metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2.D8⋊7D5, C4⋊C4.53D10, (C2×C8).30D10, C20.43(C4○D4), C4.83(C4○D20), C10.30(C4○D8), C10.Q16⋊22C2, C20.Q8⋊22C2, (C22×D5).38D4, C22.234(D4×D5), D10⋊1C8.11C2, C20.44D4⋊27C2, C2.15(D8⋊3D5), (C2×C40).244C22, (C2×C20).304C23, D10⋊2Q8.10C2, C4.31(Q8⋊2D5), (C2×Dic5).225D4, C5⋊5(C23.20D4), C2.24(Q16⋊D5), C10.72(C8.C22), C4⋊Dic5.127C22, (C2×Dic10).96C22, C2.18(D10.13D4), C10.48(C22.D4), (C5×C2.D8)⋊14C2, C4⋊C4⋊7D5.9C2, (C2×C4×D5).46C22, (C2×C10).309(C2×D4), (C5×C4⋊C4).97C22, (C2×C5⋊2C8).73C22, (C2×C4).407(C22×D5), SmallGroup(320,515)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D8⋊7D5
G = < a,b,c,d,e | a2=b8=d5=e2=1, c2=a, ebe=ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, cd=dc, ece=ab4c, ede=d-1 >
Subgroups: 382 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C5⋊2C8, C40, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C23.20D4, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C20.Q8, C10.Q16, C20.44D4, D10⋊1C8, C5×C2.D8, C4⋊C4⋊7D5, D10⋊2Q8, C2.D8⋊7D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8.C22, C22×D5, C23.20D4, C4○D20, D4×D5, Q8⋊2D5, D10.13D4, D8⋊3D5, Q16⋊D5, C2.D8⋊7D5
►On 160 points
Generators in S160
(1 77)(2 78)(3 79)(4 80)(5 73)(6 74)(7 75)(8 76)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 93)(18 94)(19 95)(20 96)(21 89)(22 90)(23 91)(24 92)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 101)(42 102)(43 103)(44 104)(45 97)(46 98)(47 99)(48 100)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 124)(66 125)(67 126)(68 127)(69 128)(70 121)(71 122)(72 123)(105 138)(106 139)(107 140)(108 141)(109 142)(110 143)(111 144)(112 137)(145 157)(146 158)(147 159)(148 160)(149 153)(150 154)(151 155)(152 156)
(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)
(1 152 77 156)(2 151 78 155)(3 150 79 154)(4 149 80 153)(5 148 73 160)(6 147 74 159)(7 146 75 158)(8 145 76 157)(9 127 117 68)(10 126 118 67)(11 125 119 66)(12 124 120 65)(13 123 113 72)(14 122 114 71)(15 121 115 70)(16 128 116 69)(17 43 93 103)(18 42 94 102)(19 41 95 101)(20 48 96 100)(21 47 89 99)(22 46 90 98)(23 45 91 97)(24 44 92 104)(25 105 36 138)(26 112 37 137)(27 111 38 144)(28 110 39 143)(29 109 40 142)(30 108 33 141)(31 107 34 140)(32 106 35 139)(49 87 129 59)(50 86 130 58)(51 85 131 57)(52 84 132 64)(53 83 133 63)(54 82 134 62)(55 81 135 61)(56 88 136 60)
(1 61 24 37 66)(2 62 17 38 67)(3 63 18 39 68)(4 64 19 40 69)(5 57 20 33 70)(6 58 21 34 71)(7 59 22 35 72)(8 60 23 36 65)(9 150 53 42 143)(10 151 54 43 144)(11 152 55 44 137)(12 145 56 45 138)(13 146 49 46 139)(14 147 50 47 140)(15 148 51 48 141)(16 149 52 41 142)(25 124 76 88 91)(26 125 77 81 92)(27 126 78 82 93)(28 127 79 83 94)(29 128 80 84 95)(30 121 73 85 96)(31 122 74 86 89)(32 123 75 87 90)(97 105 120 157 136)(98 106 113 158 129)(99 107 114 159 130)(100 108 115 160 131)(101 109 116 153 132)(102 110 117 154 133)(103 111 118 155 134)(104 112 119 156 135)
(1 66)(2 126)(3 68)(4 128)(5 70)(6 122)(7 72)(8 124)(9 158)(10 147)(11 160)(12 149)(13 154)(14 151)(15 156)(16 145)(17 93)(19 95)(21 89)(23 91)(25 60)(26 81)(27 62)(28 83)(29 64)(30 85)(31 58)(32 87)(33 57)(34 86)(35 59)(36 88)(37 61)(38 82)(39 63)(40 84)(41 45)(42 98)(43 47)(44 100)(46 102)(48 104)(49 110)(50 144)(51 112)(52 138)(53 106)(54 140)(55 108)(56 142)(65 76)(67 78)(69 80)(71 74)(73 121)(75 123)(77 125)(79 127)(97 101)(99 103)(105 132)(107 134)(109 136)(111 130)(113 150)(114 155)(115 152)(116 157)(117 146)(118 159)(119 148)(120 153)(129 143)(131 137)(133 139)(135 141)
G:=sub<Sym(160)| (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,93)(18,94)(19,95)(20,96)(21,89)(22,90)(23,91)(24,92)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,101)(42,102)(43,103)(44,104)(45,97)(46,98)(47,99)(48,100)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(65,124)(66,125)(67,126)(68,127)(69,128)(70,121)(71,122)(72,123)(105,138)(106,139)(107,140)(108,141)(109,142)(110,143)(111,144)(112,137)(145,157)(146,158)(147,159)(148,160)(149,153)(150,154)(151,155)(152,156), (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), (1,152,77,156)(2,151,78,155)(3,150,79,154)(4,149,80,153)(5,148,73,160)(6,147,74,159)(7,146,75,158)(8,145,76,157)(9,127,117,68)(10,126,118,67)(11,125,119,66)(12,124,120,65)(13,123,113,72)(14,122,114,71)(15,121,115,70)(16,128,116,69)(17,43,93,103)(18,42,94,102)(19,41,95,101)(20,48,96,100)(21,47,89,99)(22,46,90,98)(23,45,91,97)(24,44,92,104)(25,105,36,138)(26,112,37,137)(27,111,38,144)(28,110,39,143)(29,109,40,142)(30,108,33,141)(31,107,34,140)(32,106,35,139)(49,87,129,59)(50,86,130,58)(51,85,131,57)(52,84,132,64)(53,83,133,63)(54,82,134,62)(55,81,135,61)(56,88,136,60), (1,61,24,37,66)(2,62,17,38,67)(3,63,18,39,68)(4,64,19,40,69)(5,57,20,33,70)(6,58,21,34,71)(7,59,22,35,72)(8,60,23,36,65)(9,150,53,42,143)(10,151,54,43,144)(11,152,55,44,137)(12,145,56,45,138)(13,146,49,46,139)(14,147,50,47,140)(15,148,51,48,141)(16,149,52,41,142)(25,124,76,88,91)(26,125,77,81,92)(27,126,78,82,93)(28,127,79,83,94)(29,128,80,84,95)(30,121,73,85,96)(31,122,74,86,89)(32,123,75,87,90)(97,105,120,157,136)(98,106,113,158,129)(99,107,114,159,130)(100,108,115,160,131)(101,109,116,153,132)(102,110,117,154,133)(103,111,118,155,134)(104,112,119,156,135), (1,66)(2,126)(3,68)(4,128)(5,70)(6,122)(7,72)(8,124)(9,158)(10,147)(11,160)(12,149)(13,154)(14,151)(15,156)(16,145)(17,93)(19,95)(21,89)(23,91)(25,60)(26,81)(27,62)(28,83)(29,64)(30,85)(31,58)(32,87)(33,57)(34,86)(35,59)(36,88)(37,61)(38,82)(39,63)(40,84)(41,45)(42,98)(43,47)(44,100)(46,102)(48,104)(49,110)(50,144)(51,112)(52,138)(53,106)(54,140)(55,108)(56,142)(65,76)(67,78)(69,80)(71,74)(73,121)(75,123)(77,125)(79,127)(97,101)(99,103)(105,132)(107,134)(109,136)(111,130)(113,150)(114,155)(115,152)(116,157)(117,146)(118,159)(119,148)(120,153)(129,143)(131,137)(133,139)(135,141)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,117)(10,118)(11,119)(12,120)(13,113)(14,114)(15,115)(16,116)(17,93)(18,94)(19,95)(20,96)(21,89)(22,90)(23,91)(24,92)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,101)(42,102)(43,103)(44,104)(45,97)(46,98)(47,99)(48,100)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,85)(58,86)(59,87)(60,88)(61,81)(62,82)(63,83)(64,84)(65,124)(66,125)(67,126)(68,127)(69,128)(70,121)(71,122)(72,123)(105,138)(106,139)(107,140)(108,141)(109,142)(110,143)(111,144)(112,137)(145,157)(146,158)(147,159)(148,160)(149,153)(150,154)(151,155)(152,156), (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), (1,152,77,156)(2,151,78,155)(3,150,79,154)(4,149,80,153)(5,148,73,160)(6,147,74,159)(7,146,75,158)(8,145,76,157)(9,127,117,68)(10,126,118,67)(11,125,119,66)(12,124,120,65)(13,123,113,72)(14,122,114,71)(15,121,115,70)(16,128,116,69)(17,43,93,103)(18,42,94,102)(19,41,95,101)(20,48,96,100)(21,47,89,99)(22,46,90,98)(23,45,91,97)(24,44,92,104)(25,105,36,138)(26,112,37,137)(27,111,38,144)(28,110,39,143)(29,109,40,142)(30,108,33,141)(31,107,34,140)(32,106,35,139)(49,87,129,59)(50,86,130,58)(51,85,131,57)(52,84,132,64)(53,83,133,63)(54,82,134,62)(55,81,135,61)(56,88,136,60), (1,61,24,37,66)(2,62,17,38,67)(3,63,18,39,68)(4,64,19,40,69)(5,57,20,33,70)(6,58,21,34,71)(7,59,22,35,72)(8,60,23,36,65)(9,150,53,42,143)(10,151,54,43,144)(11,152,55,44,137)(12,145,56,45,138)(13,146,49,46,139)(14,147,50,47,140)(15,148,51,48,141)(16,149,52,41,142)(25,124,76,88,91)(26,125,77,81,92)(27,126,78,82,93)(28,127,79,83,94)(29,128,80,84,95)(30,121,73,85,96)(31,122,74,86,89)(32,123,75,87,90)(97,105,120,157,136)(98,106,113,158,129)(99,107,114,159,130)(100,108,115,160,131)(101,109,116,153,132)(102,110,117,154,133)(103,111,118,155,134)(104,112,119,156,135), (1,66)(2,126)(3,68)(4,128)(5,70)(6,122)(7,72)(8,124)(9,158)(10,147)(11,160)(12,149)(13,154)(14,151)(15,156)(16,145)(17,93)(19,95)(21,89)(23,91)(25,60)(26,81)(27,62)(28,83)(29,64)(30,85)(31,58)(32,87)(33,57)(34,86)(35,59)(36,88)(37,61)(38,82)(39,63)(40,84)(41,45)(42,98)(43,47)(44,100)(46,102)(48,104)(49,110)(50,144)(51,112)(52,138)(53,106)(54,140)(55,108)(56,142)(65,76)(67,78)(69,80)(71,74)(73,121)(75,123)(77,125)(79,127)(97,101)(99,103)(105,132)(107,134)(109,136)(111,130)(113,150)(114,155)(115,152)(116,157)(117,146)(118,159)(119,148)(120,153)(129,143)(131,137)(133,139)(135,141) );
G=PermutationGroup([[(1,77),(2,78),(3,79),(4,80),(5,73),(6,74),(7,75),(8,76),(9,117),(10,118),(11,119),(12,120),(13,113),(14,114),(15,115),(16,116),(17,93),(18,94),(19,95),(20,96),(21,89),(22,90),(23,91),(24,92),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35),(41,101),(42,102),(43,103),(44,104),(45,97),(46,98),(47,99),(48,100),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,85),(58,86),(59,87),(60,88),(61,81),(62,82),(63,83),(64,84),(65,124),(66,125),(67,126),(68,127),(69,128),(70,121),(71,122),(72,123),(105,138),(106,139),(107,140),(108,141),(109,142),(110,143),(111,144),(112,137),(145,157),(146,158),(147,159),(148,160),(149,153),(150,154),(151,155),(152,156)], [(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)], [(1,152,77,156),(2,151,78,155),(3,150,79,154),(4,149,80,153),(5,148,73,160),(6,147,74,159),(7,146,75,158),(8,145,76,157),(9,127,117,68),(10,126,118,67),(11,125,119,66),(12,124,120,65),(13,123,113,72),(14,122,114,71),(15,121,115,70),(16,128,116,69),(17,43,93,103),(18,42,94,102),(19,41,95,101),(20,48,96,100),(21,47,89,99),(22,46,90,98),(23,45,91,97),(24,44,92,104),(25,105,36,138),(26,112,37,137),(27,111,38,144),(28,110,39,143),(29,109,40,142),(30,108,33,141),(31,107,34,140),(32,106,35,139),(49,87,129,59),(50,86,130,58),(51,85,131,57),(52,84,132,64),(53,83,133,63),(54,82,134,62),(55,81,135,61),(56,88,136,60)], [(1,61,24,37,66),(2,62,17,38,67),(3,63,18,39,68),(4,64,19,40,69),(5,57,20,33,70),(6,58,21,34,71),(7,59,22,35,72),(8,60,23,36,65),(9,150,53,42,143),(10,151,54,43,144),(11,152,55,44,137),(12,145,56,45,138),(13,146,49,46,139),(14,147,50,47,140),(15,148,51,48,141),(16,149,52,41,142),(25,124,76,88,91),(26,125,77,81,92),(27,126,78,82,93),(28,127,79,83,94),(29,128,80,84,95),(30,121,73,85,96),(31,122,74,86,89),(32,123,75,87,90),(97,105,120,157,136),(98,106,113,158,129),(99,107,114,159,130),(100,108,115,160,131),(101,109,116,153,132),(102,110,117,154,133),(103,111,118,155,134),(104,112,119,156,135)], [(1,66),(2,126),(3,68),(4,128),(5,70),(6,122),(7,72),(8,124),(9,158),(10,147),(11,160),(12,149),(13,154),(14,151),(15,156),(16,145),(17,93),(19,95),(21,89),(23,91),(25,60),(26,81),(27,62),(28,83),(29,64),(30,85),(31,58),(32,87),(33,57),(34,86),(35,59),(36,88),(37,61),(38,82),(39,63),(40,84),(41,45),(42,98),(43,47),(44,100),(46,102),(48,104),(49,110),(50,144),(51,112),(52,138),(53,106),(54,140),(55,108),(56,142),(65,76),(67,78),(69,80),(71,74),(73,121),(75,123),(77,125),(79,127),(97,101),(99,103),(105,132),(107,134),(109,136),(111,130),(113,150),(114,155),(115,152),(116,157),(117,146),(118,159),(119,148),(120,153),(129,143),(131,137),(133,139),(135,141)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | C4○D20 | C8.C22 | Q8⋊2D5 | D4×D5 | D8⋊3D5 | Q16⋊D5 |
| kernel | C2.D8⋊7D5 | C20.Q8 | C10.Q16 | C20.44D4 | D10⋊1C8 | C5×C2.D8 | C4⋊C4⋊7D5 | D10⋊2Q8 | C2×Dic5 | C22×D5 | C2.D8 | C20 | C4⋊C4 | C2×C8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C2.D8⋊7D5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 23 | 35 | 0 | 0 |
| 6 | 18 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 34 | 14 |
| 2 | 28 | 0 | 0 |
| 13 | 39 | 0 | 0 |
| 0 | 0 | 15 | 35 |
| 0 | 0 | 10 | 26 |
| 6 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 5 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[23,6,0,0,35,18,0,0,0,0,3,34,0,0,0,14],[2,13,0,0,28,39,0,0,0,0,15,10,0,0,35,26],[6,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,5,0,0,0,40] >;
C2.D8⋊7D5 in GAP, Magma, Sage, TeX
C_2.D_8\rtimes_7D_5
% in TeX
G:=Group("C2.D8:7D5"); // GroupNames label
G:=SmallGroup(320,515);
// by ID
G=gap.SmallGroup(320,515);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,254,219,268,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=d^5=e^2=1,c^2=a,e*b*e=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,c*d=d*c,e*c*e=a*b^4*c,e*d*e=d^-1>;
// generators/relations