direct product, non-abelian, soluble
Aliases: C32×C4.A4, C4○D4⋊C33, C4.(C32×A4), C6.20(C6×A4), (C3×C12).6A4, C12.11(C3×A4), Q8.(C32×C6), SL2(𝔽3)⋊2(C3×C6), (C3×SL2(𝔽3))⋊8C6, (Q8×C32).17C6, (C32×SL2(𝔽3))⋊8C2, C2.3(A4×C3×C6), (C3×C4○D4)⋊C32, (C3×C6).27(C2×A4), (C32×C4○D4)⋊3C3, (C3×Q8).12(C3×C6), SmallGroup(432,699)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C32×C4.A4 |
Generators and relations for C32×C4.A4
G = < a,b,c,d,e,f | a3=b3=c4=f3=1, d2=e2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=c2d, fdf-1=c2de, fef-1=d >
Subgroups: 446 in 170 conjugacy classes, 74 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C32, C32, C12, C12, C2×C6, C4○D4, C3×C6, C3×C6, SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C33, C3×C12, C3×C12, C62, C4.A4, C3×C4○D4, C32×C6, C3×SL2(𝔽3), C6×C12, D4×C32, Q8×C32, C32×C12, C3×C4.A4, C32×C4○D4, C32×SL2(𝔽3), C32×C4.A4
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, C33, C3×A4, C4.A4, C32×C6, C6×A4, C32×A4, C3×C4.A4, A4×C3×C6, C32×C4.A4
►On 144 points
Generators in S144
(1 79 95)(2 80 96)(3 77 93)(4 78 94)(5 26 10)(6 27 11)(7 28 12)(8 25 9)(13 33 29)(14 34 30)(15 35 31)(16 36 32)(17 37 21)(18 38 22)(19 39 23)(20 40 24)(41 75 57)(42 76 58)(43 73 59)(44 74 60)(45 65 61)(46 66 62)(47 67 63)(48 68 64)(49 69 53)(50 70 54)(51 71 55)(52 72 56)(81 101 97)(82 102 98)(83 103 99)(84 104 100)(85 105 89)(86 106 90)(87 107 91)(88 108 92)(109 141 125)(110 142 126)(111 143 127)(112 144 128)(113 133 129)(114 134 130)(115 135 131)(116 136 132)(117 137 121)(118 138 122)(119 139 123)(120 140 124)
(1 87 83)(2 88 84)(3 85 81)(4 86 82)(5 38 34)(6 39 35)(7 40 36)(8 37 33)(9 17 13)(10 18 14)(11 19 15)(12 20 16)(21 29 25)(22 30 26)(23 31 27)(24 32 28)(41 49 45)(42 50 46)(43 51 47)(44 52 48)(53 61 57)(54 62 58)(55 63 59)(56 64 60)(65 75 69)(66 76 70)(67 73 71)(68 74 72)(77 105 101)(78 106 102)(79 107 103)(80 108 104)(89 97 93)(90 98 94)(91 99 95)(92 100 96)(109 117 113)(110 118 114)(111 119 115)(112 120 116)(121 129 125)(122 130 126)(123 131 127)(124 132 128)(133 141 137)(134 142 138)(135 143 139)(136 144 140)
(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)
(1 109 3 111)(2 110 4 112)(5 76 7 74)(6 73 8 75)(9 41 11 43)(10 42 12 44)(13 45 15 47)(14 46 16 48)(17 49 19 51)(18 50 20 52)(21 53 23 55)(22 54 24 56)(25 57 27 59)(26 58 28 60)(29 61 31 63)(30 62 32 64)(33 65 35 67)(34 66 36 68)(37 69 39 71)(38 70 40 72)(77 143 79 141)(78 144 80 142)(81 115 83 113)(82 116 84 114)(85 119 87 117)(86 120 88 118)(89 123 91 121)(90 124 92 122)(93 127 95 125)(94 128 96 126)(97 131 99 129)(98 132 100 130)(101 135 103 133)(102 136 104 134)(105 139 107 137)(106 140 108 138)
(1 9 3 11)(2 10 4 12)(5 78 7 80)(6 79 8 77)(13 81 15 83)(14 82 16 84)(17 85 19 87)(18 86 20 88)(21 89 23 91)(22 90 24 92)(25 93 27 95)(26 94 28 96)(29 97 31 99)(30 98 32 100)(33 101 35 103)(34 102 36 104)(37 105 39 107)(38 106 40 108)(41 109 43 111)(42 110 44 112)(45 113 47 115)(46 114 48 116)(49 117 51 119)(50 118 52 120)(53 121 55 123)(54 122 56 124)(57 125 59 127)(58 126 60 128)(61 129 63 131)(62 130 64 132)(65 133 67 135)(66 134 68 136)(69 137 71 139)(70 138 72 140)(73 143 75 141)(74 144 76 142)
(1 79 95)(2 80 96)(3 77 93)(4 78 94)(5 58 110)(6 59 111)(7 60 112)(8 57 109)(9 75 125)(10 76 126)(11 73 127)(12 74 128)(13 65 129)(14 66 130)(15 67 131)(16 68 132)(17 69 121)(18 70 122)(19 71 123)(20 72 124)(21 49 137)(22 50 138)(23 51 139)(24 52 140)(25 41 141)(26 42 142)(27 43 143)(28 44 144)(29 45 133)(30 46 134)(31 47 135)(32 48 136)(33 61 113)(34 62 114)(35 63 115)(36 64 116)(37 53 117)(38 54 118)(39 55 119)(40 56 120)(81 101 97)(82 102 98)(83 103 99)(84 104 100)(85 105 89)(86 106 90)(87 107 91)(88 108 92)
G:=sub<Sym(144)| (1,79,95)(2,80,96)(3,77,93)(4,78,94)(5,26,10)(6,27,11)(7,28,12)(8,25,9)(13,33,29)(14,34,30)(15,35,31)(16,36,32)(17,37,21)(18,38,22)(19,39,23)(20,40,24)(41,75,57)(42,76,58)(43,73,59)(44,74,60)(45,65,61)(46,66,62)(47,67,63)(48,68,64)(49,69,53)(50,70,54)(51,71,55)(52,72,56)(81,101,97)(82,102,98)(83,103,99)(84,104,100)(85,105,89)(86,106,90)(87,107,91)(88,108,92)(109,141,125)(110,142,126)(111,143,127)(112,144,128)(113,133,129)(114,134,130)(115,135,131)(116,136,132)(117,137,121)(118,138,122)(119,139,123)(120,140,124), (1,87,83)(2,88,84)(3,85,81)(4,86,82)(5,38,34)(6,39,35)(7,40,36)(8,37,33)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,29,25)(22,30,26)(23,31,27)(24,32,28)(41,49,45)(42,50,46)(43,51,47)(44,52,48)(53,61,57)(54,62,58)(55,63,59)(56,64,60)(65,75,69)(66,76,70)(67,73,71)(68,74,72)(77,105,101)(78,106,102)(79,107,103)(80,108,104)(89,97,93)(90,98,94)(91,99,95)(92,100,96)(109,117,113)(110,118,114)(111,119,115)(112,120,116)(121,129,125)(122,130,126)(123,131,127)(124,132,128)(133,141,137)(134,142,138)(135,143,139)(136,144,140), (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), (1,109,3,111)(2,110,4,112)(5,76,7,74)(6,73,8,75)(9,41,11,43)(10,42,12,44)(13,45,15,47)(14,46,16,48)(17,49,19,51)(18,50,20,52)(21,53,23,55)(22,54,24,56)(25,57,27,59)(26,58,28,60)(29,61,31,63)(30,62,32,64)(33,65,35,67)(34,66,36,68)(37,69,39,71)(38,70,40,72)(77,143,79,141)(78,144,80,142)(81,115,83,113)(82,116,84,114)(85,119,87,117)(86,120,88,118)(89,123,91,121)(90,124,92,122)(93,127,95,125)(94,128,96,126)(97,131,99,129)(98,132,100,130)(101,135,103,133)(102,136,104,134)(105,139,107,137)(106,140,108,138), (1,9,3,11)(2,10,4,12)(5,78,7,80)(6,79,8,77)(13,81,15,83)(14,82,16,84)(17,85,19,87)(18,86,20,88)(21,89,23,91)(22,90,24,92)(25,93,27,95)(26,94,28,96)(29,97,31,99)(30,98,32,100)(33,101,35,103)(34,102,36,104)(37,105,39,107)(38,106,40,108)(41,109,43,111)(42,110,44,112)(45,113,47,115)(46,114,48,116)(49,117,51,119)(50,118,52,120)(53,121,55,123)(54,122,56,124)(57,125,59,127)(58,126,60,128)(61,129,63,131)(62,130,64,132)(65,133,67,135)(66,134,68,136)(69,137,71,139)(70,138,72,140)(73,143,75,141)(74,144,76,142), (1,79,95)(2,80,96)(3,77,93)(4,78,94)(5,58,110)(6,59,111)(7,60,112)(8,57,109)(9,75,125)(10,76,126)(11,73,127)(12,74,128)(13,65,129)(14,66,130)(15,67,131)(16,68,132)(17,69,121)(18,70,122)(19,71,123)(20,72,124)(21,49,137)(22,50,138)(23,51,139)(24,52,140)(25,41,141)(26,42,142)(27,43,143)(28,44,144)(29,45,133)(30,46,134)(31,47,135)(32,48,136)(33,61,113)(34,62,114)(35,63,115)(36,64,116)(37,53,117)(38,54,118)(39,55,119)(40,56,120)(81,101,97)(82,102,98)(83,103,99)(84,104,100)(85,105,89)(86,106,90)(87,107,91)(88,108,92)>;
G:=Group( (1,79,95)(2,80,96)(3,77,93)(4,78,94)(5,26,10)(6,27,11)(7,28,12)(8,25,9)(13,33,29)(14,34,30)(15,35,31)(16,36,32)(17,37,21)(18,38,22)(19,39,23)(20,40,24)(41,75,57)(42,76,58)(43,73,59)(44,74,60)(45,65,61)(46,66,62)(47,67,63)(48,68,64)(49,69,53)(50,70,54)(51,71,55)(52,72,56)(81,101,97)(82,102,98)(83,103,99)(84,104,100)(85,105,89)(86,106,90)(87,107,91)(88,108,92)(109,141,125)(110,142,126)(111,143,127)(112,144,128)(113,133,129)(114,134,130)(115,135,131)(116,136,132)(117,137,121)(118,138,122)(119,139,123)(120,140,124), (1,87,83)(2,88,84)(3,85,81)(4,86,82)(5,38,34)(6,39,35)(7,40,36)(8,37,33)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,29,25)(22,30,26)(23,31,27)(24,32,28)(41,49,45)(42,50,46)(43,51,47)(44,52,48)(53,61,57)(54,62,58)(55,63,59)(56,64,60)(65,75,69)(66,76,70)(67,73,71)(68,74,72)(77,105,101)(78,106,102)(79,107,103)(80,108,104)(89,97,93)(90,98,94)(91,99,95)(92,100,96)(109,117,113)(110,118,114)(111,119,115)(112,120,116)(121,129,125)(122,130,126)(123,131,127)(124,132,128)(133,141,137)(134,142,138)(135,143,139)(136,144,140), (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), (1,109,3,111)(2,110,4,112)(5,76,7,74)(6,73,8,75)(9,41,11,43)(10,42,12,44)(13,45,15,47)(14,46,16,48)(17,49,19,51)(18,50,20,52)(21,53,23,55)(22,54,24,56)(25,57,27,59)(26,58,28,60)(29,61,31,63)(30,62,32,64)(33,65,35,67)(34,66,36,68)(37,69,39,71)(38,70,40,72)(77,143,79,141)(78,144,80,142)(81,115,83,113)(82,116,84,114)(85,119,87,117)(86,120,88,118)(89,123,91,121)(90,124,92,122)(93,127,95,125)(94,128,96,126)(97,131,99,129)(98,132,100,130)(101,135,103,133)(102,136,104,134)(105,139,107,137)(106,140,108,138), (1,9,3,11)(2,10,4,12)(5,78,7,80)(6,79,8,77)(13,81,15,83)(14,82,16,84)(17,85,19,87)(18,86,20,88)(21,89,23,91)(22,90,24,92)(25,93,27,95)(26,94,28,96)(29,97,31,99)(30,98,32,100)(33,101,35,103)(34,102,36,104)(37,105,39,107)(38,106,40,108)(41,109,43,111)(42,110,44,112)(45,113,47,115)(46,114,48,116)(49,117,51,119)(50,118,52,120)(53,121,55,123)(54,122,56,124)(57,125,59,127)(58,126,60,128)(61,129,63,131)(62,130,64,132)(65,133,67,135)(66,134,68,136)(69,137,71,139)(70,138,72,140)(73,143,75,141)(74,144,76,142), (1,79,95)(2,80,96)(3,77,93)(4,78,94)(5,58,110)(6,59,111)(7,60,112)(8,57,109)(9,75,125)(10,76,126)(11,73,127)(12,74,128)(13,65,129)(14,66,130)(15,67,131)(16,68,132)(17,69,121)(18,70,122)(19,71,123)(20,72,124)(21,49,137)(22,50,138)(23,51,139)(24,52,140)(25,41,141)(26,42,142)(27,43,143)(28,44,144)(29,45,133)(30,46,134)(31,47,135)(32,48,136)(33,61,113)(34,62,114)(35,63,115)(36,64,116)(37,53,117)(38,54,118)(39,55,119)(40,56,120)(81,101,97)(82,102,98)(83,103,99)(84,104,100)(85,105,89)(86,106,90)(87,107,91)(88,108,92) );
G=PermutationGroup([[(1,79,95),(2,80,96),(3,77,93),(4,78,94),(5,26,10),(6,27,11),(7,28,12),(8,25,9),(13,33,29),(14,34,30),(15,35,31),(16,36,32),(17,37,21),(18,38,22),(19,39,23),(20,40,24),(41,75,57),(42,76,58),(43,73,59),(44,74,60),(45,65,61),(46,66,62),(47,67,63),(48,68,64),(49,69,53),(50,70,54),(51,71,55),(52,72,56),(81,101,97),(82,102,98),(83,103,99),(84,104,100),(85,105,89),(86,106,90),(87,107,91),(88,108,92),(109,141,125),(110,142,126),(111,143,127),(112,144,128),(113,133,129),(114,134,130),(115,135,131),(116,136,132),(117,137,121),(118,138,122),(119,139,123),(120,140,124)], [(1,87,83),(2,88,84),(3,85,81),(4,86,82),(5,38,34),(6,39,35),(7,40,36),(8,37,33),(9,17,13),(10,18,14),(11,19,15),(12,20,16),(21,29,25),(22,30,26),(23,31,27),(24,32,28),(41,49,45),(42,50,46),(43,51,47),(44,52,48),(53,61,57),(54,62,58),(55,63,59),(56,64,60),(65,75,69),(66,76,70),(67,73,71),(68,74,72),(77,105,101),(78,106,102),(79,107,103),(80,108,104),(89,97,93),(90,98,94),(91,99,95),(92,100,96),(109,117,113),(110,118,114),(111,119,115),(112,120,116),(121,129,125),(122,130,126),(123,131,127),(124,132,128),(133,141,137),(134,142,138),(135,143,139),(136,144,140)], [(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)], [(1,109,3,111),(2,110,4,112),(5,76,7,74),(6,73,8,75),(9,41,11,43),(10,42,12,44),(13,45,15,47),(14,46,16,48),(17,49,19,51),(18,50,20,52),(21,53,23,55),(22,54,24,56),(25,57,27,59),(26,58,28,60),(29,61,31,63),(30,62,32,64),(33,65,35,67),(34,66,36,68),(37,69,39,71),(38,70,40,72),(77,143,79,141),(78,144,80,142),(81,115,83,113),(82,116,84,114),(85,119,87,117),(86,120,88,118),(89,123,91,121),(90,124,92,122),(93,127,95,125),(94,128,96,126),(97,131,99,129),(98,132,100,130),(101,135,103,133),(102,136,104,134),(105,139,107,137),(106,140,108,138)], [(1,9,3,11),(2,10,4,12),(5,78,7,80),(6,79,8,77),(13,81,15,83),(14,82,16,84),(17,85,19,87),(18,86,20,88),(21,89,23,91),(22,90,24,92),(25,93,27,95),(26,94,28,96),(29,97,31,99),(30,98,32,100),(33,101,35,103),(34,102,36,104),(37,105,39,107),(38,106,40,108),(41,109,43,111),(42,110,44,112),(45,113,47,115),(46,114,48,116),(49,117,51,119),(50,118,52,120),(53,121,55,123),(54,122,56,124),(57,125,59,127),(58,126,60,128),(61,129,63,131),(62,130,64,132),(65,133,67,135),(66,134,68,136),(69,137,71,139),(70,138,72,140),(73,143,75,141),(74,144,76,142)], [(1,79,95),(2,80,96),(3,77,93),(4,78,94),(5,58,110),(6,59,111),(7,60,112),(8,57,109),(9,75,125),(10,76,126),(11,73,127),(12,74,128),(13,65,129),(14,66,130),(15,67,131),(16,68,132),(17,69,121),(18,70,122),(19,71,123),(20,72,124),(21,49,137),(22,50,138),(23,51,139),(24,52,140),(25,41,141),(26,42,142),(27,43,143),(28,44,144),(29,45,133),(30,46,134),(31,47,135),(32,48,136),(33,61,113),(34,62,114),(35,63,115),(36,64,116),(37,53,117),(38,54,118),(39,55,119),(40,56,120),(81,101,97),(82,102,98),(83,103,99),(84,104,100),(85,105,89),(86,106,90),(87,107,91),(88,108,92)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 3I | ··· | 3Z | 4A | 4B | 4C | 6A | ··· | 6H | 6I | ··· | 6Z | 6AA | ··· | 6AH | 12A | ··· | 12P | 12Q | ··· | 12AZ | 12BA | ··· | 12BH |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | 1 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C4.A4 | C3×C4.A4 | A4 | C2×A4 | C3×A4 | C6×A4 |
| kernel | C32×C4.A4 | C32×SL2(𝔽3) | C3×C4.A4 | C32×C4○D4 | C3×SL2(𝔽3) | Q8×C32 | C32 | C3 | C3×C12 | C3×C6 | C12 | C6 |
| # reps | 1 | 1 | 24 | 2 | 24 | 2 | 6 | 48 | 1 | 1 | 8 | 8 |
Matrix representation of C32×C4.A4 ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 9 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 12 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 12 | 0 |
| 1 | 0 | 0 |
| 0 | 4 | 3 |
| 0 | 3 | 9 |
| 9 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 9 | 1 |
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,3],[9,0,0,0,1,0,0,0,1],[12,0,0,0,5,0,0,0,5],[1,0,0,0,0,12,0,1,0],[1,0,0,0,4,3,0,3,9],[9,0,0,0,3,9,0,0,1] >;
C32×C4.A4 in GAP, Magma, Sage, TeX
C_3^2\times C_4.A_4
% in TeX
G:=Group("C3^2xC4.A4"); // GroupNames label
G:=SmallGroup(432,699);
// by ID
G=gap.SmallGroup(432,699);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,1901,172,3414,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=f^3=1,d^2=e^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=c^2*d,f*d*f^-1=c^2*d*e,f*e*f^-1=d>;
// generators/relations