direct product, metabelian, supersoluble, monomial
Aliases: D4×C3⋊Dic3, C62.252C23, C3⋊5(D4×Dic3), C32⋊25(C4×D4), C62⋊14(C2×C4), (C3×D4)⋊3Dic3, (C6×D4).16S3, C12⋊3(C2×Dic3), C6.128(S3×D4), (D4×C32)⋊9C4, (C2×C12).152D6, (C22×C6).93D6, C62⋊5C4⋊15C2, C12⋊Dic3⋊22C2, (C6×C12).143C22, (C2×C62).70C22, C6.104(D4⋊2S3), C6.36(C22×Dic3), C2.5(C12.D6), C2.5(D4×C3⋊S3), (D4×C3×C6).9C2, C4⋊1(C2×C3⋊Dic3), (C3×C12)⋊14(C2×C4), (C4×C3⋊Dic3)⋊8C2, (C2×C6)⋊6(C2×Dic3), (C2×D4).7(C3⋊S3), (C3×C6).249(C2×D4), C23.22(C2×C3⋊S3), C22⋊2(C2×C3⋊Dic3), (C22×C3⋊Dic3)⋊8C2, C2.6(C22×C3⋊Dic3), (C3×C6).150(C4○D4), (C3×C6).124(C22×C4), (C2×C6).269(C22×S3), C22.25(C22×C3⋊S3), (C2×C3⋊Dic3).187C22, (C2×C4).48(C2×C3⋊S3), SmallGroup(288,791)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — D4×C3⋊Dic3 |
Generators and relations for D4×C3⋊Dic3
G = < a,b,c,d,e | a4=b2=c3=d6=1, e2=d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 804 in 282 conjugacy classes, 127 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×C6, C3×C6, C2×Dic3, C2×C12, C3×D4, C22×C6, C4×D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C6×D4, C2×C3⋊Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, D4×C32, C2×C62, D4×Dic3, C4×C3⋊Dic3, C12⋊Dic3, C62⋊5C4, C22×C3⋊Dic3, D4×C3×C6, D4×C3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C3⋊S3, C2×Dic3, C22×S3, C4×D4, C3⋊Dic3, C2×C3⋊S3, S3×D4, D4⋊2S3, C22×Dic3, C2×C3⋊Dic3, C22×C3⋊S3, D4×Dic3, D4×C3⋊S3, C12.D6, C22×C3⋊Dic3, D4×C3⋊Dic3
►On 144 points
Generators in S144
(1 69 33 50)(2 70 34 51)(3 71 35 52)(4 72 36 53)(5 67 31 54)(6 68 32 49)(7 131 27 134)(8 132 28 135)(9 127 29 136)(10 128 30 137)(11 129 25 138)(12 130 26 133)(13 95 57 66)(14 96 58 61)(15 91 59 62)(16 92 60 63)(17 93 55 64)(18 94 56 65)(19 110 116 80)(20 111 117 81)(21 112 118 82)(22 113 119 83)(23 114 120 84)(24 109 115 79)(37 97 43 75)(38 98 44 76)(39 99 45 77)(40 100 46 78)(41 101 47 73)(42 102 48 74)(85 144 108 121)(86 139 103 122)(87 140 104 123)(88 141 105 124)(89 142 106 125)(90 143 107 126)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 49)(7 131)(8 132)(9 127)(10 128)(11 129)(12 130)(13 66)(14 61)(15 62)(16 63)(17 64)(18 65)(19 110)(20 111)(21 112)(22 113)(23 114)(24 109)(25 138)(26 133)(27 134)(28 135)(29 136)(30 137)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 75)(38 76)(39 77)(40 78)(41 73)(42 74)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(55 93)(56 94)(57 95)(58 96)(59 91)(60 92)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
(1 41 60)(2 42 55)(3 37 56)(4 38 57)(5 39 58)(6 40 59)(7 125 21)(8 126 22)(9 121 23)(10 122 24)(11 123 19)(12 124 20)(13 36 44)(14 31 45)(15 32 46)(16 33 47)(17 34 48)(18 35 43)(25 140 116)(26 141 117)(27 142 118)(28 143 119)(29 144 120)(30 139 115)(49 78 91)(50 73 92)(51 74 93)(52 75 94)(53 76 95)(54 77 96)(61 67 99)(62 68 100)(63 69 101)(64 70 102)(65 71 97)(66 72 98)(79 137 103)(80 138 104)(81 133 105)(82 134 106)(83 135 107)(84 136 108)(85 114 127)(86 109 128)(87 110 129)(88 111 130)(89 112 131)(90 113 132)
(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 85 4 88)(2 90 5 87)(3 89 6 86)(7 78 10 75)(8 77 11 74)(9 76 12 73)(13 81 16 84)(14 80 17 83)(15 79 18 82)(19 93 22 96)(20 92 23 95)(21 91 24 94)(25 102 28 99)(26 101 29 98)(27 100 30 97)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 131 40 128)(38 130 41 127)(39 129 42 132)(43 134 46 137)(44 133 47 136)(45 138 48 135)(49 122 52 125)(50 121 53 124)(51 126 54 123)(55 113 58 110)(56 112 59 109)(57 111 60 114)(61 116 64 119)(62 115 65 118)(63 120 66 117)(67 140 70 143)(68 139 71 142)(69 144 72 141)
G:=sub<Sym(144)| (1,69,33,50)(2,70,34,51)(3,71,35,52)(4,72,36,53)(5,67,31,54)(6,68,32,49)(7,131,27,134)(8,132,28,135)(9,127,29,136)(10,128,30,137)(11,129,25,138)(12,130,26,133)(13,95,57,66)(14,96,58,61)(15,91,59,62)(16,92,60,63)(17,93,55,64)(18,94,56,65)(19,110,116,80)(20,111,117,81)(21,112,118,82)(22,113,119,83)(23,114,120,84)(24,109,115,79)(37,97,43,75)(38,98,44,76)(39,99,45,77)(40,100,46,78)(41,101,47,73)(42,102,48,74)(85,144,108,121)(86,139,103,122)(87,140,104,123)(88,141,105,124)(89,142,106,125)(90,143,107,126), (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,131)(8,132)(9,127)(10,128)(11,129)(12,130)(13,66)(14,61)(15,62)(16,63)(17,64)(18,65)(19,110)(20,111)(21,112)(22,113)(23,114)(24,109)(25,138)(26,133)(27,134)(28,135)(29,136)(30,137)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,75)(38,76)(39,77)(40,78)(41,73)(42,74)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(55,93)(56,94)(57,95)(58,96)(59,91)(60,92)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144), (1,41,60)(2,42,55)(3,37,56)(4,38,57)(5,39,58)(6,40,59)(7,125,21)(8,126,22)(9,121,23)(10,122,24)(11,123,19)(12,124,20)(13,36,44)(14,31,45)(15,32,46)(16,33,47)(17,34,48)(18,35,43)(25,140,116)(26,141,117)(27,142,118)(28,143,119)(29,144,120)(30,139,115)(49,78,91)(50,73,92)(51,74,93)(52,75,94)(53,76,95)(54,77,96)(61,67,99)(62,68,100)(63,69,101)(64,70,102)(65,71,97)(66,72,98)(79,137,103)(80,138,104)(81,133,105)(82,134,106)(83,135,107)(84,136,108)(85,114,127)(86,109,128)(87,110,129)(88,111,130)(89,112,131)(90,113,132), (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), (1,85,4,88)(2,90,5,87)(3,89,6,86)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,81,16,84)(14,80,17,83)(15,79,18,82)(19,93,22,96)(20,92,23,95)(21,91,24,94)(25,102,28,99)(26,101,29,98)(27,100,30,97)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,131,40,128)(38,130,41,127)(39,129,42,132)(43,134,46,137)(44,133,47,136)(45,138,48,135)(49,122,52,125)(50,121,53,124)(51,126,54,123)(55,113,58,110)(56,112,59,109)(57,111,60,114)(61,116,64,119)(62,115,65,118)(63,120,66,117)(67,140,70,143)(68,139,71,142)(69,144,72,141)>;
G:=Group( (1,69,33,50)(2,70,34,51)(3,71,35,52)(4,72,36,53)(5,67,31,54)(6,68,32,49)(7,131,27,134)(8,132,28,135)(9,127,29,136)(10,128,30,137)(11,129,25,138)(12,130,26,133)(13,95,57,66)(14,96,58,61)(15,91,59,62)(16,92,60,63)(17,93,55,64)(18,94,56,65)(19,110,116,80)(20,111,117,81)(21,112,118,82)(22,113,119,83)(23,114,120,84)(24,109,115,79)(37,97,43,75)(38,98,44,76)(39,99,45,77)(40,100,46,78)(41,101,47,73)(42,102,48,74)(85,144,108,121)(86,139,103,122)(87,140,104,123)(88,141,105,124)(89,142,106,125)(90,143,107,126), (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,131)(8,132)(9,127)(10,128)(11,129)(12,130)(13,66)(14,61)(15,62)(16,63)(17,64)(18,65)(19,110)(20,111)(21,112)(22,113)(23,114)(24,109)(25,138)(26,133)(27,134)(28,135)(29,136)(30,137)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,75)(38,76)(39,77)(40,78)(41,73)(42,74)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(55,93)(56,94)(57,95)(58,96)(59,91)(60,92)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144), (1,41,60)(2,42,55)(3,37,56)(4,38,57)(5,39,58)(6,40,59)(7,125,21)(8,126,22)(9,121,23)(10,122,24)(11,123,19)(12,124,20)(13,36,44)(14,31,45)(15,32,46)(16,33,47)(17,34,48)(18,35,43)(25,140,116)(26,141,117)(27,142,118)(28,143,119)(29,144,120)(30,139,115)(49,78,91)(50,73,92)(51,74,93)(52,75,94)(53,76,95)(54,77,96)(61,67,99)(62,68,100)(63,69,101)(64,70,102)(65,71,97)(66,72,98)(79,137,103)(80,138,104)(81,133,105)(82,134,106)(83,135,107)(84,136,108)(85,114,127)(86,109,128)(87,110,129)(88,111,130)(89,112,131)(90,113,132), (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), (1,85,4,88)(2,90,5,87)(3,89,6,86)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,81,16,84)(14,80,17,83)(15,79,18,82)(19,93,22,96)(20,92,23,95)(21,91,24,94)(25,102,28,99)(26,101,29,98)(27,100,30,97)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,131,40,128)(38,130,41,127)(39,129,42,132)(43,134,46,137)(44,133,47,136)(45,138,48,135)(49,122,52,125)(50,121,53,124)(51,126,54,123)(55,113,58,110)(56,112,59,109)(57,111,60,114)(61,116,64,119)(62,115,65,118)(63,120,66,117)(67,140,70,143)(68,139,71,142)(69,144,72,141) );
