direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8.D10, C40.8C23, C20.59C24, C23.53D20, M4(2)⋊18D10, Dic20⋊8C22, D20.22C23, Dic10.22C23, C4.49(C2×D20), (C2×C4).58D20, C8.8(C22×D5), C20.293(C2×D4), (C2×C20).204D4, (C2×C8).101D10, C40⋊C2⋊9C22, (C2×M4(2))⋊4D5, C4.56(C23×D5), (C2×Dic20)⋊14C2, C10⋊1(C8.C22), (C10×M4(2))⋊4C2, (C2×C40).69C22, C22.74(C2×D20), C2.28(C22×D20), C10.26(C22×D4), (C2×C20).512C23, C4○D20.50C22, (C22×C4).266D10, (C22×C10).119D4, (C22×Dic10)⋊18C2, (C2×Dic10)⋊63C22, (C2×D20).238C22, (C5×M4(2))⋊20C22, (C22×C20).267C22, C5⋊1(C2×C8.C22), (C2×C40⋊C2)⋊5C2, (C2×C10).63(C2×D4), (C2×C4○D20).23C2, (C2×C4).224(C22×D5), SmallGroup(320,1419)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8.D10
G = < a,b,c,d | a2=b8=1, c10=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c9 >
Subgroups: 958 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×13], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×10], C4○D4 [×6], Dic5 [×6], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C40 [×4], Dic10 [×6], Dic10 [×7], C4×D5 [×4], D20 [×2], D20, C2×Dic5 [×7], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5, C22×C10, C2×C8.C22, C40⋊C2 [×8], Dic20 [×8], C2×C40 [×2], C5×M4(2) [×4], C2×Dic10, C2×Dic10 [×6], C2×Dic10 [×3], C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C22×Dic5, C2×C5⋊D4, C22×C20, C2×C40⋊C2 [×2], C2×Dic20 [×2], C8.D10 [×8], C10×M4(2), C22×Dic10, C2×C4○D20, C2×C8.D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, D20 [×4], C22×D5 [×7], C2×C8.C22, C2×D20 [×6], C23×D5, C8.D10 [×2], C22×D20, C2×C8.D10
►On 160 points
Generators in S160
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 113)(81 149)(82 150)(83 151)(84 152)(85 153)(86 154)(87 155)(88 156)(89 157)(90 158)(91 159)(92 160)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)
(1 157 38 103 11 147 28 113)(2 148 39 114 12 158 29 104)(3 159 40 105 13 149 30 115)(4 150 21 116 14 160 31 106)(5 141 22 107 15 151 32 117)(6 152 23 118 16 142 33 108)(7 143 24 109 17 153 34 119)(8 154 25 120 18 144 35 110)(9 145 26 111 19 155 36 101)(10 156 27 102 20 146 37 112)(41 66 90 137 51 76 100 127)(42 77 91 128 52 67 81 138)(43 68 92 139 53 78 82 129)(44 79 93 130 54 69 83 140)(45 70 94 121 55 80 84 131)(46 61 95 132 56 71 85 122)(47 72 96 123 57 62 86 133)(48 63 97 134 58 73 87 124)(49 74 98 125 59 64 88 135)(50 65 99 136 60 75 89 126)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 24 31 34)(22 33 32 23)(25 40 35 30)(26 29 36 39)(27 38 37 28)(41 97 51 87)(42 86 52 96)(43 95 53 85)(44 84 54 94)(45 93 55 83)(46 82 56 92)(47 91 57 81)(48 100 58 90)(49 89 59 99)(50 98 60 88)(61 78 71 68)(62 67 72 77)(63 76 73 66)(64 65 74 75)(69 70 79 80)(101 148 111 158)(102 157 112 147)(103 146 113 156)(104 155 114 145)(105 144 115 154)(106 153 116 143)(107 142 117 152)(108 151 118 141)(109 160 119 150)(110 149 120 159)(121 140 131 130)(122 129 132 139)(123 138 133 128)(124 127 134 137)(125 136 135 126)
G:=sub<Sym(160)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148), (1,157,38,103,11,147,28,113)(2,148,39,114,12,158,29,104)(3,159,40,105,13,149,30,115)(4,150,21,116,14,160,31,106)(5,141,22,107,15,151,32,117)(6,152,23,118,16,142,33,108)(7,143,24,109,17,153,34,119)(8,154,25,120,18,144,35,110)(9,145,26,111,19,155,36,101)(10,156,27,102,20,146,37,112)(41,66,90,137,51,76,100,127)(42,77,91,128,52,67,81,138)(43,68,92,139,53,78,82,129)(44,79,93,130,54,69,83,140)(45,70,94,121,55,80,84,131)(46,61,95,132,56,71,85,122)(47,72,96,123,57,62,86,133)(48,63,97,134,58,73,87,124)(49,74,98,125,59,64,88,135)(50,65,99,136,60,75,89,126), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,97,51,87)(42,86,52,96)(43,95,53,85)(44,84,54,94)(45,93,55,83)(46,82,56,92)(47,91,57,81)(48,100,58,90)(49,89,59,99)(50,98,60,88)(61,78,71,68)(62,67,72,77)(63,76,73,66)(64,65,74,75)(69,70,79,80)(101,148,111,158)(102,157,112,147)(103,146,113,156)(104,155,114,145)(105,144,115,154)(106,153,116,143)(107,142,117,152)(108,151,118,141)(109,160,119,150)(110,149,120,159)(121,140,131,130)(122,129,132,139)(123,138,133,128)(124,127,134,137)(125,136,135,126)>;
G:=Group( (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148), (1,157,38,103,11,147,28,113)(2,148,39,114,12,158,29,104)(3,159,40,105,13,149,30,115)(4,150,21,116,14,160,31,106)(5,141,22,107,15,151,32,117)(6,152,23,118,16,142,33,108)(7,143,24,109,17,153,34,119)(8,154,25,120,18,144,35,110)(9,145,26,111,19,155,36,101)(10,156,27,102,20,146,37,112)(41,66,90,137,51,76,100,127)(42,77,91,128,52,67,81,138)(43,68,92,139,53,78,82,129)(44,79,93,130,54,69,83,140)(45,70,94,121,55,80,84,131)(46,61,95,132,56,71,85,122)(47,72,96,123,57,62,86,133)(48,63,97,134,58,73,87,124)(49,74,98,125,59,64,88,135)(50,65,99,136,60,75,89,126), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,97,51,87)(42,86,52,96)(43,95,53,85)(44,84,54,94)(45,93,55,83)(46,82,56,92)(47,91,57,81)(48,100,58,90)(49,89,59,99)(50,98,60,88)(61,78,71,68)(62,67,72,77)(63,76,73,66)(64,65,74,75)(69,70,79,80)(101,148,111,158)(102,157,112,147)(103,146,113,156)(104,155,114,145)(105,144,115,154)(106,153,116,143)(107,142,117,152)(108,151,118,141)(109,160,119,150)(110,149,120,159)(121,140,131,130)(122,129,132,139)(123,138,133,128)(124,127,134,137)(125,136,135,126) );
G=PermutationGroup([(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,129),(22,130),(23,131),(24,132),(25,133),(26,134),(27,135),(28,136),(29,137),(30,138),(31,139),(32,140),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,101),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,109),(57,110),(58,111),(59,112),(60,113),(81,149),(82,150),(83,151),(84,152),(85,153),(86,154),(87,155),(88,156),(89,157),(90,158),(91,159),(92,160),(93,141),(94,142),(95,143),(96,144),(97,145),(98,146),(99,147),(100,148)], [(1,157,38,103,11,147,28,113),(2,148,39,114,12,158,29,104),(3,159,40,105,13,149,30,115),(4,150,21,116,14,160,31,106),(5,141,22,107,15,151,32,117),(6,152,23,118,16,142,33,108),(7,143,24,109,17,153,34,119),(8,154,25,120,18,144,35,110),(9,145,26,111,19,155,36,101),(10,156,27,102,20,146,37,112),(41,66,90,137,51,76,100,127),(42,77,91,128,52,67,81,138),(43,68,92,139,53,78,82,129),(44,79,93,130,54,69,83,140),(45,70,94,121,55,80,84,131),(46,61,95,132,56,71,85,122),(47,72,96,123,57,62,86,133),(48,63,97,134,58,73,87,124),(49,74,98,125,59,64,88,135),(50,65,99,136,60,75,89,126)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,24,31,34),(22,33,32,23),(25,40,35,30),(26,29,36,39),(27,38,37,28),(41,97,51,87),(42,86,52,96),(43,95,53,85),(44,84,54,94),(45,93,55,83),(46,82,56,92),(47,91,57,81),(48,100,58,90),(49,89,59,99),(50,98,60,88),(61,78,71,68),(62,67,72,77),(63,76,73,66),(64,65,74,75),(69,70,79,80),(101,148,111,158),(102,157,112,147),(103,146,113,156),(104,155,114,145),(105,144,115,154),(106,153,116,143),(107,142,117,152),(108,151,118,141),(109,160,119,150),(110,149,120,159),(121,140,131,130),(122,129,132,139),(123,138,133,128),(124,127,134,137),(125,136,135,126)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | C8.C22 | C8.D10 |
| kernel | C2×C8.D10 | C2×C40⋊C2 | C2×Dic20 | C8.D10 | C10×M4(2) | C22×Dic10 | C2×C4○D20 | C2×C20 | C22×C10 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 3 | 1 | 2 | 4 | 8 | 2 | 12 | 4 | 2 | 8 |
Matrix representation of C2×C8.D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 28 | 0 | 0 |
| 0 | 0 | 13 | 39 | 0 | 0 |
| 0 | 34 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 34 | 8 |
| 0 | 0 | 6 | 1 | 33 | 27 |
| 0 | 0 | 33 | 7 | 35 | 6 |
| 0 | 0 | 34 | 32 | 35 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 5 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 35 | 32 |
| 0 | 0 | 1 | 0 | 27 | 6 |
| 0 | 0 | 35 | 32 | 28 | 2 |
| 0 | 0 | 27 | 6 | 39 | 13 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,1,0,0,0,0,0,0,1,0,0],[0,6,0,0,0,0,34,6,0,0,0,0,0,0,6,6,33,34,0,0,35,1,7,32,0,0,34,33,35,35,0,0,8,27,6,40],[1,5,0,0,0,0,0,40,0,0,0,0,0,0,0,1,35,27,0,0,1,0,32,6,0,0,35,27,28,39,0,0,32,6,2,13] >;
C2×C8.D10 in GAP, Magma, Sage, TeX
C_2\times C_8.D_{10} % in TeX
G:=Group("C2xC8.D10"); // GroupNames label
G:=SmallGroup(320,1419);
// by ID
G=gap.SmallGroup(320,1419);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,80,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^10=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations