direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C4.Q8, D10.22SD16, (C8×D5)⋊3C4, C8⋊14(C4×D5), C40⋊24(C2×C4), C4.23(Q8×D5), C40⋊6C4⋊24C2, (C4×D5).13Q8, C20.12(C2×Q8), C2.6(D5×SD16), C4⋊C4.160D10, (C2×C8).258D10, C22.83(D4×D5), D10.36(C4⋊C4), C20.Q8⋊14C2, C10.37(C2×SD16), Dic5.15(C4⋊C4), C20.101(C22×C4), (C2×C40).159C22, (C2×C20).275C23, (C2×Dic5).143D4, (C22×D5).154D4, C4⋊Dic5.107C22, C5⋊2(C2×C4.Q8), (D5×C2×C8).7C2, C4.76(C2×C4×D5), (D5×C4⋊C4).4C2, C2.11(D5×C4⋊C4), (C5×C4.Q8)⋊7C2, C5⋊2C8⋊27(C2×C4), C10.33(C2×C4⋊C4), (C4×D5).73(C2×C4), (C2×C10).280(C2×D4), (C5×C4⋊C4).68C22, (C2×C4×D5).301C22, (C2×C4).378(C22×D5), (C2×C5⋊2C8).234C22, SmallGroup(320,486)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C4.Q8
G = < a,b,c,d,e | a5=b2=c4=1, d4=c2, e2=c-1d2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >
Subgroups: 478 in 130 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, C20, D10, C2×C10, C4.Q8, C4.Q8, C2×C4⋊C4, C22×C8, C5⋊2C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C2×C4.Q8, C8×D5, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C20.Q8, C40⋊6C4, C5×C4.Q8, D5×C4⋊C4, D5×C2×C8, D5×C4.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, D10, C4.Q8, C2×C4⋊C4, C2×SD16, C4×D5, C22×D5, C2×C4.Q8, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D5×SD16, D5×C4.Q8
►On 160 points
Generators in S160
(1 113 23 133 39)(2 114 24 134 40)(3 115 17 135 33)(4 116 18 136 34)(5 117 19 129 35)(6 118 20 130 36)(7 119 21 131 37)(8 120 22 132 38)(9 47 88 55 91)(10 48 81 56 92)(11 41 82 49 93)(12 42 83 50 94)(13 43 84 51 95)(14 44 85 52 96)(15 45 86 53 89)(16 46 87 54 90)(25 66 60 108 145)(26 67 61 109 146)(27 68 62 110 147)(28 69 63 111 148)(29 70 64 112 149)(30 71 57 105 150)(31 72 58 106 151)(32 65 59 107 152)(73 137 159 124 98)(74 138 160 125 99)(75 139 153 126 100)(76 140 154 127 101)(77 141 155 128 102)(78 142 156 121 103)(79 143 157 122 104)(80 144 158 123 97)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 149)(26 150)(27 151)(28 152)(29 145)(30 146)(31 147)(32 148)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 86)(50 87)(51 88)(52 81)(53 82)(54 83)(55 84)(56 85)(57 61)(58 62)(59 63)(60 64)(65 111)(66 112)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 128)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 127)(97 101)(98 102)(99 103)(100 104)(113 129)(114 130)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 153)(144 154)
(1 30 5 26)(2 31 6 27)(3 32 7 28)(4 25 8 29)(9 104 13 100)(10 97 14 101)(11 98 15 102)(12 99 16 103)(17 59 21 63)(18 60 22 64)(19 61 23 57)(20 62 24 58)(33 152 37 148)(34 145 38 149)(35 146 39 150)(36 147 40 151)(41 73 45 77)(42 74 46 78)(43 75 47 79)(44 76 48 80)(49 159 53 155)(50 160 54 156)(51 153 55 157)(52 154 56 158)(65 119 69 115)(66 120 70 116)(67 113 71 117)(68 114 72 118)(81 144 85 140)(82 137 86 141)(83 138 87 142)(84 139 88 143)(89 128 93 124)(90 121 94 125)(91 122 95 126)(92 123 96 127)(105 129 109 133)(106 130 110 134)(107 131 111 135)(108 132 112 136)
(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 55 28 155)(2 50 29 158)(3 53 30 153)(4 56 31 156)(5 51 32 159)(6 54 25 154)(7 49 26 157)(8 52 27 160)(9 63 102 23)(10 58 103 18)(11 61 104 21)(12 64 97 24)(13 59 98 19)(14 62 99 22)(15 57 100 17)(16 60 101 20)(33 86 150 139)(34 81 151 142)(35 84 152 137)(36 87 145 140)(37 82 146 143)(38 85 147 138)(39 88 148 141)(40 83 149 144)(41 109 79 131)(42 112 80 134)(43 107 73 129)(44 110 74 132)(45 105 75 135)(46 108 76 130)(47 111 77 133)(48 106 78 136)(65 124 117 95)(66 127 118 90)(67 122 119 93)(68 125 120 96)(69 128 113 91)(70 123 114 94)(71 126 115 89)(72 121 116 92)
G:=sub<Sym(160)| (1,113,23,133,39)(2,114,24,134,40)(3,115,17,135,33)(4,116,18,136,34)(5,117,19,129,35)(6,118,20,130,36)(7,119,21,131,37)(8,120,22,132,38)(9,47,88,55,91)(10,48,81,56,92)(11,41,82,49,93)(12,42,83,50,94)(13,43,84,51,95)(14,44,85,52,96)(15,45,86,53,89)(16,46,87,54,90)(25,66,60,108,145)(26,67,61,109,146)(27,68,62,110,147)(28,69,63,111,148)(29,70,64,112,149)(30,71,57,105,150)(31,72,58,106,151)(32,65,59,107,152)(73,137,159,124,98)(74,138,160,125,99)(75,139,153,126,100)(76,140,154,127,101)(77,141,155,128,102)(78,142,156,121,103)(79,143,157,122,104)(80,144,158,123,97), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,149)(26,150)(27,151)(28,152)(29,145)(30,146)(31,147)(32,148)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,86)(50,87)(51,88)(52,81)(53,82)(54,83)(55,84)(56,85)(57,61)(58,62)(59,63)(60,64)(65,111)(66,112)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,128)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,127)(97,101)(98,102)(99,103)(100,104)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(137,155)(138,156)(139,157)(140,158)(141,159)(142,160)(143,153)(144,154), (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,104,13,100)(10,97,14,101)(11,98,15,102)(12,99,16,103)(17,59,21,63)(18,60,22,64)(19,61,23,57)(20,62,24,58)(33,152,37,148)(34,145,38,149)(35,146,39,150)(36,147,40,151)(41,73,45,77)(42,74,46,78)(43,75,47,79)(44,76,48,80)(49,159,53,155)(50,160,54,156)(51,153,55,157)(52,154,56,158)(65,119,69,115)(66,120,70,116)(67,113,71,117)(68,114,72,118)(81,144,85,140)(82,137,86,141)(83,138,87,142)(84,139,88,143)(89,128,93,124)(90,121,94,125)(91,122,95,126)(92,123,96,127)(105,129,109,133)(106,130,110,134)(107,131,111,135)(108,132,112,136), (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), (1,55,28,155)(2,50,29,158)(3,53,30,153)(4,56,31,156)(5,51,32,159)(6,54,25,154)(7,49,26,157)(8,52,27,160)(9,63,102,23)(10,58,103,18)(11,61,104,21)(12,64,97,24)(13,59,98,19)(14,62,99,22)(15,57,100,17)(16,60,101,20)(33,86,150,139)(34,81,151,142)(35,84,152,137)(36,87,145,140)(37,82,146,143)(38,85,147,138)(39,88,148,141)(40,83,149,144)(41,109,79,131)(42,112,80,134)(43,107,73,129)(44,110,74,132)(45,105,75,135)(46,108,76,130)(47,111,77,133)(48,106,78,136)(65,124,117,95)(66,127,118,90)(67,122,119,93)(68,125,120,96)(69,128,113,91)(70,123,114,94)(71,126,115,89)(72,121,116,92)>;
G:=Group( (1,113,23,133,39)(2,114,24,134,40)(3,115,17,135,33)(4,116,18,136,34)(5,117,19,129,35)(6,118,20,130,36)(7,119,21,131,37)(8,120,22,132,38)(9,47,88,55,91)(10,48,81,56,92)(11,41,82,49,93)(12,42,83,50,94)(13,43,84,51,95)(14,44,85,52,96)(15,45,86,53,89)(16,46,87,54,90)(25,66,60,108,145)(26,67,61,109,146)(27,68,62,110,147)(28,69,63,111,148)(29,70,64,112,149)(30,71,57,105,150)(31,72,58,106,151)(32,65,59,107,152)(73,137,159,124,98)(74,138,160,125,99)(75,139,153,126,100)(76,140,154,127,101)(77,141,155,128,102)(78,142,156,121,103)(79,143,157,122,104)(80,144,158,123,97), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,149)(26,150)(27,151)(28,152)(29,145)(30,146)(31,147)(32,148)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,86)(50,87)(51,88)(52,81)(53,82)(54,83)(55,84)(56,85)(57,61)(58,62)(59,63)(60,64)(65,111)(66,112)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,128)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,127)(97,101)(98,102)(99,103)(100,104)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(137,155)(138,156)(139,157)(140,158)(141,159)(142,160)(143,153)(144,154), (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,104,13,100)(10,97,14,101)(11,98,15,102)(12,99,16,103)(17,59,21,63)(18,60,22,64)(19,61,23,57)(20,62,24,58)(33,152,37,148)(34,145,38,149)(35,146,39,150)(36,147,40,151)(41,73,45,77)(42,74,46,78)(43,75,47,79)(44,76,48,80)(49,159,53,155)(50,160,54,156)(51,153,55,157)(52,154,56,158)(65,119,69,115)(66,120,70,116)(67,113,71,117)(68,114,72,118)(81,144,85,140)(82,137,86,141)(83,138,87,142)(84,139,88,143)(89,128,93,124)(90,121,94,125)(91,122,95,126)(92,123,96,127)(105,129,109,133)(106,130,110,134)(107,131,111,135)(108,132,112,136), (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), (1,55,28,155)(2,50,29,158)(3,53,30,153)(4,56,31,156)(5,51,32,159)(6,54,25,154)(7,49,26,157)(8,52,27,160)(9,63,102,23)(10,58,103,18)(11,61,104,21)(12,64,97,24)(13,59,98,19)(14,62,99,22)(15,57,100,17)(16,60,101,20)(33,86,150,139)(34,81,151,142)(35,84,152,137)(36,87,145,140)(37,82,146,143)(38,85,147,138)(39,88,148,141)(40,83,149,144)(41,109,79,131)(42,112,80,134)(43,107,73,129)(44,110,74,132)(45,105,75,135)(46,108,76,130)(47,111,77,133)(48,106,78,136)(65,124,117,95)(66,127,118,90)(67,122,119,93)(68,125,120,96)(69,128,113,91)(70,123,114,94)(71,126,115,89)(72,121,116,92) );
G=PermutationGroup([[(1,113,23,133,39),(2,114,24,134,40),(3,115,17,135,33),(4,116,18,136,34),(5,117,19,129,35),(6,118,20,130,36),(7,119,21,131,37),(8,120,22,132,38),(9,47,88,55,91),(10,48,81,56,92),(11,41,82,49,93),(12,42,83,50,94),(13,43,84,51,95),(14,44,85,52,96),(15,45,86,53,89),(16,46,87,54,90),(25,66,60,108,145),(26,67,61,109,146),(27,68,62,110,147),(28,69,63,111,148),(29,70,64,112,149),(30,71,57,105,150),(31,72,58,106,151),(32,65,59,107,152),(73,137,159,124,98),(74,138,160,125,99),(75,139,153,126,100),(76,140,154,127,101),(77,141,155,128,102),(78,142,156,121,103),(79,143,157,122,104),(80,144,158,123,97)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,149),(26,150),(27,151),(28,152),(29,145),(30,146),(31,147),(32,148),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,86),(50,87),(51,88),(52,81),(53,82),(54,83),(55,84),(56,85),(57,61),(58,62),(59,63),(60,64),(65,111),(66,112),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110),(73,128),(74,121),(75,122),(76,123),(77,124),(78,125),(79,126),(80,127),(97,101),(98,102),(99,103),(100,104),(113,129),(114,130),(115,131),(116,132),(117,133),(118,134),(119,135),(120,136),(137,155),(138,156),(139,157),(140,158),(141,159),(142,160),(143,153),(144,154)], [(1,30,5,26),(2,31,6,27),(3,32,7,28),(4,25,8,29),(9,104,13,100),(10,97,14,101),(11,98,15,102),(12,99,16,103),(17,59,21,63),(18,60,22,64),(19,61,23,57),(20,62,24,58),(33,152,37,148),(34,145,38,149),(35,146,39,150),(36,147,40,151),(41,73,45,77),(42,74,46,78),(43,75,47,79),(44,76,48,80),(49,159,53,155),(50,160,54,156),(51,153,55,157),(52,154,56,158),(65,119,69,115),(66,120,70,116),(67,113,71,117),(68,114,72,118),(81,144,85,140),(82,137,86,141),(83,138,87,142),(84,139,88,143),(89,128,93,124),(90,121,94,125),(91,122,95,126),(92,123,96,127),(105,129,109,133),(106,130,110,134),(107,131,111,135),(108,132,112,136)], [(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)], [(1,55,28,155),(2,50,29,158),(3,53,30,153),(4,56,31,156),(5,51,32,159),(6,54,25,154),(7,49,26,157),(8,52,27,160),(9,63,102,23),(10,58,103,18),(11,61,104,21),(12,64,97,24),(13,59,98,19),(14,62,99,22),(15,57,100,17),(16,60,101,20),(33,86,150,139),(34,81,151,142),(35,84,152,137),(36,87,145,140),(37,82,146,143),(38,85,147,138),(39,88,148,141),(40,83,149,144),(41,109,79,131),(42,112,80,134),(43,107,73,129),(44,110,74,132),(45,105,75,135),(46,108,76,130),(47,111,77,133),(48,106,78,136),(65,124,117,95),(66,127,118,90),(67,122,119,93),(68,125,120,96),(69,128,113,91),(70,123,114,94),(71,126,115,89),(72,121,116,92)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D4 | D5 | SD16 | D10 | D10 | C4×D5 | Q8×D5 | D4×D5 | D5×SD16 |
| kernel | D5×C4.Q8 | C20.Q8 | C40⋊6C4 | C5×C4.Q8 | D5×C4⋊C4 | D5×C2×C8 | C8×D5 | C4×D5 | C2×Dic5 | C22×D5 | C4.Q8 | D10 | C4⋊C4 | C2×C8 | C8 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 2 | 8 | 4 | 2 | 8 | 2 | 2 | 8 |
Matrix representation of D5×C4.Q8 ►in GL4(𝔽41) generated by
| 7 | 1 | 0 | 0 |
| 33 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 37 |
| 0 | 0 | 21 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 19 |
| 0 | 0 | 13 | 30 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 33 | 27 |
| 0 | 0 | 25 | 8 |
G:=sub<GL(4,GF(41))| [7,33,0,0,1,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,40,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,21,0,0,37,40],[40,0,0,0,0,40,0,0,0,0,0,13,0,0,19,30],[32,0,0,0,0,32,0,0,0,0,33,25,0,0,27,8] >;
D5×C4.Q8 in GAP, Magma, Sage, TeX
D_5\times C_4.Q_8
% in TeX
G:=Group("D5xC4.Q8"); // GroupNames label
G:=SmallGroup(320,486);
// by ID
G=gap.SmallGroup(320,486);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,555,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=1,d^4=c^2,e^2=c^-1*d^2,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^3>;
// generators/relations