G=PermutationGroup([[(1,69,33,50),(2,70,34,51),(3,71,35,52),(4,72,36,53),(5,67,31,54),(6,68,32,49),(7,131,27,134),(8,132,28,135),(9,127,29,136),(10,128,30,137),(11,129,25,138),(12,130,26,133),(13,95,57,66),(14,96,58,61),(15,91,59,62),(16,92,60,63),(17,93,55,64),(18,94,56,65),(19,110,116,80),(20,111,117,81),(21,112,118,82),(22,113,119,83),(23,114,120,84),(24,109,115,79),(37,97,43,75),(38,98,44,76),(39,99,45,77),(40,100,46,78),(41,101,47,73),(42,102,48,74),(85,144,108,121),(86,139,103,122),(87,140,104,123),(88,141,105,124),(89,142,106,125),(90,143,107,126)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,49),(7,131),(8,132),(9,127),(10,128),(11,129),(12,130),(13,66),(14,61),(15,62),(16,63),(17,64),(18,65),(19,110),(20,111),(21,112),(22,113),(23,114),(24,109),(25,138),(26,133),(27,134),(28,135),(29,136),(30,137),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,75),(38,76),(39,77),(40,78),(41,73),(42,74),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(55,93),(56,94),(57,95),(58,96),(59,91),(60,92),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(85,121),(86,122),(87,123),(88,124),(89,125),(90,126),(103,139),(104,140),(105,141),(106,142),(107,143),(108,144)], [(1,41,60),(2,42,55),(3,37,56),(4,38,57),(5,39,58),(6,40,59),(7,125,21),(8,126,22),(9,121,23),(10,122,24),(11,123,19),(12,124,20),(13,36,44),(14,31,45),(15,32,46),(16,33,47),(17,34,48),(18,35,43),(25,140,116),(26,141,117),(27,142,118),(28,143,119),(29,144,120),(30,139,115),(49,78,91),(50,73,92),(51,74,93),(52,75,94),(53,76,95),(54,77,96),(61,67,99),(62,68,100),(63,69,101),(64,70,102),(65,71,97),(66,72,98),(79,137,103),(80,138,104),(81,133,105),(82,134,106),(83,135,107),(84,136,108),(85,114,127),(86,109,128),(87,110,129),(88,111,130),(89,112,131),(90,113,132)], [(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)], [(1,85,4,88),(2,90,5,87),(3,89,6,86),(7,78,10,75),(8,77,11,74),(9,76,12,73),(13,81,16,84),(14,80,17,83),(15,79,18,82),(19,93,22,96),(20,92,23,95),(21,91,24,94),(25,102,28,99),(26,101,29,98),(27,100,30,97),(31,104,34,107),(32,103,35,106),(33,108,36,105),(37,131,40,128),(38,130,41,127),(39,129,42,132),(43,134,46,137),(44,133,47,136),(45,138,48,135),(49,122,52,125),(50,121,53,124),(51,126,54,123),(55,113,58,110),(56,112,59,109),(57,111,60,114),(61,116,64,119),(62,115,65,118),(63,120,66,117),(67,140,70,143),(68,139,71,142),(69,144,72,141)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | ··· | 6L | 6M | ··· | 6AB | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | Dic3 | D6 | C4○D4 | S3×D4 | D4⋊2S3 |
| kernel | D4×C3⋊Dic3 | C4×C3⋊Dic3 | C12⋊Dic3 | C62⋊5C4 | C22×C3⋊Dic3 | D4×C3×C6 | D4×C32 | C6×D4 | C3⋊Dic3 | C2×C12 | C3×D4 | C22×C6 | C3×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 4 | 2 | 4 | 16 | 8 | 2 | 4 | 4 |
Matrix representation of D4×C3⋊Dic3 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 6 | 3 | 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[9,6,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[5,0,0,0,0,0,8,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;
D4×C3⋊Dic3 in GAP, Magma, Sage, TeX
D_4\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("D4xC3:Dic3"); // GroupNames label
G:=SmallGroup(288,791);
// by ID
G=gap.SmallGroup(288,791);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,219,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^3=d^6=1,e^2=d^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations