metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D8).3D5, (C5×D4).6D4, (D4×Dic5)⋊6C2, (C10×D8).8C2, (C2×C8).34D10, (C2×D4).61D10, C20.163(C2×D4), D4.1(C5⋊D4), C5⋊6(D4.2D4), C20.92(C4○D4), C10.32(C4○D8), C4.8(D4⋊2D5), D4⋊Dic5⋊27C2, C20.17D4⋊3C2, C20.8Q8⋊28C2, (C2×Dic5).71D4, C22.253(D4×D5), C2.16(D8⋊3D5), C20.44D4⋊28C2, C2.27(D8⋊D5), C10.48(C8⋊C22), (C2×C40).248C22, (C2×C20).430C23, (D4×C10).80C22, C10.108(C4⋊D4), C4⋊Dic5.164C22, (C4×Dic5).50C22, C2.23(Dic5⋊D4), (C2×Dic10).123C22, C4.35(C2×C5⋊D4), (C2×D4.D5)⋊16C2, (C2×C10).343(C2×D4), (C2×C4).520(C22×D5), (C2×C5⋊2C8).147C22, SmallGroup(320,780)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D8).D5
G = < a,b,c,d,e | a2=b8=c2=d5=1, e2=ab4, ebe-1=ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 470 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C5×D4, C22×C10, D4.2D4, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D4.D5, C23.D5, C2×C40, C5×D8, C2×Dic10, C22×Dic5, D4×C10, C20.8Q8, C20.44D4, D4⋊Dic5, C2×D4.D5, D4×Dic5, C20.17D4, C10×D8, (C2×D8).D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4.2D4, D4×D5, D4⋊2D5, C2×C5⋊D4, D8⋊D5, D8⋊3D5, Dic5⋊D4, (C2×D8).D5
►On 160 points
Generators in S160
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 113)(8 114)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 109)(18 110)(19 111)(20 112)(21 105)(22 106)(23 107)(24 108)(25 133)(26 134)(27 135)(28 136)(29 129)(30 130)(31 131)(32 132)(33 102)(34 103)(35 104)(36 97)(37 98)(38 99)(39 100)(40 101)(41 138)(42 139)(43 140)(44 141)(45 142)(46 143)(47 144)(48 137)(49 125)(50 126)(51 127)(52 128)(53 121)(54 122)(55 123)(56 124)(57 88)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 153)(73 145)(74 146)(75 147)(76 148)(77 149)(78 150)(79 151)(80 152)
(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)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 120)(8 119)(9 96)(10 95)(11 94)(12 93)(13 92)(14 91)(15 90)(16 89)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 104)(41 44)(42 43)(45 48)(46 47)(49 128)(50 127)(51 126)(52 125)(53 124)(54 123)(55 122)(56 121)(57 62)(58 61)(59 60)(63 64)(65 68)(66 67)(69 72)(70 71)(73 152)(74 151)(75 150)(76 149)(77 148)(78 147)(79 146)(80 145)(81 84)(82 83)(85 88)(86 87)(105 106)(107 112)(108 111)(109 110)(129 136)(130 135)(131 134)(132 133)(137 142)(138 141)(139 140)(143 144)(153 158)(154 157)(155 156)(159 160)
(1 9 97 121 79)(2 10 98 122 80)(3 11 99 123 73)(4 12 100 124 74)(5 13 101 125 75)(6 14 102 126 76)(7 15 103 127 77)(8 16 104 128 78)(17 136 143 66 59)(18 129 144 67 60)(19 130 137 68 61)(20 131 138 69 62)(21 132 139 70 63)(22 133 140 71 64)(23 134 141 72 57)(24 135 142 65 58)(25 43 160 87 106)(26 44 153 88 107)(27 45 154 81 108)(28 46 155 82 109)(29 47 156 83 110)(30 48 157 84 111)(31 41 158 85 112)(32 42 159 86 105)(33 50 148 120 90)(34 51 149 113 91)(35 52 150 114 92)(36 53 151 115 93)(37 54 152 116 94)(38 55 145 117 95)(39 56 146 118 96)(40 49 147 119 89)
(1 154 119 69)(2 66 120 159)(3 156 113 71)(4 68 114 153)(5 158 115 65)(6 70 116 155)(7 160 117 67)(8 72 118 157)(9 45 89 138)(10 143 90 42)(11 47 91 140)(12 137 92 44)(13 41 93 142)(14 139 94 46)(15 43 95 144)(16 141 96 48)(17 50 105 122)(18 127 106 55)(19 52 107 124)(20 121 108 49)(21 54 109 126)(22 123 110 51)(23 56 111 128)(24 125 112 53)(25 38 129 103)(26 100 130 35)(27 40 131 97)(28 102 132 37)(29 34 133 99)(30 104 134 39)(31 36 135 101)(32 98 136 33)(57 146 84 78)(58 75 85 151)(59 148 86 80)(60 77 87 145)(61 150 88 74)(62 79 81 147)(63 152 82 76)(64 73 83 149)
G:=sub<Sym(160)| (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,113)(8,114)(9,93)(10,94)(11,95)(12,96)(13,89)(14,90)(15,91)(16,92)(17,109)(18,110)(19,111)(20,112)(21,105)(22,106)(23,107)(24,108)(25,133)(26,134)(27,135)(28,136)(29,129)(30,130)(31,131)(32,132)(33,102)(34,103)(35,104)(36,97)(37,98)(38,99)(39,100)(40,101)(41,138)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,137)(49,125)(50,126)(51,127)(52,128)(53,121)(54,122)(55,123)(56,124)(57,88)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,153)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152), (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), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,120)(8,119)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,90)(16,89)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,104)(41,44)(42,43)(45,48)(46,47)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,62)(58,61)(59,60)(63,64)(65,68)(66,67)(69,72)(70,71)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145)(81,84)(82,83)(85,88)(86,87)(105,106)(107,112)(108,111)(109,110)(129,136)(130,135)(131,134)(132,133)(137,142)(138,141)(139,140)(143,144)(153,158)(154,157)(155,156)(159,160), (1,9,97,121,79)(2,10,98,122,80)(3,11,99,123,73)(4,12,100,124,74)(5,13,101,125,75)(6,14,102,126,76)(7,15,103,127,77)(8,16,104,128,78)(17,136,143,66,59)(18,129,144,67,60)(19,130,137,68,61)(20,131,138,69,62)(21,132,139,70,63)(22,133,140,71,64)(23,134,141,72,57)(24,135,142,65,58)(25,43,160,87,106)(26,44,153,88,107)(27,45,154,81,108)(28,46,155,82,109)(29,47,156,83,110)(30,48,157,84,111)(31,41,158,85,112)(32,42,159,86,105)(33,50,148,120,90)(34,51,149,113,91)(35,52,150,114,92)(36,53,151,115,93)(37,54,152,116,94)(38,55,145,117,95)(39,56,146,118,96)(40,49,147,119,89), (1,154,119,69)(2,66,120,159)(3,156,113,71)(4,68,114,153)(5,158,115,65)(6,70,116,155)(7,160,117,67)(8,72,118,157)(9,45,89,138)(10,143,90,42)(11,47,91,140)(12,137,92,44)(13,41,93,142)(14,139,94,46)(15,43,95,144)(16,141,96,48)(17,50,105,122)(18,127,106,55)(19,52,107,124)(20,121,108,49)(21,54,109,126)(22,123,110,51)(23,56,111,128)(24,125,112,53)(25,38,129,103)(26,100,130,35)(27,40,131,97)(28,102,132,37)(29,34,133,99)(30,104,134,39)(31,36,135,101)(32,98,136,33)(57,146,84,78)(58,75,85,151)(59,148,86,80)(60,77,87,145)(61,150,88,74)(62,79,81,147)(63,152,82,76)(64,73,83,149)>;
G:=Group( (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,113)(8,114)(9,93)(10,94)(11,95)(12,96)(13,89)(14,90)(15,91)(16,92)(17,109)(18,110)(19,111)(20,112)(21,105)(22,106)(23,107)(24,108)(25,133)(26,134)(27,135)(28,136)(29,129)(30,130)(31,131)(32,132)(33,102)(34,103)(35,104)(36,97)(37,98)(38,99)(39,100)(40,101)(41,138)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,137)(49,125)(50,126)(51,127)(52,128)(53,121)(54,122)(55,123)(56,124)(57,88)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,153)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152), (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), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,120)(8,119)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,90)(16,89)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,104)(41,44)(42,43)(45,48)(46,47)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,62)(58,61)(59,60)(63,64)(65,68)(66,67)(69,72)(70,71)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145)(81,84)(82,83)(85,88)(86,87)(105,106)(107,112)(108,111)(109,110)(129,136)(130,135)(131,134)(132,133)(137,142)(138,141)(139,140)(143,144)(153,158)(154,157)(155,156)(159,160), (1,9,97,121,79)(2,10,98,122,80)(3,11,99,123,73)(4,12,100,124,74)(5,13,101,125,75)(6,14,102,126,76)(7,15,103,127,77)(8,16,104,128,78)(17,136,143,66,59)(18,129,144,67,60)(19,130,137,68,61)(20,131,138,69,62)(21,132,139,70,63)(22,133,140,71,64)(23,134,141,72,57)(24,135,142,65,58)(25,43,160,87,106)(26,44,153,88,107)(27,45,154,81,108)(28,46,155,82,109)(29,47,156,83,110)(30,48,157,84,111)(31,41,158,85,112)(32,42,159,86,105)(33,50,148,120,90)(34,51,149,113,91)(35,52,150,114,92)(36,53,151,115,93)(37,54,152,116,94)(38,55,145,117,95)(39,56,146,118,96)(40,49,147,119,89), (1,154,119,69)(2,66,120,159)(3,156,113,71)(4,68,114,153)(5,158,115,65)(6,70,116,155)(7,160,117,67)(8,72,118,157)(9,45,89,138)(10,143,90,42)(11,47,91,140)(12,137,92,44)(13,41,93,142)(14,139,94,46)(15,43,95,144)(16,141,96,48)(17,50,105,122)(18,127,106,55)(19,52,107,124)(20,121,108,49)(21,54,109,126)(22,123,110,51)(23,56,111,128)(24,125,112,53)(25,38,129,103)(26,100,130,35)(27,40,131,97)(28,102,132,37)(29,34,133,99)(30,104,134,39)(31,36,135,101)(32,98,136,33)(57,146,84,78)(58,75,85,151)(59,148,86,80)(60,77,87,145)(61,150,88,74)(62,79,81,147)(63,152,82,76)(64,73,83,149) );
G=PermutationGroup([[(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,113),(8,114),(9,93),(10,94),(11,95),(12,96),(13,89),(14,90),(15,91),(16,92),(17,109),(18,110),(19,111),(20,112),(21,105),(22,106),(23,107),(24,108),(25,133),(26,134),(27,135),(28,136),(29,129),(30,130),(31,131),(32,132),(33,102),(34,103),(35,104),(36,97),(37,98),(38,99),(39,100),(40,101),(41,138),(42,139),(43,140),(44,141),(45,142),(46,143),(47,144),(48,137),(49,125),(50,126),(51,127),(52,128),(53,121),(54,122),(55,123),(56,124),(57,88),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,154),(66,155),(67,156),(68,157),(69,158),(70,159),(71,160),(72,153),(73,145),(74,146),(75,147),(76,148),(77,149),(78,150),(79,151),(80,152)], [(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)], [(1,118),(2,117),(3,116),(4,115),(5,114),(6,113),(7,120),(8,119),(9,96),(10,95),(11,94),(12,93),(13,92),(14,91),(15,90),(16,89),(17,18),(19,24),(20,23),(21,22),(25,32),(26,31),(27,30),(28,29),(33,103),(34,102),(35,101),(36,100),(37,99),(38,98),(39,97),(40,104),(41,44),(42,43),(45,48),(46,47),(49,128),(50,127),(51,126),(52,125),(53,124),(54,123),(55,122),(56,121),(57,62),(58,61),(59,60),(63,64),(65,68),(66,67),(69,72),(70,71),(73,152),(74,151),(75,150),(76,149),(77,148),(78,147),(79,146),(80,145),(81,84),(82,83),(85,88),(86,87),(105,106),(107,112),(108,111),(109,110),(129,136),(130,135),(131,134),(132,133),(137,142),(138,141),(139,140),(143,144),(153,158),(154,157),(155,156),(159,160)], [(1,9,97,121,79),(2,10,98,122,80),(3,11,99,123,73),(4,12,100,124,74),(5,13,101,125,75),(6,14,102,126,76),(7,15,103,127,77),(8,16,104,128,78),(17,136,143,66,59),(18,129,144,67,60),(19,130,137,68,61),(20,131,138,69,62),(21,132,139,70,63),(22,133,140,71,64),(23,134,141,72,57),(24,135,142,65,58),(25,43,160,87,106),(26,44,153,88,107),(27,45,154,81,108),(28,46,155,82,109),(29,47,156,83,110),(30,48,157,84,111),(31,41,158,85,112),(32,42,159,86,105),(33,50,148,120,90),(34,51,149,113,91),(35,52,150,114,92),(36,53,151,115,93),(37,54,152,116,94),(38,55,145,117,95),(39,56,146,118,96),(40,49,147,119,89)], [(1,154,119,69),(2,66,120,159),(3,156,113,71),(4,68,114,153),(5,158,115,65),(6,70,116,155),(7,160,117,67),(8,72,118,157),(9,45,89,138),(10,143,90,42),(11,47,91,140),(12,137,92,44),(13,41,93,142),(14,139,94,46),(15,43,95,144),(16,141,96,48),(17,50,105,122),(18,127,106,55),(19,52,107,124),(20,121,108,49),(21,54,109,126),(22,123,110,51),(23,56,111,128),(24,125,112,53),(25,38,129,103),(26,100,130,35),(27,40,131,97),(28,102,132,37),(29,34,133,99),(30,104,134,39),(31,36,135,101),(32,98,136,33),(57,146,84,78),(58,75,85,151),(59,148,86,80),(60,77,87,145),(61,150,88,74),(62,79,81,147),(63,152,82,76),(64,73,83,149)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 2 | 2 | 10 | 10 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 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 | C5⋊D4 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊D5 | D8⋊3D5 |
| kernel | (C2×D8).D5 | C20.8Q8 | C20.44D4 | D4⋊Dic5 | C2×D4.D5 | D4×Dic5 | C20.17D4 | C10×D8 | C2×Dic5 | C5×D4 | C2×D8 | C20 | C2×C8 | C2×D4 | C10 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of (C2×D8).D5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 24 | 0 | 0 |
| 29 | 24 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 17 | 0 | 0 |
| 29 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 37 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 37 |
| 0 | 0 | 31 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,29,0,0,24,24,0,0,0,0,40,0,0,0,0,1],[0,29,0,0,17,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,37],[32,0,0,0,0,32,0,0,0,0,0,31,0,0,37,0] >;
(C2×D8).D5 in GAP, Magma, Sage, TeX
(C_2\times D_8).D_5
% in TeX
G:=Group("(C2xD8).D5"); // GroupNames label
G:=SmallGroup(320,780);
// by ID
G=gap.SmallGroup(320,780);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1094,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^5=1,e^2=a*b^4,e*b*e^-1=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,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations