direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4⋊D10, C60.220D4, C60.206C23, D4⋊D5⋊6C6, D4⋊4(C6×D5), Q8⋊D5⋊6C6, Q8⋊5(C6×D5), (C2×D20)⋊10C6, (C6×D20)⋊26C2, (C3×D4)⋊26D10, (C3×Q8)⋊23D10, C10.59(C6×D4), C20.49(C3×D4), (C2×C30).85D4, C4.Dic5⋊9C6, C15⋊38(C8⋊C22), D20.11(C2×C6), C30.416(C2×D4), (C2×C12).246D10, (D4×C15)⋊28C22, C20.17(C22×C6), (Q8×C15)⋊25C22, C12.117(C5⋊D4), (C2×C60).302C22, (C3×D20).50C22, C12.206(C22×D5), C5⋊5(C3×C8⋊C22), C4.17(D5×C2×C6), C5⋊2C8⋊4(C2×C6), (C3×C4○D4)⋊6D5, C4○D4⋊3(C3×D5), (C5×C4○D4)⋊5C6, (C5×D4)⋊4(C2×C6), (C5×Q8)⋊6(C2×C6), (C3×D4⋊D5)⋊14C2, (C15×C4○D4)⋊6C2, (C3×Q8⋊D5)⋊14C2, (C2×C10).8(C3×D4), (C2×C4).17(C6×D5), C2.23(C6×C5⋊D4), C4.24(C3×C5⋊D4), (C2×C20).39(C2×C6), C6.144(C2×C5⋊D4), C22.5(C3×C5⋊D4), (C3×C5⋊2C8)⋊26C22, (C2×C6).41(C5⋊D4), (C3×C4.Dic5)⋊21C2, SmallGroup(480,742)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊D10
G = < a,b,c,d,e | a3=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >
Subgroups: 512 in 136 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C8⋊C22, C5⋊2C8, D20, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C3×M4(2), C3×D8, C3×SD16, C6×D4, C3×C4○D4, C60, C60, C6×D5, C2×C30, C2×C30, C4.Dic5, D4⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, C3×C8⋊C22, C3×C5⋊2C8, C3×D20, C3×D20, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D5×C2×C6, D4⋊D10, C3×C4.Dic5, C3×D4⋊D5, C3×Q8⋊D5, C6×D20, C15×C4○D4, C3×D4⋊D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8⋊C22, C5⋊D4, C22×D5, C6×D4, C6×D5, C2×C5⋊D4, C3×C8⋊C22, C3×C5⋊D4, D5×C2×C6, D4⋊D10, C6×C5⋊D4, C3×D4⋊D10
►On 120 points
Generators in S120
(1 45 25)(2 41 21)(3 42 22)(4 43 23)(5 44 24)(6 49 27)(7 50 28)(8 46 29)(9 47 30)(10 48 26)(11 54 34)(12 55 35)(13 51 31)(14 52 32)(15 53 33)(16 57 37)(17 58 38)(18 59 39)(19 60 40)(20 56 36)(61 107 87)(62 108 88)(63 109 89)(64 110 90)(65 101 81)(66 102 82)(67 103 83)(68 104 84)(69 105 85)(70 106 86)(71 111 95)(72 112 96)(73 113 97)(74 114 98)(75 115 99)(76 116 100)(77 117 91)(78 118 92)(79 119 93)(80 120 94)
(1 19 7 12)(2 20 8 13)(3 16 9 14)(4 17 10 15)(5 18 6 11)(21 36 29 31)(22 37 30 32)(23 38 26 33)(24 39 27 34)(25 40 28 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 77 66 72)(62 78 67 73)(63 79 68 74)(64 80 69 75)(65 71 70 76)(81 95 86 100)(82 96 87 91)(83 97 88 92)(84 98 89 93)(85 99 90 94)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)
(1 72)(2 78)(3 74)(4 80)(5 76)(6 71)(7 77)(8 73)(9 79)(10 75)(11 65)(12 61)(13 67)(14 63)(15 69)(16 68)(17 64)(18 70)(19 66)(20 62)(21 92)(22 98)(23 94)(24 100)(25 96)(26 99)(27 95)(28 91)(29 97)(30 93)(31 83)(32 89)(33 85)(34 81)(35 87)(36 88)(37 84)(38 90)(39 86)(40 82)(41 118)(42 114)(43 120)(44 116)(45 112)(46 113)(47 119)(48 115)(49 111)(50 117)(51 103)(52 109)(53 105)(54 101)(55 107)(56 108)(57 104)(58 110)(59 106)(60 102)
(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)
(1 5)(2 4)(6 7)(8 10)(11 19)(12 18)(13 17)(14 16)(15 20)(21 23)(24 25)(26 29)(27 28)(31 38)(32 37)(33 36)(34 40)(35 39)(41 43)(44 45)(46 48)(49 50)(51 58)(52 57)(53 56)(54 60)(55 59)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 80)(68 79)(69 78)(70 77)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 100)(88 99)(89 98)(90 97)(101 112)(102 111)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)
G:=sub<Sym(120)| (1,45,25)(2,41,21)(3,42,22)(4,43,23)(5,44,24)(6,49,27)(7,50,28)(8,46,29)(9,47,30)(10,48,26)(11,54,34)(12,55,35)(13,51,31)(14,52,32)(15,53,33)(16,57,37)(17,58,38)(18,59,39)(19,60,40)(20,56,36)(61,107,87)(62,108,88)(63,109,89)(64,110,90)(65,101,81)(66,102,82)(67,103,83)(68,104,84)(69,105,85)(70,106,86)(71,111,95)(72,112,96)(73,113,97)(74,114,98)(75,115,99)(76,116,100)(77,117,91)(78,118,92)(79,119,93)(80,120,94), (1,19,7,12)(2,20,8,13)(3,16,9,14)(4,17,10,15)(5,18,6,11)(21,36,29,31)(22,37,30,32)(23,38,26,33)(24,39,27,34)(25,40,28,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,77,66,72)(62,78,67,73)(63,79,68,74)(64,80,69,75)(65,71,70,76)(81,95,86,100)(82,96,87,91)(83,97,88,92)(84,98,89,93)(85,99,90,94)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120), (1,72)(2,78)(3,74)(4,80)(5,76)(6,71)(7,77)(8,73)(9,79)(10,75)(11,65)(12,61)(13,67)(14,63)(15,69)(16,68)(17,64)(18,70)(19,66)(20,62)(21,92)(22,98)(23,94)(24,100)(25,96)(26,99)(27,95)(28,91)(29,97)(30,93)(31,83)(32,89)(33,85)(34,81)(35,87)(36,88)(37,84)(38,90)(39,86)(40,82)(41,118)(42,114)(43,120)(44,116)(45,112)(46,113)(47,119)(48,115)(49,111)(50,117)(51,103)(52,109)(53,105)(54,101)(55,107)(56,108)(57,104)(58,110)(59,106)(60,102), (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), (1,5)(2,4)(6,7)(8,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,23)(24,25)(26,29)(27,28)(31,38)(32,37)(33,36)(34,40)(35,39)(41,43)(44,45)(46,48)(49,50)(51,58)(52,57)(53,56)(54,60)(55,59)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,100)(88,99)(89,98)(90,97)(101,112)(102,111)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)>;
G:=Group( (1,45,25)(2,41,21)(3,42,22)(4,43,23)(5,44,24)(6,49,27)(7,50,28)(8,46,29)(9,47,30)(10,48,26)(11,54,34)(12,55,35)(13,51,31)(14,52,32)(15,53,33)(16,57,37)(17,58,38)(18,59,39)(19,60,40)(20,56,36)(61,107,87)(62,108,88)(63,109,89)(64,110,90)(65,101,81)(66,102,82)(67,103,83)(68,104,84)(69,105,85)(70,106,86)(71,111,95)(72,112,96)(73,113,97)(74,114,98)(75,115,99)(76,116,100)(77,117,91)(78,118,92)(79,119,93)(80,120,94), (1,19,7,12)(2,20,8,13)(3,16,9,14)(4,17,10,15)(5,18,6,11)(21,36,29,31)(22,37,30,32)(23,38,26,33)(24,39,27,34)(25,40,28,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,77,66,72)(62,78,67,73)(63,79,68,74)(64,80,69,75)(65,71,70,76)(81,95,86,100)(82,96,87,91)(83,97,88,92)(84,98,89,93)(85,99,90,94)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120), (1,72)(2,78)(3,74)(4,80)(5,76)(6,71)(7,77)(8,73)(9,79)(10,75)(11,65)(12,61)(13,67)(14,63)(15,69)(16,68)(17,64)(18,70)(19,66)(20,62)(21,92)(22,98)(23,94)(24,100)(25,96)(26,99)(27,95)(28,91)(29,97)(30,93)(31,83)(32,89)(33,85)(34,81)(35,87)(36,88)(37,84)(38,90)(39,86)(40,82)(41,118)(42,114)(43,120)(44,116)(45,112)(46,113)(47,119)(48,115)(49,111)(50,117)(51,103)(52,109)(53,105)(54,101)(55,107)(56,108)(57,104)(58,110)(59,106)(60,102), (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), (1,5)(2,4)(6,7)(8,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,23)(24,25)(26,29)(27,28)(31,38)(32,37)(33,36)(34,40)(35,39)(41,43)(44,45)(46,48)(49,50)(51,58)(52,57)(53,56)(54,60)(55,59)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,100)(88,99)(89,98)(90,97)(101,112)(102,111)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113) );
