direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D8.D5, D8.7D10, C10⋊2SD32, C20.22D8, C40.21D4, C40.25C23, Dic20⋊12C22, C5⋊3(C2×SD32), (C2×D8).2D5, C4.9(D4⋊D5), (C10×D8).3C2, (C2×C10).43D8, C10.64(C2×D8), C5⋊2C16⋊8C22, C20.161(C2×D4), (C2×C8).234D10, (C2×C20).181D4, C8.14(C5⋊D4), (C5×D8).7C22, C8.31(C22×D5), (C2×Dic20)⋊17C2, (C2×C40).86C22, C22.22(D4⋊D5), (C2×C5⋊2C16)⋊7C2, C4.3(C2×C5⋊D4), C2.19(C2×D4⋊D5), (C2×C4).143(C5⋊D4), SmallGroup(320,775)
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=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=d-1 >
Subgroups: 350 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C16, C2×C8, D8, D8, Q16, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2×C16, SD32, C2×D8, C2×Q16, C40, Dic10, C2×Dic5, C2×C20, C5×D4, C22×C10, C2×SD32, C5⋊2C16, Dic20, Dic20, C2×C40, C5×D8, C5×D8, C2×Dic10, D4×C10, C2×C5⋊2C16, D8.D5, C2×Dic20, C10×D8, C2×D8.D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, SD32, C2×D8, C5⋊D4, C22×D5, C2×SD32, D4⋊D5, C2×C5⋊D4, D8.D5, C2×D4⋊D5, C2×D8.D5
►On 160 points
Generators in S160
(1 102)(2 103)(3 104)(4 97)(5 98)(6 99)(7 100)(8 101)(9 148)(10 149)(11 150)(12 151)(13 152)(14 145)(15 146)(16 147)(17 144)(18 137)(19 138)(20 139)(21 140)(22 141)(23 142)(24 143)(25 132)(26 133)(27 134)(28 135)(29 136)(30 129)(31 130)(32 131)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 121)(41 78)(42 79)(43 80)(44 73)(45 74)(46 75)(47 76)(48 77)(49 87)(50 88)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 95)(58 96)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(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)
(1 101)(2 100)(3 99)(4 98)(5 97)(6 104)(7 103)(8 102)(9 152)(10 151)(11 150)(12 149)(13 148)(14 147)(15 146)(16 145)(17 138)(18 137)(19 144)(20 143)(21 142)(22 141)(23 140)(24 139)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 124)(34 123)(35 122)(36 121)(37 128)(38 127)(39 126)(40 125)(41 73)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 84)(50 83)(51 82)(52 81)(53 88)(54 87)(55 86)(56 85)(57 90)(58 89)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 107)(66 106)(67 105)(68 112)(69 111)(70 110)(71 109)(72 108)(113 157)(114 156)(115 155)(116 154)(117 153)(118 160)(119 159)(120 158)
(1 72 47 56 63)(2 65 48 49 64)(3 66 41 50 57)(4 67 42 51 58)(5 68 43 52 59)(6 69 44 53 60)(7 70 45 54 61)(8 71 46 55 62)(9 24 25 40 153)(10 17 26 33 154)(11 18 27 34 155)(12 19 28 35 156)(13 20 29 36 157)(14 21 30 37 158)(15 22 31 38 159)(16 23 32 39 160)(73 83 90 99 106)(74 84 91 100 107)(75 85 92 101 108)(76 86 93 102 109)(77 87 94 103 110)(78 88 95 104 111)(79 81 96 97 112)(80 82 89 98 105)(113 148 143 132 121)(114 149 144 133 122)(115 150 137 134 123)(116 151 138 135 124)(117 152 139 136 125)(118 145 140 129 126)(119 146 141 130 127)(120 147 142 131 128)
(1 147 5 151)(2 146 6 150)(3 145 7 149)(4 152 8 148)(9 97 13 101)(10 104 14 100)(11 103 15 99)(12 102 16 98)(17 95 21 91)(18 94 22 90)(19 93 23 89)(20 92 24 96)(25 81 29 85)(26 88 30 84)(27 87 31 83)(28 86 32 82)(33 78 37 74)(34 77 38 73)(35 76 39 80)(36 75 40 79)(41 126 45 122)(42 125 46 121)(43 124 47 128)(44 123 48 127)(49 130 53 134)(50 129 54 133)(51 136 55 132)(52 135 56 131)(57 140 61 144)(58 139 62 143)(59 138 63 142)(60 137 64 141)(65 119 69 115)(66 118 70 114)(67 117 71 113)(68 116 72 120)(105 156 109 160)(106 155 110 159)(107 154 111 158)(108 153 112 157)
G:=sub<Sym(160)| (1,102)(2,103)(3,104)(4,97)(5,98)(6,99)(7,100)(8,101)(9,148)(10,149)(11,150)(12,151)(13,152)(14,145)(15,146)(16,147)(17,144)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,132)(26,133)(27,134)(28,135)(29,136)(30,129)(31,130)(32,131)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,121)(41,78)(42,79)(43,80)(44,73)(45,74)(46,75)(47,76)(48,77)(49,87)(50,88)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,95)(58,96)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (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), (1,101)(2,100)(3,99)(4,98)(5,97)(6,104)(7,103)(8,102)(9,152)(10,151)(11,150)(12,149)(13,148)(14,147)(15,146)(16,145)(17,138)(18,137)(19,144)(20,143)(21,142)(22,141)(23,140)(24,139)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(31,130)(32,129)(33,124)(34,123)(35,122)(36,121)(37,128)(38,127)(39,126)(40,125)(41,73)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,84)(50,83)(51,82)(52,81)(53,88)(54,87)(55,86)(56,85)(57,90)(58,89)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,107)(66,106)(67,105)(68,112)(69,111)(70,110)(71,109)(72,108)(113,157)(114,156)(115,155)(116,154)(117,153)(118,160)(119,159)(120,158), (1,72,47,56,63)(2,65,48,49,64)(3,66,41,50,57)(4,67,42,51,58)(5,68,43,52,59)(6,69,44,53,60)(7,70,45,54,61)(8,71,46,55,62)(9,24,25,40,153)(10,17,26,33,154)(11,18,27,34,155)(12,19,28,35,156)(13,20,29,36,157)(14,21,30,37,158)(15,22,31,38,159)(16,23,32,39,160)(73,83,90,99,106)(74,84,91,100,107)(75,85,92,101,108)(76,86,93,102,109)(77,87,94,103,110)(78,88,95,104,111)(79,81,96,97,112)(80,82,89,98,105)(113,148,143,132,121)(114,149,144,133,122)(115,150,137,134,123)(116,151,138,135,124)(117,152,139,136,125)(118,145,140,129,126)(119,146,141,130,127)(120,147,142,131,128), (1,147,5,151)(2,146,6,150)(3,145,7,149)(4,152,8,148)(9,97,13,101)(10,104,14,100)(11,103,15,99)(12,102,16,98)(17,95,21,91)(18,94,22,90)(19,93,23,89)(20,92,24,96)(25,81,29,85)(26,88,30,84)(27,87,31,83)(28,86,32,82)(33,78,37,74)(34,77,38,73)(35,76,39,80)(36,75,40,79)(41,126,45,122)(42,125,46,121)(43,124,47,128)(44,123,48,127)(49,130,53,134)(50,129,54,133)(51,136,55,132)(52,135,56,131)(57,140,61,144)(58,139,62,143)(59,138,63,142)(60,137,64,141)(65,119,69,115)(66,118,70,114)(67,117,71,113)(68,116,72,120)(105,156,109,160)(106,155,110,159)(107,154,111,158)(108,153,112,157)>;
G:=Group( (1,102)(2,103)(3,104)(4,97)(5,98)(6,99)(7,100)(8,101)(9,148)(10,149)(11,150)(12,151)(13,152)(14,145)(15,146)(16,147)(17,144)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,132)(26,133)(27,134)(28,135)(29,136)(30,129)(31,130)(32,131)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,121)(41,78)(42,79)(43,80)(44,73)(45,74)(46,75)(47,76)(48,77)(49,87)(50,88)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,95)(58,96)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,110)(66,111)(67,112)(68,105)(69,106)(70,107)(71,108)(72,109)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (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), (1,101)(2,100)(3,99)(4,98)(5,97)(6,104)(7,103)(8,102)(9,152)(10,151)(11,150)(12,149)(13,148)(14,147)(15,146)(16,145)(17,138)(18,137)(19,144)(20,143)(21,142)(22,141)(23,140)(24,139)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(31,130)(32,129)(33,124)(34,123)(35,122)(36,121)(37,128)(38,127)(39,126)(40,125)(41,73)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,84)(50,83)(51,82)(52,81)(53,88)(54,87)(55,86)(56,85)(57,90)(58,89)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,107)(66,106)(67,105)(68,112)(69,111)(70,110)(71,109)(72,108)(113,157)(114,156)(115,155)(116,154)(117,153)(118,160)(119,159)(120,158), (1,72,47,56,63)(2,65,48,49,64)(3,66,41,50,57)(4,67,42,51,58)(5,68,43,52,59)(6,69,44,53,60)(7,70,45,54,61)(8,71,46,55,62)(9,24,25,40,153)(10,17,26,33,154)(11,18,27,34,155)(12,19,28,35,156)(13,20,29,36,157)(14,21,30,37,158)(15,22,31,38,159)(16,23,32,39,160)(73,83,90,99,106)(74,84,91,100,107)(75,85,92,101,108)(76,86,93,102,109)(77,87,94,103,110)(78,88,95,104,111)(79,81,96,97,112)(80,82,89,98,105)(113,148,143,132,121)(114,149,144,133,122)(115,150,137,134,123)(116,151,138,135,124)(117,152,139,136,125)(118,145,140,129,126)(119,146,141,130,127)(120,147,142,131,128), (1,147,5,151)(2,146,6,150)(3,145,7,149)(4,152,8,148)(9,97,13,101)(10,104,14,100)(11,103,15,99)(12,102,16,98)(17,95,21,91)(18,94,22,90)(19,93,23,89)(20,92,24,96)(25,81,29,85)(26,88,30,84)(27,87,31,83)(28,86,32,82)(33,78,37,74)(34,77,38,73)(35,76,39,80)(36,75,40,79)(41,126,45,122)(42,125,46,121)(43,124,47,128)(44,123,48,127)(49,130,53,134)(50,129,54,133)(51,136,55,132)(52,135,56,131)(57,140,61,144)(58,139,62,143)(59,138,63,142)(60,137,64,141)(65,119,69,115)(66,118,70,114)(67,117,71,113)(68,116,72,120)(105,156,109,160)(106,155,110,159)(107,154,111,158)(108,153,112,157) );
G=PermutationGroup([[(1,102),(2,103),(3,104),(4,97),(5,98),(6,99),(7,100),(8,101),(9,148),(10,149),(11,150),(12,151),(13,152),(14,145),(15,146),(16,147),(17,144),(18,137),(19,138),(20,139),(21,140),(22,141),(23,142),(24,143),(25,132),(26,133),(27,134),(28,135),(29,136),(30,129),(31,130),(32,131),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,121),(41,78),(42,79),(43,80),(44,73),(45,74),(46,75),(47,76),(48,77),(49,87),(50,88),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,95),(58,96),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,110),(66,111),(67,112),(68,105),(69,106),(70,107),(71,108),(72,109),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160)], [(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)], [(1,101),(2,100),(3,99),(4,98),(5,97),(6,104),(7,103),(8,102),(9,152),(10,151),(11,150),(12,149),(13,148),(14,147),(15,146),(16,145),(17,138),(18,137),(19,144),(20,143),(21,142),(22,141),(23,140),(24,139),(25,136),(26,135),(27,134),(28,133),(29,132),(30,131),(31,130),(32,129),(33,124),(34,123),(35,122),(36,121),(37,128),(38,127),(39,126),(40,125),(41,73),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,84),(50,83),(51,82),(52,81),(53,88),(54,87),(55,86),(56,85),(57,90),(58,89),(59,96),(60,95),(61,94),(62,93),(63,92),(64,91),(65,107),(66,106),(67,105),(68,112),(69,111),(70,110),(71,109),(72,108),(113,157),(114,156),(115,155),(116,154),(117,153),(118,160),(119,159),(120,158)], [(1,72,47,56,63),(2,65,48,49,64),(3,66,41,50,57),(4,67,42,51,58),(5,68,43,52,59),(6,69,44,53,60),(7,70,45,54,61),(8,71,46,55,62),(9,24,25,40,153),(10,17,26,33,154),(11,18,27,34,155),(12,19,28,35,156),(13,20,29,36,157),(14,21,30,37,158),(15,22,31,38,159),(16,23,32,39,160),(73,83,90,99,106),(74,84,91,100,107),(75,85,92,101,108),(76,86,93,102,109),(77,87,94,103,110),(78,88,95,104,111),(79,81,96,97,112),(80,82,89,98,105),(113,148,143,132,121),(114,149,144,133,122),(115,150,137,134,123),(116,151,138,135,124),(117,152,139,136,125),(118,145,140,129,126),(119,146,141,130,127),(120,147,142,131,128)], [(1,147,5,151),(2,146,6,150),(3,145,7,149),(4,152,8,148),(9,97,13,101),(10,104,14,100),(11,103,15,99),(12,102,16,98),(17,95,21,91),(18,94,22,90),(19,93,23,89),(20,92,24,96),(25,81,29,85),(26,88,30,84),(27,87,31,83),(28,86,32,82),(33,78,37,74),(34,77,38,73),(35,76,39,80),(36,75,40,79),(41,126,45,122),(42,125,46,121),(43,124,47,128),(44,123,48,127),(49,130,53,134),(50,129,54,133),(51,136,55,132),(52,135,56,131),(57,140,61,144),(58,139,62,143),(59,138,63,142),(60,137,64,141),(65,119,69,115),(66,118,70,114),(67,117,71,113),(68,116,72,120),(105,156,109,160),(106,155,110,159),(107,154,111,158),(108,153,112,157)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | SD32 | C5⋊D4 | C5⋊D4 | D4⋊D5 | D4⋊D5 | D8.D5 |
| kernel | C2×D8.D5 | C2×C5⋊2C16 | D8.D5 | C2×Dic20 | C10×D8 | C40 | C2×C20 | C2×D8 | C20 | C2×C10 | C2×C8 | D8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 2 | 2 | 8 |
Matrix representation of C2×D8.D5 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 230 | 11 |
| 0 | 0 | 230 | 230 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 230 | 11 |
| 0 | 0 | 11 | 11 |
| 52 | 240 | 0 | 0 |
| 53 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 239 | 175 | 0 | 0 |
| 11 | 2 | 0 | 0 |
| 0 | 0 | 103 | 41 |
| 0 | 0 | 41 | 138 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,230,230,0,0,11,230],[240,0,0,0,0,240,0,0,0,0,230,11,0,0,11,11],[52,53,0,0,240,240,0,0,0,0,1,0,0,0,0,1],[239,11,0,0,175,2,0,0,0,0,103,41,0,0,41,138] >;
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(320,775);
// by ID
G=gap.SmallGroup(320,775);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,675,185,192,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^5=1,e^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
// generators/relations