metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.12D8, C2.9(D5×D8), C40⋊5C4⋊8C2, (C2×C8).9D10, D4⋊C4⋊5D5, C10.23(C2×D8), D10⋊1C8⋊5C2, C4⋊C4.135D10, (C2×D4).23D10, C10.D8⋊5C2, D4⋊Dic5⋊7C2, C20⋊2D4.2C2, (C2×C40).9C22, C4.51(C4○D20), (C2×Dic5).27D4, C22.172(D4×D5), C5⋊2(C22.D8), C20.149(C4○D4), C4.78(D4⋊2D5), (C2×C20).214C23, (C22×D5).108D4, (D4×C10).35C22, C4⋊Dic5.69C22, C2.11(SD16⋊D5), C10.29(C8.C22), C2.12(D10.12D4), C10.20(C22.D4), (D5×C4⋊C4)⋊3C2, (C5×D4⋊C4)⋊5C2, (C2×C4×D5).12C22, (C2×C10).227(C2×D4), (C5×C4⋊C4).16C22, (C2×C5⋊2C8).15C22, (C2×C4).321(C22×D5), SmallGroup(320,401)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.12D8
G = < a,b,c,d | a10=b2=c8=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=a5c-1 >
Subgroups: 494 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.D8, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×C5⋊D4, D4×C10, C10.D8, C40⋊5C4, D10⋊1C8, D4⋊Dic5, C5×D4⋊C4, D5×C4⋊C4, C20⋊2D4, D10.12D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C22.D4, C2×D8, C8.C22, C22×D5, C22.D8, C4○D20, D4×D5, D4⋊2D5, D10.12D4, D5×D8, SD16⋊D5, D10.12D8
►On 160 points
Generators in S160
(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 32)(2 31)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 50)(21 140)(22 139)(23 138)(24 137)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(51 84)(52 83)(53 82)(54 81)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 71)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(91 129)(92 128)(93 127)(94 126)(95 125)(96 124)(97 123)(98 122)(99 121)(100 130)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 120)(108 119)(109 118)(110 117)(141 151)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)
(1 136 16 153 33 21 45 145)(2 137 17 154 34 22 46 146)(3 138 18 155 35 23 47 147)(4 139 19 156 36 24 48 148)(5 140 20 157 37 25 49 149)(6 131 11 158 38 26 50 150)(7 132 12 159 39 27 41 141)(8 133 13 160 40 28 42 142)(9 134 14 151 31 29 43 143)(10 135 15 152 32 30 44 144)(51 110 70 130 90 118 78 91)(52 101 61 121 81 119 79 92)(53 102 62 122 82 120 80 93)(54 103 63 123 83 111 71 94)(55 104 64 124 84 112 72 95)(56 105 65 125 85 113 73 96)(57 106 66 126 86 114 74 97)(58 107 67 127 87 115 75 98)(59 108 68 128 88 116 76 99)(60 109 69 129 89 117 77 100)
(1 110 6 105)(2 101 7 106)(3 102 8 107)(4 103 9 108)(5 104 10 109)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(21 85 26 90)(22 86 27 81)(23 87 28 82)(24 88 29 83)(25 89 30 84)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
G:=sub<Sym(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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,50)(21,140)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(51,84)(52,83)(53,82)(54,81)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,71)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,130)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,120)(108,119)(109,118)(110,117)(141,151)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152), (1,136,16,153,33,21,45,145)(2,137,17,154,34,22,46,146)(3,138,18,155,35,23,47,147)(4,139,19,156,36,24,48,148)(5,140,20,157,37,25,49,149)(6,131,11,158,38,26,50,150)(7,132,12,159,39,27,41,141)(8,133,13,160,40,28,42,142)(9,134,14,151,31,29,43,143)(10,135,15,152,32,30,44,144)(51,110,70,130,90,118,78,91)(52,101,61,121,81,119,79,92)(53,102,62,122,82,120,80,93)(54,103,63,123,83,111,71,94)(55,104,64,124,84,112,72,95)(56,105,65,125,85,113,73,96)(57,106,66,126,86,114,74,97)(58,107,67,127,87,115,75,98)(59,108,68,128,88,116,76,99)(60,109,69,129,89,117,77,100), (1,110,6,105)(2,101,7,106)(3,102,8,107)(4,103,9,108)(5,104,10,109)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(21,85,26,90)(22,86,27,81)(23,87,28,82)(24,88,29,83)(25,89,30,84)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155)>;
G:=Group( (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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,50)(21,140)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(51,84)(52,83)(53,82)(54,81)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,71)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,130)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,120)(108,119)(109,118)(110,117)(141,151)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152), (1,136,16,153,33,21,45,145)(2,137,17,154,34,22,46,146)(3,138,18,155,35,23,47,147)(4,139,19,156,36,24,48,148)(5,140,20,157,37,25,49,149)(6,131,11,158,38,26,50,150)(7,132,12,159,39,27,41,141)(8,133,13,160,40,28,42,142)(9,134,14,151,31,29,43,143)(10,135,15,152,32,30,44,144)(51,110,70,130,90,118,78,91)(52,101,61,121,81,119,79,92)(53,102,62,122,82,120,80,93)(54,103,63,123,83,111,71,94)(55,104,64,124,84,112,72,95)(56,105,65,125,85,113,73,96)(57,106,66,126,86,114,74,97)(58,107,67,127,87,115,75,98)(59,108,68,128,88,116,76,99)(60,109,69,129,89,117,77,100), (1,110,6,105)(2,101,7,106)(3,102,8,107)(4,103,9,108)(5,104,10,109)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(21,85,26,90)(22,86,27,81)(23,87,28,82)(24,88,29,83)(25,89,30,84)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155) );
G=PermutationGroup([[(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)], [(1,32),(2,31),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,50),(21,140),(22,139),(23,138),(24,137),(25,136),(26,135),(27,134),(28,133),(29,132),(30,131),(51,84),(52,83),(53,82),(54,81),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85),(61,71),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(91,129),(92,128),(93,127),(94,126),(95,125),(96,124),(97,123),(98,122),(99,121),(100,130),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,120),(108,119),(109,118),(110,117),(141,151),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,154),(149,153),(150,152)], [(1,136,16,153,33,21,45,145),(2,137,17,154,34,22,46,146),(3,138,18,155,35,23,47,147),(4,139,19,156,36,24,48,148),(5,140,20,157,37,25,49,149),(6,131,11,158,38,26,50,150),(7,132,12,159,39,27,41,141),(8,133,13,160,40,28,42,142),(9,134,14,151,31,29,43,143),(10,135,15,152,32,30,44,144),(51,110,70,130,90,118,78,91),(52,101,61,121,81,119,79,92),(53,102,62,122,82,120,80,93),(54,103,63,123,83,111,71,94),(55,104,64,124,84,112,72,95),(56,105,65,125,85,113,73,96),(57,106,66,126,86,114,74,97),(58,107,67,127,87,115,75,98),(59,108,68,128,88,116,76,99),(60,109,69,129,89,117,77,100)], [(1,110,6,105),(2,101,7,106),(3,102,8,107),(4,103,9,108),(5,104,10,109),(11,96,16,91),(12,97,17,92),(13,98,18,93),(14,99,19,94),(15,100,20,95),(21,85,26,90),(22,86,27,81),(23,87,28,82),(24,88,29,83),(25,89,30,84),(31,116,36,111),(32,117,37,112),(33,118,38,113),(34,119,39,114),(35,120,40,115),(41,126,46,121),(42,127,47,122),(43,128,48,123),(44,129,49,124),(45,130,50,125),(51,136,56,131),(52,137,57,132),(53,138,58,133),(54,139,59,134),(55,140,60,135),(61,146,66,141),(62,147,67,142),(63,148,68,143),(64,149,69,144),(65,150,70,145),(71,156,76,151),(72,157,77,152),(73,158,78,153),(74,159,79,154),(75,160,80,155)]])
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 | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 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 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 10 | 10 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | D8 | D10 | D10 | D10 | C4○D20 | C8.C22 | D4⋊2D5 | D4×D5 | D5×D8 | SD16⋊D5 |
| kernel | D10.12D8 | C10.D8 | C40⋊5C4 | D10⋊1C8 | D4⋊Dic5 | C5×D4⋊C4 | D5×C4⋊C4 | C20⋊2D4 | C2×Dic5 | C22×D5 | D4⋊C4 | C20 | D10 | C4⋊C4 | C2×C8 | C2×D4 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10.12D8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 35 | 0 | 0 |
| 0 | 0 | 8 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 32 | 18 | 0 | 0 | 0 | 0 |
| 32 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 38 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 9 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 38 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,0,0,0,0,35,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,7,8,0,0,0,0,35,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,32,0,0,0,0,18,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,27,0,0,0,0,0,0,38],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,38,0,0,0,0,27,0] >;
D10.12D8 in GAP, Magma, Sage, TeX
D_{10}._{12}D_8 % in TeX
G:=Group("D10.12D8"); // GroupNames label
G:=SmallGroup(320,401);
// by ID
G=gap.SmallGroup(320,401);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,254,219,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;
// generators/relations