G=PermutationGroup([[(1,45,25),(2,41,21),(3,42,22),(4,43,23),(5,44,24),(6,49,27),(7,50,28),(8,46,29),(9,47,30),(10,48,26),(11,54,34),(12,55,35),(13,51,31),(14,52,32),(15,53,33),(16,57,37),(17,58,38),(18,59,39),(19,60,40),(20,56,36),(61,107,87),(62,108,88),(63,109,89),(64,110,90),(65,101,81),(66,102,82),(67,103,83),(68,104,84),(69,105,85),(70,106,86),(71,111,95),(72,112,96),(73,113,97),(74,114,98),(75,115,99),(76,116,100),(77,117,91),(78,118,92),(79,119,93),(80,120,94)], [(1,19,7,12),(2,20,8,13),(3,16,9,14),(4,17,10,15),(5,18,6,11),(21,36,29,31),(22,37,30,32),(23,38,26,33),(24,39,27,34),(25,40,28,35),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,77,66,72),(62,78,67,73),(63,79,68,74),(64,80,69,75),(65,71,70,76),(81,95,86,100),(82,96,87,91),(83,97,88,92),(84,98,89,93),(85,99,90,94),(101,111,106,116),(102,112,107,117),(103,113,108,118),(104,114,109,119),(105,115,110,120)], [(1,72),(2,78),(3,74),(4,80),(5,76),(6,71),(7,77),(8,73),(9,79),(10,75),(11,65),(12,61),(13,67),(14,63),(15,69),(16,68),(17,64),(18,70),(19,66),(20,62),(21,92),(22,98),(23,94),(24,100),(25,96),(26,99),(27,95),(28,91),(29,97),(30,93),(31,83),(32,89),(33,85),(34,81),(35,87),(36,88),(37,84),(38,90),(39,86),(40,82),(41,118),(42,114),(43,120),(44,116),(45,112),(46,113),(47,119),(48,115),(49,111),(50,117),(51,103),(52,109),(53,105),(54,101),(55,107),(56,108),(57,104),(58,110),(59,106),(60,102)], [(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)], [(1,5),(2,4),(6,7),(8,10),(11,19),(12,18),(13,17),(14,16),(15,20),(21,23),(24,25),(26,29),(27,28),(31,38),(32,37),(33,36),(34,40),(35,39),(41,43),(44,45),(46,48),(49,50),(51,58),(52,57),(53,56),(54,60),(55,59),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,80),(68,79),(69,78),(70,77),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,100),(88,99),(89,98),(90,97),(101,112),(102,111),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 20 | 20 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | D10 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C5⋊D4 | C5⋊D4 | C6×D5 | C6×D5 | C6×D5 | C3×C5⋊D4 | C3×C5⋊D4 | C8⋊C22 | C3×C8⋊C22 | D4⋊D10 | C3×D4⋊D10 |
| kernel | C3×D4⋊D10 | C3×C4.Dic5 | C3×D4⋊D5 | C3×Q8⋊D5 | C6×D20 | C15×C4○D4 | D4⋊D10 | C4.Dic5 | D4⋊D5 | Q8⋊D5 | C2×D20 | C5×C4○D4 | C60 | C2×C30 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C20 | C2×C10 | C4○D4 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C3×D4⋊D10 ►in GL6(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 197 | 156 | 0 | 0 |
| 0 | 0 | 88 | 44 | 0 | 0 |
| 0 | 0 | 128 | 128 | 41 | 85 |
| 0 | 0 | 113 | 0 | 156 | 200 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 213 | 128 | 79 | 126 |
| 0 | 0 | 0 | 198 | 47 | 194 |
| 0 | 0 | 82 | 88 | 156 | 113 |
| 0 | 0 | 82 | 235 | 113 | 156 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 189 | 0 | 0 |
| 0 | 0 | 51 | 51 | 0 | 0 |
| 0 | 0 | 239 | 48 | 1 | 52 |
| 0 | 0 | 3 | 191 | 189 | 189 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 50 | 240 | 0 | 0 |
| 0 | 0 | 159 | 42 | 41 | 119 |
| 0 | 0 | 68 | 156 | 156 | 200 |
G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,197,88,128,113,0,0,156,44,128,0,0,0,0,0,41,156,0,0,0,0,85,200],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,213,0,82,82,0,0,128,198,88,235,0,0,79,47,156,113,0,0,126,194,113,156],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,51,239,3,0,0,189,51,48,191,0,0,0,0,1,189,0,0,0,0,52,189],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,50,159,68,0,0,0,240,42,156,0,0,0,0,41,156,0,0,0,0,119,200] >;
C3×D4⋊D10 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes D_{10} % in TeX
G:=Group("C3xD4:D10"); // GroupNames label
G:=SmallGroup(480,742);
// by ID
G=gap.SmallGroup(480,742);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,555,2524,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^10=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations