metabelian, supersoluble, monomial
Aliases: He3⋊4Q16, C32⋊3Dic12, C32⋊5Q16⋊C3, C24.3(C3×S3), (C3×C24).1C6, (C3×C24).1S3, C8.(C32⋊C6), C6.4(C3×D12), C12.67(S3×C6), (C3×C12).40D6, (C3×C6).13D12, (C8×He3).1C2, C32⋊2(C3×Q16), (C2×He3).17D4, He3⋊3Q8.4C2, C3.2(C3×Dic12), C32⋊4Q8.1C6, C2.3(He3⋊4D4), (C4×He3).32C22, (C3×C6).6(C3×D4), (C3×C12).8(C2×C6), C4.8(C2×C32⋊C6), SmallGroup(432,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊4Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, bd=db, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 321 in 71 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C32, C32, Dic3, C12, C12, Q16, C3×C6, C3×C6, C24, C24, Dic6, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, Dic12, C3×Q16, C2×He3, C3×C24, C3×C24, C3×Dic6, C32⋊4Q8, C32⋊C12, C4×He3, C3×Dic12, C32⋊5Q16, C8×He3, He3⋊3Q8, He3⋊4Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, D12, C3×D4, S3×C6, Dic12, C3×Q16, C32⋊C6, C3×D12, C2×C32⋊C6, C3×Dic12, He3⋊4D4, He3⋊4Q16
►On 144 points
Generators in S144
(9 131 122)(10 132 123)(11 133 124)(12 134 125)(13 135 126)(14 136 127)(15 129 128)(16 130 121)(25 51 78)(26 52 79)(27 53 80)(28 54 73)(29 55 74)(30 56 75)(31 49 76)(32 50 77)(33 46 105)(34 47 106)(35 48 107)(36 41 108)(37 42 109)(38 43 110)(39 44 111)(40 45 112)(57 89 116)(58 90 117)(59 91 118)(60 92 119)(61 93 120)(62 94 113)(63 95 114)(64 96 115)
(1 67 17)(2 68 18)(3 69 19)(4 70 20)(5 71 21)(6 72 22)(7 65 23)(8 66 24)(9 131 122)(10 132 123)(11 133 124)(12 134 125)(13 135 126)(14 136 127)(15 129 128)(16 130 121)(25 51 78)(26 52 79)(27 53 80)(28 54 73)(29 55 74)(30 56 75)(31 49 76)(32 50 77)(33 105 46)(34 106 47)(35 107 48)(36 108 41)(37 109 42)(38 110 43)(39 111 44)(40 112 45)(57 116 89)(58 117 90)(59 118 91)(60 119 92)(61 120 93)(62 113 94)(63 114 95)(64 115 96)(81 142 98)(82 143 99)(83 144 100)(84 137 101)(85 138 102)(86 139 103)(87 140 104)(88 141 97)
(1 117 32)(2 118 25)(3 119 26)(4 120 27)(5 113 28)(6 114 29)(7 115 30)(8 116 31)(9 86 105)(10 87 106)(11 88 107)(12 81 108)(13 82 109)(14 83 110)(15 84 111)(16 85 112)(17 58 77)(18 59 78)(19 60 79)(20 61 80)(21 62 73)(22 63 74)(23 64 75)(24 57 76)(33 122 103)(34 123 104)(35 124 97)(36 125 98)(37 126 99)(38 127 100)(39 128 101)(40 121 102)(41 134 142)(42 135 143)(43 136 144)(44 129 137)(45 130 138)(46 131 139)(47 132 140)(48 133 141)(49 66 89)(50 67 90)(51 68 91)(52 69 92)(53 70 93)(54 71 94)(55 72 95)(56 65 96)
(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)
(1 101 5 97)(2 100 6 104)(3 99 7 103)(4 98 8 102)(9 79 13 75)(10 78 14 74)(11 77 15 73)(12 76 16 80)(17 84 21 88)(18 83 22 87)(19 82 23 86)(20 81 24 85)(25 127 29 123)(26 126 30 122)(27 125 31 121)(28 124 32 128)(33 119 37 115)(34 118 38 114)(35 117 39 113)(36 116 40 120)(41 89 45 93)(42 96 46 92)(43 95 47 91)(44 94 48 90)(49 130 53 134)(50 129 54 133)(51 136 55 132)(52 135 56 131)(57 112 61 108)(58 111 62 107)(59 110 63 106)(60 109 64 105)(65 139 69 143)(66 138 70 142)(67 137 71 141)(68 144 72 140)
G:=sub<Sym(144)| (9,131,122)(10,132,123)(11,133,124)(12,134,125)(13,135,126)(14,136,127)(15,129,128)(16,130,121)(25,51,78)(26,52,79)(27,53,80)(28,54,73)(29,55,74)(30,56,75)(31,49,76)(32,50,77)(33,46,105)(34,47,106)(35,48,107)(36,41,108)(37,42,109)(38,43,110)(39,44,111)(40,45,112)(57,89,116)(58,90,117)(59,91,118)(60,92,119)(61,93,120)(62,94,113)(63,95,114)(64,96,115), (1,67,17)(2,68,18)(3,69,19)(4,70,20)(5,71,21)(6,72,22)(7,65,23)(8,66,24)(9,131,122)(10,132,123)(11,133,124)(12,134,125)(13,135,126)(14,136,127)(15,129,128)(16,130,121)(25,51,78)(26,52,79)(27,53,80)(28,54,73)(29,55,74)(30,56,75)(31,49,76)(32,50,77)(33,105,46)(34,106,47)(35,107,48)(36,108,41)(37,109,42)(38,110,43)(39,111,44)(40,112,45)(57,116,89)(58,117,90)(59,118,91)(60,119,92)(61,120,93)(62,113,94)(63,114,95)(64,115,96)(81,142,98)(82,143,99)(83,144,100)(84,137,101)(85,138,102)(86,139,103)(87,140,104)(88,141,97), (1,117,32)(2,118,25)(3,119,26)(4,120,27)(5,113,28)(6,114,29)(7,115,30)(8,116,31)(9,86,105)(10,87,106)(11,88,107)(12,81,108)(13,82,109)(14,83,110)(15,84,111)(16,85,112)(17,58,77)(18,59,78)(19,60,79)(20,61,80)(21,62,73)(22,63,74)(23,64,75)(24,57,76)(33,122,103)(34,123,104)(35,124,97)(36,125,98)(37,126,99)(38,127,100)(39,128,101)(40,121,102)(41,134,142)(42,135,143)(43,136,144)(44,129,137)(45,130,138)(46,131,139)(47,132,140)(48,133,141)(49,66,89)(50,67,90)(51,68,91)(52,69,92)(53,70,93)(54,71,94)(55,72,95)(56,65,96), (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), (1,101,5,97)(2,100,6,104)(3,99,7,103)(4,98,8,102)(9,79,13,75)(10,78,14,74)(11,77,15,73)(12,76,16,80)(17,84,21,88)(18,83,22,87)(19,82,23,86)(20,81,24,85)(25,127,29,123)(26,126,30,122)(27,125,31,121)(28,124,32,128)(33,119,37,115)(34,118,38,114)(35,117,39,113)(36,116,40,120)(41,89,45,93)(42,96,46,92)(43,95,47,91)(44,94,48,90)(49,130,53,134)(50,129,54,133)(51,136,55,132)(52,135,56,131)(57,112,61,108)(58,111,62,107)(59,110,63,106)(60,109,64,105)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)>;
G:=Group( (9,131,122)(10,132,123)(11,133,124)(12,134,125)(13,135,126)(14,136,127)(15,129,128)(16,130,121)(25,51,78)(26,52,79)(27,53,80)(28,54,73)(29,55,74)(30,56,75)(31,49,76)(32,50,77)(33,46,105)(34,47,106)(35,48,107)(36,41,108)(37,42,109)(38,43,110)(39,44,111)(40,45,112)(57,89,116)(58,90,117)(59,91,118)(60,92,119)(61,93,120)(62,94,113)(63,95,114)(64,96,115), (1,67,17)(2,68,18)(3,69,19)(4,70,20)(5,71,21)(6,72,22)(7,65,23)(8,66,24)(9,131,122)(10,132,123)(11,133,124)(12,134,125)(13,135,126)(14,136,127)(15,129,128)(16,130,121)(25,51,78)(26,52,79)(27,53,80)(28,54,73)(29,55,74)(30,56,75)(31,49,76)(32,50,77)(33,105,46)(34,106,47)(35,107,48)(36,108,41)(37,109,42)(38,110,43)(39,111,44)(40,112,45)(57,116,89)(58,117,90)(59,118,91)(60,119,92)(61,120,93)(62,113,94)(63,114,95)(64,115,96)(81,142,98)(82,143,99)(83,144,100)(84,137,101)(85,138,102)(86,139,103)(87,140,104)(88,141,97), (1,117,32)(2,118,25)(3,119,26)(4,120,27)(5,113,28)(6,114,29)(7,115,30)(8,116,31)(9,86,105)(10,87,106)(11,88,107)(12,81,108)(13,82,109)(14,83,110)(15,84,111)(16,85,112)(17,58,77)(18,59,78)(19,60,79)(20,61,80)(21,62,73)(22,63,74)(23,64,75)(24,57,76)(33,122,103)(34,123,104)(35,124,97)(36,125,98)(37,126,99)(38,127,100)(39,128,101)(40,121,102)(41,134,142)(42,135,143)(43,136,144)(44,129,137)(45,130,138)(46,131,139)(47,132,140)(48,133,141)(49,66,89)(50,67,90)(51,68,91)(52,69,92)(53,70,93)(54,71,94)(55,72,95)(56,65,96), (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), (1,101,5,97)(2,100,6,104)(3,99,7,103)(4,98,8,102)(9,79,13,75)(10,78,14,74)(11,77,15,73)(12,76,16,80)(17,84,21,88)(18,83,22,87)(19,82,23,86)(20,81,24,85)(25,127,29,123)(26,126,30,122)(27,125,31,121)(28,124,32,128)(33,119,37,115)(34,118,38,114)(35,117,39,113)(36,116,40,120)(41,89,45,93)(42,96,46,92)(43,95,47,91)(44,94,48,90)(49,130,53,134)(50,129,54,133)(51,136,55,132)(52,135,56,131)(57,112,61,108)(58,111,62,107)(59,110,63,106)(60,109,64,105)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140) );
G=PermutationGroup([[(9,131,122),(10,132,123),(11,133,124),(12,134,125),(13,135,126),(14,136,127),(15,129,128),(16,130,121),(25,51,78),(26,52,79),(27,53,80),(28,54,73),(29,55,74),(30,56,75),(31,49,76),(32,50,77),(33,46,105),(34,47,106),(35,48,107),(36,41,108),(37,42,109),(38,43,110),(39,44,111),(40,45,112),(57,89,116),(58,90,117),(59,91,118),(60,92,119),(61,93,120),(62,94,113),(63,95,114),(64,96,115)], [(1,67,17),(2,68,18),(3,69,19),(4,70,20),(5,71,21),(6,72,22),(7,65,23),(8,66,24),(9,131,122),(10,132,123),(11,133,124),(12,134,125),(13,135,126),(14,136,127),(15,129,128),(16,130,121),(25,51,78),(26,52,79),(27,53,80),(28,54,73),(29,55,74),(30,56,75),(31,49,76),(32,50,77),(33,105,46),(34,106,47),(35,107,48),(36,108,41),(37,109,42),(38,110,43),(39,111,44),(40,112,45),(57,116,89),(58,117,90),(59,118,91),(60,119,92),(61,120,93),(62,113,94),(63,114,95),(64,115,96),(81,142,98),(82,143,99),(83,144,100),(84,137,101),(85,138,102),(86,139,103),(87,140,104),(88,141,97)], [(1,117,32),(2,118,25),(3,119,26),(4,120,27),(5,113,28),(6,114,29),(7,115,30),(8,116,31),(9,86,105),(10,87,106),(11,88,107),(12,81,108),(13,82,109),(14,83,110),(15,84,111),(16,85,112),(17,58,77),(18,59,78),(19,60,79),(20,61,80),(21,62,73),(22,63,74),(23,64,75),(24,57,76),(33,122,103),(34,123,104),(35,124,97),(36,125,98),(37,126,99),(38,127,100),(39,128,101),(40,121,102),(41,134,142),(42,135,143),(43,136,144),(44,129,137),(45,130,138),(46,131,139),(47,132,140),(48,133,141),(49,66,89),(50,67,90),(51,68,91),(52,69,92),(53,70,93),(54,71,94),(55,72,95),(56,65,96)], [(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)], [(1,101,5,97),(2,100,6,104),(3,99,7,103),(4,98,8,102),(9,79,13,75),(10,78,14,74),(11,77,15,73),(12,76,16,80),(17,84,21,88),(18,83,22,87),(19,82,23,86),(20,81,24,85),(25,127,29,123),(26,126,30,122),(27,125,31,121),(28,124,32,128),(33,119,37,115),(34,118,38,114),(35,117,39,113),(36,116,40,120),(41,89,45,93),(42,96,46,92),(43,95,47,91),(44,94,48,90),(49,130,53,134),(50,129,54,133),(51,136,55,132),(52,135,56,131),(57,112,61,108),(58,111,62,107),(59,110,63,106),(60,109,64,105),(65,139,69,143),(66,138,70,142),(67,137,71,141),(68,144,72,140)]])
53 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | ··· | 12J | 12K | 12L | 12M | 12N | 24A | 24B | 24C | 24D | 24E | ··· | 24T |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 36 | 36 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 6 | ··· | 6 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | - | + | - | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | Q16 | C3×S3 | D12 | C3×D4 | S3×C6 | Dic12 | C3×Q16 | C3×D12 | C3×Dic12 | C32⋊C6 | C2×C32⋊C6 | He3⋊4D4 | He3⋊4Q16 |
| kernel | He3⋊4Q16 | C8×He3 | He3⋊3Q8 | C32⋊5Q16 | C3×C24 | C32⋊4Q8 | C3×C24 | C2×He3 | C3×C12 | He3 | C24 | C3×C6 | C3×C6 | C12 | C32 | C32 | C6 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 4 |
Matrix representation of He3⋊4Q16 ►in GL8(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 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 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 18 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 68 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 22 | 63 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 51 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 66 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 66 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 59 |
G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[18,68,0,0,0,0,0,0,5,23,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[22,12,0,0,0,0,0,0,63,51,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,66,59,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,66,59,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,66,59] >;
He3⋊4Q16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_4Q_{16} % in TeX
G:=Group("He3:4Q16"); // GroupNames label
G:=SmallGroup(432,114);
// by ID
G=gap.SmallGroup(432,114);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,260,1011,80,4037,2035,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations