direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5×Q16, C40.33C23, C20.10C24, Dic20⋊15C22, Dic10.6C23, C10⋊2(C2×Q16), C4.46(D4×D5), C5⋊2(C22×Q16), (C10×Q16)⋊7C2, (C4×D5).69D4, C20.85(C2×D4), (C2×C8).246D10, (C5×Q16)⋊8C22, C5⋊Q16⋊8C22, C8.39(C22×D5), C4.10(C23×D5), (C2×Dic20)⋊20C2, D10.113(C2×D4), (C5×Q8).4C23, (Q8×D5).8C22, Q8.4(C22×D5), (C2×C40).98C22, C5⋊2C8.22C23, (C2×Q8).152D10, Dic5.25(C2×D4), (C8×D5).44C22, (C4×D5).63C23, C22.142(D4×D5), (C2×C20).527C23, (C2×Dic5).168D4, (C22×D5).160D4, C10.111(C22×D4), (Q8×C10).149C22, (C2×Dic10).204C22, (D5×C2×C8).6C2, C2.84(C2×D4×D5), (C2×Q8×D5).8C2, (C2×C5⋊Q16)⋊27C2, (C2×C10).400(C2×D4), (C2×C4×D5).329C22, (C2×C4).615(C22×D5), (C2×C5⋊2C8).293C22, SmallGroup(320,1435)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 862 in 258 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×17], Q8 [×4], Q8 [×16], C23, D5 [×4], C10, C10 [×2], C2×C8, C2×C8 [×5], Q16 [×4], Q16 [×12], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×16], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], C2×C10, C22×C8, C2×Q16, C2×Q16 [×11], C22×Q8 [×2], C5⋊2C8 [×2], C40 [×2], Dic10 [×4], Dic10 [×10], C4×D5 [×4], C4×D5 [×8], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], C22×D5, C22×Q16, C8×D5 [×4], Dic20 [×4], C2×C5⋊2C8, C5⋊Q16 [×8], C2×C40, C5×Q16 [×4], C2×Dic10 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], Q8×D5 [×8], Q8×D5 [×4], Q8×C10 [×2], D5×C2×C8, C2×Dic20, D5×Q16 [×8], C2×C5⋊Q16 [×2], C10×Q16, C2×Q8×D5 [×2], C2×D5×Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, Q16 [×4], C2×D4 [×6], C24, D10 [×7], C2×Q16 [×6], C22×D4, C22×D5 [×7], C22×Q16, D4×D5 [×2], C23×D5, D5×Q16 [×2], C2×D4×D5, C2×D5×Q16
Generators and relations
G = < a,b,c,d,e | a2=b5=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►On 160 points
Generators in S160
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 97)(24 98)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(33 71)(34 72)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 150)(42 151)(43 152)(44 145)(45 146)(46 147)(47 148)(48 149)(49 143)(50 144)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(81 116)(82 117)(83 118)(84 119)(85 120)(86 113)(87 114)(88 115)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(105 156)(106 157)(107 158)(108 159)(109 160)(110 153)(111 154)(112 155)
(1 138 15 81 157)(2 139 16 82 158)(3 140 9 83 159)(4 141 10 84 160)(5 142 11 85 153)(6 143 12 86 154)(7 144 13 87 155)(8 137 14 88 156)(17 80 149 96 72)(18 73 150 89 65)(19 74 151 90 66)(20 75 152 91 67)(21 76 145 92 68)(22 77 146 93 69)(23 78 147 94 70)(24 79 148 95 71)(25 43 131 37 102)(26 44 132 38 103)(27 45 133 39 104)(28 46 134 40 97)(29 47 135 33 98)(30 48 136 34 99)(31 41 129 35 100)(32 42 130 36 101)(49 64 113 111 124)(50 57 114 112 125)(51 58 115 105 126)(52 59 116 106 127)(53 60 117 107 128)(54 61 118 108 121)(55 62 119 109 122)(56 63 120 110 123)
(1 157)(2 158)(3 159)(4 160)(5 153)(6 154)(7 155)(8 156)(17 96)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(73 150)(74 151)(75 152)(76 145)(77 146)(78 147)(79 148)(80 149)(81 138)(82 139)(83 140)(84 141)(85 142)(86 143)(87 144)(88 137)(97 134)(98 135)(99 136)(100 129)(101 130)(102 131)(103 132)(104 133)(105 126)(106 127)(107 128)(108 121)(109 122)(110 123)(111 124)(112 125)
(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 43 5 47)(2 42 6 46)(3 41 7 45)(4 48 8 44)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 115 21 119)(18 114 22 118)(19 113 23 117)(20 120 24 116)(25 153 29 157)(26 160 30 156)(27 159 31 155)(28 158 32 154)(49 94 53 90)(50 93 54 89)(51 92 55 96)(52 91 56 95)(57 69 61 65)(58 68 62 72)(59 67 63 71)(60 66 64 70)(73 112 77 108)(74 111 78 107)(75 110 79 106)(76 109 80 105)(81 102 85 98)(82 101 86 97)(83 100 87 104)(84 99 88 103)(121 150 125 146)(122 149 126 145)(123 148 127 152)(124 147 128 151)(129 144 133 140)(130 143 134 139)(131 142 135 138)(132 141 136 137)
G:=sub<Sym(160)| (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,97)(24,98)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,150)(42,151)(43,152)(44,145)(45,146)(46,147)(47,148)(48,149)(49,143)(50,144)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(81,116)(82,117)(83,118)(84,119)(85,120)(86,113)(87,114)(88,115)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(105,156)(106,157)(107,158)(108,159)(109,160)(110,153)(111,154)(112,155), (1,138,15,81,157)(2,139,16,82,158)(3,140,9,83,159)(4,141,10,84,160)(5,142,11,85,153)(6,143,12,86,154)(7,144,13,87,155)(8,137,14,88,156)(17,80,149,96,72)(18,73,150,89,65)(19,74,151,90,66)(20,75,152,91,67)(21,76,145,92,68)(22,77,146,93,69)(23,78,147,94,70)(24,79,148,95,71)(25,43,131,37,102)(26,44,132,38,103)(27,45,133,39,104)(28,46,134,40,97)(29,47,135,33,98)(30,48,136,34,99)(31,41,129,35,100)(32,42,130,36,101)(49,64,113,111,124)(50,57,114,112,125)(51,58,115,105,126)(52,59,116,106,127)(53,60,117,107,128)(54,61,118,108,121)(55,62,119,109,122)(56,63,120,110,123), (1,157)(2,158)(3,159)(4,160)(5,153)(6,154)(7,155)(8,156)(17,96)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,41)(32,42)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(73,150)(74,151)(75,152)(76,145)(77,146)(78,147)(79,148)(80,149)(81,138)(82,139)(83,140)(84,141)(85,142)(86,143)(87,144)(88,137)(97,134)(98,135)(99,136)(100,129)(101,130)(102,131)(103,132)(104,133)(105,126)(106,127)(107,128)(108,121)(109,122)(110,123)(111,124)(112,125), (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), (1,43,5,47)(2,42,6,46)(3,41,7,45)(4,48,8,44)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,115,21,119)(18,114,22,118)(19,113,23,117)(20,120,24,116)(25,153,29,157)(26,160,30,156)(27,159,31,155)(28,158,32,154)(49,94,53,90)(50,93,54,89)(51,92,55,96)(52,91,56,95)(57,69,61,65)(58,68,62,72)(59,67,63,71)(60,66,64,70)(73,112,77,108)(74,111,78,107)(75,110,79,106)(76,109,80,105)(81,102,85,98)(82,101,86,97)(83,100,87,104)(84,99,88,103)(121,150,125,146)(122,149,126,145)(123,148,127,152)(124,147,128,151)(129,144,133,140)(130,143,134,139)(131,142,135,138)(132,141,136,137)>;
G:=Group( (1,127)(2,128)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,97)(24,98)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,150)(42,151)(43,152)(44,145)(45,146)(46,147)(47,148)(48,149)(49,143)(50,144)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(81,116)(82,117)(83,118)(84,119)(85,120)(86,113)(87,114)(88,115)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(105,156)(106,157)(107,158)(108,159)(109,160)(110,153)(111,154)(112,155), (1,138,15,81,157)(2,139,16,82,158)(3,140,9,83,159)(4,141,10,84,160)(5,142,11,85,153)(6,143,12,86,154)(7,144,13,87,155)(8,137,14,88,156)(17,80,149,96,72)(18,73,150,89,65)(19,74,151,90,66)(20,75,152,91,67)(21,76,145,92,68)(22,77,146,93,69)(23,78,147,94,70)(24,79,148,95,71)(25,43,131,37,102)(26,44,132,38,103)(27,45,133,39,104)(28,46,134,40,97)(29,47,135,33,98)(30,48,136,34,99)(31,41,129,35,100)(32,42,130,36,101)(49,64,113,111,124)(50,57,114,112,125)(51,58,115,105,126)(52,59,116,106,127)(53,60,117,107,128)(54,61,118,108,121)(55,62,119,109,122)(56,63,120,110,123), (1,157)(2,158)(3,159)(4,160)(5,153)(6,154)(7,155)(8,156)(17,96)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,41)(32,42)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(73,150)(74,151)(75,152)(76,145)(77,146)(78,147)(79,148)(80,149)(81,138)(82,139)(83,140)(84,141)(85,142)(86,143)(87,144)(88,137)(97,134)(98,135)(99,136)(100,129)(101,130)(102,131)(103,132)(104,133)(105,126)(106,127)(107,128)(108,121)(109,122)(110,123)(111,124)(112,125), (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), (1,43,5,47)(2,42,6,46)(3,41,7,45)(4,48,8,44)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,115,21,119)(18,114,22,118)(19,113,23,117)(20,120,24,116)(25,153,29,157)(26,160,30,156)(27,159,31,155)(28,158,32,154)(49,94,53,90)(50,93,54,89)(51,92,55,96)(52,91,56,95)(57,69,61,65)(58,68,62,72)(59,67,63,71)(60,66,64,70)(73,112,77,108)(74,111,78,107)(75,110,79,106)(76,109,80,105)(81,102,85,98)(82,101,86,97)(83,100,87,104)(84,99,88,103)(121,150,125,146)(122,149,126,145)(123,148,127,152)(124,147,128,151)(129,144,133,140)(130,143,134,139)(131,142,135,138)(132,141,136,137) );
G=PermutationGroup([(1,127),(2,128),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,61),(10,62),(11,63),(12,64),(13,57),(14,58),(15,59),(16,60),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,97),(24,98),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,73),(32,74),(33,71),(34,72),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,150),(42,151),(43,152),(44,145),(45,146),(46,147),(47,148),(48,149),(49,143),(50,144),(51,137),(52,138),(53,139),(54,140),(55,141),(56,142),(81,116),(82,117),(83,118),(84,119),(85,120),(86,113),(87,114),(88,115),(89,129),(90,130),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(105,156),(106,157),(107,158),(108,159),(109,160),(110,153),(111,154),(112,155)], [(1,138,15,81,157),(2,139,16,82,158),(3,140,9,83,159),(4,141,10,84,160),(5,142,11,85,153),(6,143,12,86,154),(7,144,13,87,155),(8,137,14,88,156),(17,80,149,96,72),(18,73,150,89,65),(19,74,151,90,66),(20,75,152,91,67),(21,76,145,92,68),(22,77,146,93,69),(23,78,147,94,70),(24,79,148,95,71),(25,43,131,37,102),(26,44,132,38,103),(27,45,133,39,104),(28,46,134,40,97),(29,47,135,33,98),(30,48,136,34,99),(31,41,129,35,100),(32,42,130,36,101),(49,64,113,111,124),(50,57,114,112,125),(51,58,115,105,126),(52,59,116,106,127),(53,60,117,107,128),(54,61,118,108,121),(55,62,119,109,122),(56,63,120,110,123)], [(1,157),(2,158),(3,159),(4,160),(5,153),(6,154),(7,155),(8,156),(17,96),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,41),(32,42),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(73,150),(74,151),(75,152),(76,145),(77,146),(78,147),(79,148),(80,149),(81,138),(82,139),(83,140),(84,141),(85,142),(86,143),(87,144),(88,137),(97,134),(98,135),(99,136),(100,129),(101,130),(102,131),(103,132),(104,133),(105,126),(106,127),(107,128),(108,121),(109,122),(110,123),(111,124),(112,125)], [(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)], [(1,43,5,47),(2,42,6,46),(3,41,7,45),(4,48,8,44),(9,35,13,39),(10,34,14,38),(11,33,15,37),(12,40,16,36),(17,115,21,119),(18,114,22,118),(19,113,23,117),(20,120,24,116),(25,153,29,157),(26,160,30,156),(27,159,31,155),(28,158,32,154),(49,94,53,90),(50,93,54,89),(51,92,55,96),(52,91,56,95),(57,69,61,65),(58,68,62,72),(59,67,63,71),(60,66,64,70),(73,112,77,108),(74,111,78,107),(75,110,79,106),(76,109,80,105),(81,102,85,98),(82,101,86,97),(83,100,87,104),(84,99,88,103),(121,150,125,146),(122,149,126,145),(123,148,127,152),(124,147,128,151),(129,144,133,140),(130,143,134,139),(131,142,135,138),(132,141,136,137)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 1 |
| 0 | 0 | 33 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 33 | 1 |
| 24 | 35 | 0 | 0 |
| 7 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 37 | 40 | 0 | 0 |
| 17 | 4 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,33,0,0,1,7],[1,0,0,0,0,1,0,0,0,0,40,33,0,0,0,1],[24,7,0,0,35,0,0,0,0,0,40,0,0,0,0,40],[37,17,0,0,40,4,0,0,0,0,40,0,0,0,0,40] >;
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 | C2 | D4 | D4 | D4 | D5 | Q16 | D10 | D10 | D10 | D4×D5 | D4×D5 | D5×Q16 |
| kernel | C2×D5×Q16 | D5×C2×C8 | C2×Dic20 | D5×Q16 | C2×C5⋊Q16 | C10×Q16 | C2×Q8×D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×Q16 | D10 | C2×C8 | Q16 | C2×Q8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 8 | 2 | 8 | 4 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_5\times Q_{16} % in TeX
G:=Group("C2xD5xQ16"); // GroupNames label
G:=SmallGroup(320,1435);
// by ID
G=gap.SmallGroup(320,1435);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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