metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊6C4⋊15C2, C4⋊C4.154D10, Q8⋊C4⋊13D5, (C2×C8).125D10, (C2×Q8).21D10, D10⋊1C8.6C2, C10.50(C4○D8), C4.58(C4○D20), Q8⋊Dic5⋊10C2, C10.D8⋊12C2, D10⋊3Q8.6C2, (C22×D5).29D4, C22.205(D4×D5), C20.164(C4○D4), C4.89(D4⋊2D5), (C2×C20).255C23, (C2×C40).136C22, (C2×Dic5).213D4, C5⋊3(C23.20D4), C4⋊Dic5.99C22, (Q8×C10).38C22, C2.17(Q16⋊D5), C10.63(C8.C22), C2.19(SD16⋊3D5), C2.19(D10.12D4), C10.27(C22.D4), C4⋊C4⋊7D5.3C2, (C2×C4×D5).31C22, (C5×Q8⋊C4)⋊13C2, (C2×C10).268(C2×D4), (C5×C4⋊C4).56C22, (C2×C5⋊2C8).45C22, (C2×C4).362(C22×D5), SmallGroup(320,442)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊1C8.C2
G = < a,b,c,d,e | a2=b4=1, c2=b2, d10=ab2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=ab-1c, ebe-1=b-1c, dcd-1=ece-1=b2c, ede-1=b2d9 >
Subgroups: 366 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, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.20D4, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, Q8×C10, C10.D8, C40⋊6C4, D10⋊1C8, Q8⋊Dic5, C5×Q8⋊C4, C4⋊C4⋊7D5, D10⋊3Q8, D10⋊1C8.C2
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, D4⋊2D5, D10.12D4, SD16⋊3D5, Q16⋊D5, D10⋊1C8.C2
►On 160 points
Generators in S160
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 121)(19 122)(20 123)(21 160)(22 141)(23 142)(24 143)(25 144)(26 145)(27 146)(28 147)(29 148)(30 149)(31 150)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(41 96)(42 97)(43 98)(44 99)(45 100)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 117)(62 118)(63 119)(64 120)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)
(1 63 134 109)(2 151 135 22)(3 65 136 111)(4 153 137 24)(5 67 138 113)(6 155 139 26)(7 69 140 115)(8 157 121 28)(9 71 122 117)(10 159 123 30)(11 73 124 119)(12 141 125 32)(13 75 126 101)(14 143 127 34)(15 77 128 103)(16 145 129 36)(17 79 130 105)(18 147 131 38)(19 61 132 107)(20 149 133 40)(21 52 150 97)(23 54 152 99)(25 56 154 81)(27 58 156 83)(29 60 158 85)(31 42 160 87)(33 44 142 89)(35 46 144 91)(37 48 146 93)(39 50 148 95)(41 118 86 72)(43 120 88 74)(45 102 90 76)(47 104 92 78)(49 106 94 80)(51 108 96 62)(53 110 98 64)(55 112 100 66)(57 114 82 68)(59 116 84 70)
(1 87 134 42)(2 43 135 88)(3 89 136 44)(4 45 137 90)(5 91 138 46)(6 47 139 92)(7 93 140 48)(8 49 121 94)(9 95 122 50)(10 51 123 96)(11 97 124 52)(12 53 125 98)(13 99 126 54)(14 55 127 100)(15 81 128 56)(16 57 129 82)(17 83 130 58)(18 59 131 84)(19 85 132 60)(20 41 133 86)(21 73 150 119)(22 120 151 74)(23 75 152 101)(24 102 153 76)(25 77 154 103)(26 104 155 78)(27 79 156 105)(28 106 157 80)(29 61 158 107)(30 108 159 62)(31 63 160 109)(32 110 141 64)(33 65 142 111)(34 112 143 66)(35 67 144 113)(36 114 145 68)(37 69 146 115)(38 116 147 70)(39 71 148 117)(40 118 149 72)
(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 10 124 133)(2 132 125 9)(3 8 126 131)(4 130 127 7)(5 6 128 129)(11 20 134 123)(12 122 135 19)(13 18 136 121)(14 140 137 17)(15 16 138 139)(21 149 160 30)(22 29 141 148)(23 147 142 28)(24 27 143 146)(25 145 144 26)(31 159 150 40)(32 39 151 158)(33 157 152 38)(34 37 153 156)(35 155 154 36)(41 87 96 52)(42 51 97 86)(43 85 98 50)(44 49 99 84)(45 83 100 48)(46 47 81 82)(53 95 88 60)(54 59 89 94)(55 93 90 58)(56 57 91 92)(61 110 117 74)(62 73 118 109)(63 108 119 72)(64 71 120 107)(65 106 101 70)(66 69 102 105)(67 104 103 68)(75 116 111 80)(76 79 112 115)(77 114 113 78)
G:=sub<Sym(160)| (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,121)(19,122)(20,123)(21,160)(22,141)(23,142)(24,143)(25,144)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(41,96)(42,97)(43,98)(44,99)(45,100)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,117)(62,118)(63,119)(64,120)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116), (1,63,134,109)(2,151,135,22)(3,65,136,111)(4,153,137,24)(5,67,138,113)(6,155,139,26)(7,69,140,115)(8,157,121,28)(9,71,122,117)(10,159,123,30)(11,73,124,119)(12,141,125,32)(13,75,126,101)(14,143,127,34)(15,77,128,103)(16,145,129,36)(17,79,130,105)(18,147,131,38)(19,61,132,107)(20,149,133,40)(21,52,150,97)(23,54,152,99)(25,56,154,81)(27,58,156,83)(29,60,158,85)(31,42,160,87)(33,44,142,89)(35,46,144,91)(37,48,146,93)(39,50,148,95)(41,118,86,72)(43,120,88,74)(45,102,90,76)(47,104,92,78)(49,106,94,80)(51,108,96,62)(53,110,98,64)(55,112,100,66)(57,114,82,68)(59,116,84,70), (1,87,134,42)(2,43,135,88)(3,89,136,44)(4,45,137,90)(5,91,138,46)(6,47,139,92)(7,93,140,48)(8,49,121,94)(9,95,122,50)(10,51,123,96)(11,97,124,52)(12,53,125,98)(13,99,126,54)(14,55,127,100)(15,81,128,56)(16,57,129,82)(17,83,130,58)(18,59,131,84)(19,85,132,60)(20,41,133,86)(21,73,150,119)(22,120,151,74)(23,75,152,101)(24,102,153,76)(25,77,154,103)(26,104,155,78)(27,79,156,105)(28,106,157,80)(29,61,158,107)(30,108,159,62)(31,63,160,109)(32,110,141,64)(33,65,142,111)(34,112,143,66)(35,67,144,113)(36,114,145,68)(37,69,146,115)(38,116,147,70)(39,71,148,117)(40,118,149,72), (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), (1,10,124,133)(2,132,125,9)(3,8,126,131)(4,130,127,7)(5,6,128,129)(11,20,134,123)(12,122,135,19)(13,18,136,121)(14,140,137,17)(15,16,138,139)(21,149,160,30)(22,29,141,148)(23,147,142,28)(24,27,143,146)(25,145,144,26)(31,159,150,40)(32,39,151,158)(33,157,152,38)(34,37,153,156)(35,155,154,36)(41,87,96,52)(42,51,97,86)(43,85,98,50)(44,49,99,84)(45,83,100,48)(46,47,81,82)(53,95,88,60)(54,59,89,94)(55,93,90,58)(56,57,91,92)(61,110,117,74)(62,73,118,109)(63,108,119,72)(64,71,120,107)(65,106,101,70)(66,69,102,105)(67,104,103,68)(75,116,111,80)(76,79,112,115)(77,114,113,78)>;
G:=Group( (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,121)(19,122)(20,123)(21,160)(22,141)(23,142)(24,143)(25,144)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(41,96)(42,97)(43,98)(44,99)(45,100)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,117)(62,118)(63,119)(64,120)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116), (1,63,134,109)(2,151,135,22)(3,65,136,111)(4,153,137,24)(5,67,138,113)(6,155,139,26)(7,69,140,115)(8,157,121,28)(9,71,122,117)(10,159,123,30)(11,73,124,119)(12,141,125,32)(13,75,126,101)(14,143,127,34)(15,77,128,103)(16,145,129,36)(17,79,130,105)(18,147,131,38)(19,61,132,107)(20,149,133,40)(21,52,150,97)(23,54,152,99)(25,56,154,81)(27,58,156,83)(29,60,158,85)(31,42,160,87)(33,44,142,89)(35,46,144,91)(37,48,146,93)(39,50,148,95)(41,118,86,72)(43,120,88,74)(45,102,90,76)(47,104,92,78)(49,106,94,80)(51,108,96,62)(53,110,98,64)(55,112,100,66)(57,114,82,68)(59,116,84,70), (1,87,134,42)(2,43,135,88)(3,89,136,44)(4,45,137,90)(5,91,138,46)(6,47,139,92)(7,93,140,48)(8,49,121,94)(9,95,122,50)(10,51,123,96)(11,97,124,52)(12,53,125,98)(13,99,126,54)(14,55,127,100)(15,81,128,56)(16,57,129,82)(17,83,130,58)(18,59,131,84)(19,85,132,60)(20,41,133,86)(21,73,150,119)(22,120,151,74)(23,75,152,101)(24,102,153,76)(25,77,154,103)(26,104,155,78)(27,79,156,105)(28,106,157,80)(29,61,158,107)(30,108,159,62)(31,63,160,109)(32,110,141,64)(33,65,142,111)(34,112,143,66)(35,67,144,113)(36,114,145,68)(37,69,146,115)(38,116,147,70)(39,71,148,117)(40,118,149,72), (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), (1,10,124,133)(2,132,125,9)(3,8,126,131)(4,130,127,7)(5,6,128,129)(11,20,134,123)(12,122,135,19)(13,18,136,121)(14,140,137,17)(15,16,138,139)(21,149,160,30)(22,29,141,148)(23,147,142,28)(24,27,143,146)(25,145,144,26)(31,159,150,40)(32,39,151,158)(33,157,152,38)(34,37,153,156)(35,155,154,36)(41,87,96,52)(42,51,97,86)(43,85,98,50)(44,49,99,84)(45,83,100,48)(46,47,81,82)(53,95,88,60)(54,59,89,94)(55,93,90,58)(56,57,91,92)(61,110,117,74)(62,73,118,109)(63,108,119,72)(64,71,120,107)(65,106,101,70)(66,69,102,105)(67,104,103,68)(75,116,111,80)(76,79,112,115)(77,114,113,78) );
G=PermutationGroup([[(1,124),(2,125),(3,126),(4,127),(5,128),(6,129),(7,130),(8,131),(9,132),(10,133),(11,134),(12,135),(13,136),(14,137),(15,138),(16,139),(17,140),(18,121),(19,122),(20,123),(21,160),(22,141),(23,142),(24,143),(25,144),(26,145),(27,146),(28,147),(29,148),(30,149),(31,150),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(41,96),(42,97),(43,98),(44,99),(45,100),(46,81),(47,82),(48,83),(49,84),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(57,92),(58,93),(59,94),(60,95),(61,117),(62,118),(63,119),(64,120),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(79,115),(80,116)], [(1,63,134,109),(2,151,135,22),(3,65,136,111),(4,153,137,24),(5,67,138,113),(6,155,139,26),(7,69,140,115),(8,157,121,28),(9,71,122,117),(10,159,123,30),(11,73,124,119),(12,141,125,32),(13,75,126,101),(14,143,127,34),(15,77,128,103),(16,145,129,36),(17,79,130,105),(18,147,131,38),(19,61,132,107),(20,149,133,40),(21,52,150,97),(23,54,152,99),(25,56,154,81),(27,58,156,83),(29,60,158,85),(31,42,160,87),(33,44,142,89),(35,46,144,91),(37,48,146,93),(39,50,148,95),(41,118,86,72),(43,120,88,74),(45,102,90,76),(47,104,92,78),(49,106,94,80),(51,108,96,62),(53,110,98,64),(55,112,100,66),(57,114,82,68),(59,116,84,70)], [(1,87,134,42),(2,43,135,88),(3,89,136,44),(4,45,137,90),(5,91,138,46),(6,47,139,92),(7,93,140,48),(8,49,121,94),(9,95,122,50),(10,51,123,96),(11,97,124,52),(12,53,125,98),(13,99,126,54),(14,55,127,100),(15,81,128,56),(16,57,129,82),(17,83,130,58),(18,59,131,84),(19,85,132,60),(20,41,133,86),(21,73,150,119),(22,120,151,74),(23,75,152,101),(24,102,153,76),(25,77,154,103),(26,104,155,78),(27,79,156,105),(28,106,157,80),(29,61,158,107),(30,108,159,62),(31,63,160,109),(32,110,141,64),(33,65,142,111),(34,112,143,66),(35,67,144,113),(36,114,145,68),(37,69,146,115),(38,116,147,70),(39,71,148,117),(40,118,149,72)], [(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)], [(1,10,124,133),(2,132,125,9),(3,8,126,131),(4,130,127,7),(5,6,128,129),(11,20,134,123),(12,122,135,19),(13,18,136,121),(14,140,137,17),(15,16,138,139),(21,149,160,30),(22,29,141,148),(23,147,142,28),(24,27,143,146),(25,145,144,26),(31,159,150,40),(32,39,151,158),(33,157,152,38),(34,37,153,156),(35,155,154,36),(41,87,96,52),(42,51,97,86),(43,85,98,50),(44,49,99,84),(45,83,100,48),(46,47,81,82),(53,95,88,60),(54,59,89,94),(55,93,90,58),(56,57,91,92),(61,110,117,74),(62,73,118,109),(63,108,119,72),(64,71,120,107),(65,106,101,70),(66,69,102,105),(67,104,103,68),(75,116,111,80),(76,79,112,115),(77,114,113,78)]])
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 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | C8.C22 | D4⋊2D5 | D4×D5 | SD16⋊3D5 | Q16⋊D5 |
| kernel | D10⋊1C8.C2 | C10.D8 | C40⋊6C4 | D10⋊1C8 | Q8⋊Dic5 | C5×Q8⋊C4 | C4⋊C4⋊7D5 | D10⋊3Q8 | C2×Dic5 | C22×D5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊1C8.C2 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 24 | 1 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 33 | 24 |
| 0 | 0 | 40 | 8 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 21 |
| 0 | 0 | 0 | 9 |
| 19 | 19 | 0 | 0 |
| 22 | 9 | 0 | 0 |
| 0 | 0 | 30 | 9 |
| 0 | 0 | 14 | 11 |
| 9 | 22 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 30 | 3 |
| 0 | 0 | 14 | 11 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[24,40,0,0,1,17,0,0,0,0,33,40,0,0,24,8],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,21,9],[19,22,0,0,19,9,0,0,0,0,30,14,0,0,9,11],[9,0,0,0,22,32,0,0,0,0,30,14,0,0,3,11] >;
D10⋊1C8.C2 in GAP, Magma, Sage, TeX
D_{10}\rtimes_1C_8.C_2 % in TeX
G:=Group("D10:1C8.C2"); // GroupNames label
G:=SmallGroup(320,442);
// by ID
G=gap.SmallGroup(320,442);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,926,219,184,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=b^2,d^10=a*b^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=a*b^-1*c,e*b*e^-1=b^-1*c,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=b^2*d^9>;
// generators/relations