)(117,159,124,166)(118,160,125,167)(119,161,126,168), (1,97,13,90)(2,98,14,91)(3,92,8,85)(4,93,9,86)(5,94,10,87)(6,95,11,88)(7,96,12,89)(15,106,22,99)(16,107,23,100)(17,108,24,101)(18,109,25,102)(19,110,26,103)(20,111,27,104)(21,112,28,105)(29,120,36,113)(30,121,37,114)(31,122,38,115)(32,123,39,116)(33,124,40,117)(34,125,41,118)(35,126,42,119)(43,134,50,127)(44,135,51,128)(45,136,52,129)(46,137,53,130)(47,138,54,131)(48,139,55,132)(49,140,56,133)(57,148,64,141)(58,149,65,142)(59,150,66,143)(60,151,67,144)(61,152,68,145)(62,153,69,146)(63,154,70,147)(71,162,78,155)(72,163,79,156)(73,164,80,157)(74,165,81,158)(75,166,82,159)(76,167,83,160)(77,168,84,161), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109)(113,122)(114,121)(115,120)(116,126)(117,125)(118,124)(119,123)(127,136)(128,135)(129,134)(130,140)(131,139)(132,138)(133,137)(141,150)(142,149)(143,148)(144,154)(145,153)(146,152)(147,151)(155,164)(156,163)(157,162)(158,168)(159,167)(160,166)(161,165)>;
G:=Group( (1,34,20)(2,35,21)(3,29,15)(4,30,16)(5,31,17)(6,32,18)(7,33,19)(8,36,22)(9,37,23)(10,38,24)(11,39,25)(12,40,26)(13,41,27)(14,42,28)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70)(85,113,99)(86,114,100)(87,115,101)(88,116,102)(89,117,103)(90,118,104)(91,119,105)(92,120,106)(93,121,107)(94,122,108)(95,123,109)(96,124,110)(97,125,111)(98,126,112)(127,155,141)(128,156,142)(129,157,143)(130,158,144)(131,159,145)(132,160,146)(133,161,147)(134,162,148)(135,163,149)(136,164,150)(137,165,151)(138,166,152)(139,167,153)(140,168,154), (1,55,13,48)(2,56,14,49)(3,50,8,43)(4,51,9,44)(5,52,10,45)(6,53,11,46)(7,54,12,47)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,78,36,71)(30,79,37,72)(31,80,38,73)(32,81,39,74)(33,82,40,75)(34,83,41,76)(35,84,42,77)(85,127,92,134)(86,128,93,135)(87,129,94,136)(88,130,95,137)(89,131,96,138)(90,132,97,139)(91,133,98,140)(99,141,106,148)(100,142,107,149)(101,143,108,150)(102,144,109,151)(103,145,110,152)(104,146,111,153)(105,147,112,154)(113,155,120,162)(114,156,121,163)(115,157,122,164)(116,158,123,165)(117,159,124,166)(118,160,125,167)(119,161,126,168), (1,97,13,90)(2,98,14,91)(3,92,8,85)(4,93,9,86)(5,94,10,87)(6,95,11,88)(7,96,12,89)(15,106,22,99)(16,107,23,100)(17,108,24,101)(18,109,25,102)(19,110,26,103)(20,111,27,104)(21,112,28,105)(29,120,36,113)(30,121,37,114)(31,122,38,115)(32,123,39,116)(33,124,40,117)(34,125,41,118)(35,126,42,119)(43,134,50,127)(44,135,51,128)(45,136,52,129)(46,137,53,130)(47,138,54,131)(48,139,55,132)(49,140,56,133)(57,148,64,141)(58,149,65,142)(59,150,66,143)(60,151,67,144)(61,152,68,145)(62,153,69,146)(63,154,70,147)(71,162,78,155)(72,163,79,156)(73,164,80,157)(74,165,81,158)(75,166,82,159)(76,167,83,160)(77,168,84,161), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109)(113,122)(114,121)(115,120)(116,126)(117,125)(118,124)(119,123)(127,136)(128,135)(129,134)(130,140)(131,139)(132,138)(133,137)(141,150)(142,149)(143,148)(144,154)(145,153)(146,152)(147,151)(155,164)(156,163)(157,162)(158,168)(159,167)(160,166)(161,165) );
G=PermutationGroup([[(1,34,20),(2,35,21),(3,29,15),(4,30,16),(5,31,17),(6,32,18),(7,33,19),(8,36,22),(9,37,23),(10,38,24),(11,39,25),(12,40,26),(13,41,27),(14,42,28),(43,71,57),(44,72,58),(45,73,59),(46,74,60),(47,75,61),(48,76,62),(49,77,63),(50,78,64),(51,79,65),(52,80,66),(53,81,67),(54,82,68),(55,83,69),(56,84,70),(85,113,99),(86,114,100),(87,115,101),(88,116,102),(89,117,103),(90,118,104),(91,119,105),(92,120,106),(93,121,107),(94,122,108),(95,123,109),(96,124,110),(97,125,111),(98,126,112),(127,155,141),(128,156,142),(129,157,143),(130,158,144),(131,159,145),(132,160,146),(133,161,147),(134,162,148),(135,163,149),(136,164,150),(137,165,151),(138,166,152),(139,167,153),(140,168,154)], [(1,55,13,48),(2,56,14,49),(3,50,8,43),(4,51,9,44),(5,52,10,45),(6,53,11,46),(7,54,12,47),(15,64,22,57),(16,65,23,58),(17,66,24,59),(18,67,25,60),(19,68,26,61),(20,69,27,62),(21,70,28,63),(29,78,36,71),(30,79,37,72),(31,80,38,73),(32,81,39,74),(33,82,40,75),(34,83,41,76),(35,84,42,77),(85,127,92,134),(86,128,93,135),(87,129,94,136),(88,130,95,137),(89,131,96,138),(90,132,97,139),(91,133,98,140),(99,141,106,148),(100,142,107,149),(101,143,108,150),(102,144,109,151),(103,145,110,152),(104,146,111,153),(105,147,112,154),(113,155,120,162),(114,156,121,163),(115,157,122,164),(116,158,123,165),(117,159,124,166),(118,160,125,167),(119,161,126,168)], [(1,97,13,90),(2,98,14,91),(3,92,8,85),(4,93,9,86),(5,94,10,87),(6,95,11,88),(7,96,12,89),(15,106,22,99),(16,107,23,100),(17,108,24,101),(18,109,25,102),(19,110,26,103),(20,111,27,104),(21,112,28,105),(29,120,36,113),(30,121,37,114),(31,122,38,115),(32,123,39,116),(33,124,40,117),(34,125,41,118),(35,126,42,119),(43,134,50,127),(44,135,51,128),(45,136,52,129),(46,137,53,130),(47,138,54,131),(48,139,55,132),(49,140,56,133),(57,148,64,141),(58,149,65,142),(59,150,66,143),(60,151,67,144),(61,152,68,145),(62,153,69,146),(63,154,70,147),(71,162,78,155),(72,163,79,156),(73,164,80,157),(74,165,81,158),(75,166,82,159),(76,167,83,160),(77,168,84,161)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53),(57,66),(58,65),(59,64),(60,70),(61,69),(62,68),(63,67),(71,80),(72,79),(73,78),(74,84),(75,83),(76,82),(77,81),(85,94),(86,93),(87,92),(88,98),(89,97),(90,96),(91,95),(99,108),(100,107),(101,106),(102,112),(103,111),(104,110),(105,109),(113,122),(114,121),(115,120),(116,126),(117,125),(118,124),(119,123),(127,136),(128,135),(129,134),(130,140),(131,139),(132,138),(133,137),(141,150),(142,149),(143,148),(144,154),(145,153),(146,152),(147,151),(155,164),(156,163),(157,162),(158,168),(159,167),(160,166),(161,165)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 12A | ··· | 12F | 12G | ··· | 12L | 14A | 14B | 14C | 21A | ··· | 21F | 28A | ··· | 28I | 42A | ··· | 42F | 84A | ··· | 84R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | ··· | 12 | 12 | ··· | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 2 | 2 | 2 | 14 | 14 | 14 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | Q8 | D7 | C3×Q8 | D14 | C3×D7 | C6×D7 | Q8×D7 | C3×Q8×D7 |
| kernel | C3×Q8×D7 | C3×Dic14 | C12×D7 | Q8×C21 | Q8×D7 | Dic14 | C4×D7 | C7×Q8 | C3×D7 | C3×Q8 | D7 | C12 | Q8 | C4 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 2 | 3 | 4 | 9 | 6 | 18 | 3 | 6 |
Matrix representation of C3×Q8×D7 ►in GL4(𝔽337) generated by
| 128 | 0 | 0 | 0 |
| 0 | 128 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 316 |
| 0 | 0 | 305 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 76 | 30 |
| 0 | 0 | 268 | 261 |
| 194 | 1 | 0 | 0 |
| 251 | 109 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 109 | 336 | 0 | 0 |
| 85 | 228 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
G:=sub<GL(4,GF(337))| [128,0,0,0,0,128,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,336,305,0,0,316,1],[1,0,0,0,0,1,0,0,0,0,76,268,0,0,30,261],[194,251,0,0,1,109,0,0,0,0,1,0,0,0,0,1],[109,85,0,0,336,228,0,0,0,0,336,0,0,0,0,336] >;
C3×Q8×D7 in GAP, Magma, Sage, TeX
C_3\times Q_8\times D_7
% in TeX
G:=Group("C3xQ8xD7"); // GroupNames label
G:=SmallGroup(336,180);
// by ID
G=gap.SmallGroup(336,180);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,260,122,10373]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^7=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations