metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊3D8, (C2×D8)⋊2D5, (C5×D4)⋊4D4, C5⋊5(C4⋊D8), C2.26(D5×D8), (C10×D8)⋊12C2, C20⋊D4⋊3C2, D4⋊1(C5⋊D4), (D4×Dic5)⋊4C2, (C2×C8).33D10, C10.43(C2×D8), C20.162(C2×D4), D20⋊5C4⋊28C2, (C2×D4).140D10, C20.90(C4○D4), D4⋊Dic5⋊25C2, C4.7(D4⋊2D5), C20.8Q8⋊27C2, C22.251(D4×D5), C2.26(D8⋊D5), C10.46(C8⋊C22), (C2×C40).247C22, (C2×C20).427C23, (C2×Dic5).235D4, (D4×C10).77C22, C10.107(C4⋊D4), (C2×D20).117C22, C4⋊Dic5.162C22, (C4×Dic5).48C22, C2.22(Dic5⋊D4), (C2×D4⋊D5)⋊16C2, C4.34(C2×C5⋊D4), (C2×C10).340(C2×D4), (C2×C4).517(C22×D5), (C2×C5⋊2C8).145C22, SmallGroup(320,777)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊D8
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c-1 >
Subgroups: 662 in 140 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, C2×D8, C5⋊2C8, C40, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C4⋊D8, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D4⋊D5, C23.D5, C2×C40, C5×D8, C2×D20, C22×Dic5, C2×C5⋊D4, D4×C10, C20.8Q8, D20⋊5C4, D4⋊Dic5, C2×D4⋊D5, D4×Dic5, C20⋊D4, C10×D8, Dic5⋊D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C8⋊C22, C5⋊D4, C22×D5, C4⋊D8, D4×D5, D4⋊2D5, C2×C5⋊D4, D5×D8, D8⋊D5, Dic5⋊D4, Dic5⋊D8
►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 60 6 55)(2 59 7 54)(3 58 8 53)(4 57 9 52)(5 56 10 51)(11 118 16 113)(12 117 17 112)(13 116 18 111)(14 115 19 120)(15 114 20 119)(21 65 26 70)(22 64 27 69)(23 63 28 68)(24 62 29 67)(25 61 30 66)(31 87 36 82)(32 86 37 81)(33 85 38 90)(34 84 39 89)(35 83 40 88)(41 77 46 72)(42 76 47 71)(43 75 48 80)(44 74 49 79)(45 73 50 78)(91 139 96 134)(92 138 97 133)(93 137 98 132)(94 136 99 131)(95 135 100 140)(101 145 106 150)(102 144 107 149)(103 143 108 148)(104 142 109 147)(105 141 110 146)(121 157 126 152)(122 156 127 151)(123 155 128 160)(124 154 129 159)(125 153 130 158)
(1 160 40 148 28 14 48 140)(2 151 31 149 29 15 49 131)(3 152 32 150 30 16 50 132)(4 153 33 141 21 17 41 133)(5 154 34 142 22 18 42 134)(6 155 35 143 23 19 43 135)(7 156 36 144 24 20 44 136)(8 157 37 145 25 11 45 137)(9 158 38 146 26 12 46 138)(10 159 39 147 27 13 47 139)(51 129 89 109 69 111 71 91)(52 130 90 110 70 112 72 92)(53 121 81 101 61 113 73 93)(54 122 82 102 62 114 74 94)(55 123 83 103 63 115 75 95)(56 124 84 104 64 116 76 96)(57 125 85 105 65 117 77 97)(58 126 86 106 66 118 78 98)(59 127 87 107 67 119 79 99)(60 128 88 108 68 120 80 100)
(1 100)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
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), (1,60,6,55)(2,59,7,54)(3,58,8,53)(4,57,9,52)(5,56,10,51)(11,118,16,113)(12,117,17,112)(13,116,18,111)(14,115,19,120)(15,114,20,119)(21,65,26,70)(22,64,27,69)(23,63,28,68)(24,62,29,67)(25,61,30,66)(31,87,36,82)(32,86,37,81)(33,85,38,90)(34,84,39,89)(35,83,40,88)(41,77,46,72)(42,76,47,71)(43,75,48,80)(44,74,49,79)(45,73,50,78)(91,139,96,134)(92,138,97,133)(93,137,98,132)(94,136,99,131)(95,135,100,140)(101,145,106,150)(102,144,107,149)(103,143,108,148)(104,142,109,147)(105,141,110,146)(121,157,126,152)(122,156,127,151)(123,155,128,160)(124,154,129,159)(125,153,130,158), (1,160,40,148,28,14,48,140)(2,151,31,149,29,15,49,131)(3,152,32,150,30,16,50,132)(4,153,33,141,21,17,41,133)(5,154,34,142,22,18,42,134)(6,155,35,143,23,19,43,135)(7,156,36,144,24,20,44,136)(8,157,37,145,25,11,45,137)(9,158,38,146,26,12,46,138)(10,159,39,147,27,13,47,139)(51,129,89,109,69,111,71,91)(52,130,90,110,70,112,72,92)(53,121,81,101,61,113,73,93)(54,122,82,102,62,114,74,94)(55,123,83,103,63,115,75,95)(56,124,84,104,64,116,76,96)(57,125,85,105,65,117,77,97)(58,126,86,106,66,118,78,98)(59,127,87,107,67,119,79,99)(60,128,88,108,68,120,80,100), (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)>;
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), (1,60,6,55)(2,59,7,54)(3,58,8,53)(4,57,9,52)(5,56,10,51)(11,118,16,113)(12,117,17,112)(13,116,18,111)(14,115,19,120)(15,114,20,119)(21,65,26,70)(22,64,27,69)(23,63,28,68)(24,62,29,67)(25,61,30,66)(31,87,36,82)(32,86,37,81)(33,85,38,90)(34,84,39,89)(35,83,40,88)(41,77,46,72)(42,76,47,71)(43,75,48,80)(44,74,49,79)(45,73,50,78)(91,139,96,134)(92,138,97,133)(93,137,98,132)(94,136,99,131)(95,135,100,140)(101,145,106,150)(102,144,107,149)(103,143,108,148)(104,142,109,147)(105,141,110,146)(121,157,126,152)(122,156,127,151)(123,155,128,160)(124,154,129,159)(125,153,130,158), (1,160,40,148,28,14,48,140)(2,151,31,149,29,15,49,131)(3,152,32,150,30,16,50,132)(4,153,33,141,21,17,41,133)(5,154,34,142,22,18,42,134)(6,155,35,143,23,19,43,135)(7,156,36,144,24,20,44,136)(8,157,37,145,25,11,45,137)(9,158,38,146,26,12,46,138)(10,159,39,147,27,13,47,139)(51,129,89,109,69,111,71,91)(52,130,90,110,70,112,72,92)(53,121,81,101,61,113,73,93)(54,122,82,102,62,114,74,94)(55,123,83,103,63,115,75,95)(56,124,84,104,64,116,76,96)(57,125,85,105,65,117,77,97)(58,126,86,106,66,118,78,98)(59,127,87,107,67,119,79,99)(60,128,88,108,68,120,80,100), (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160) );
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)], [(1,60,6,55),(2,59,7,54),(3,58,8,53),(4,57,9,52),(5,56,10,51),(11,118,16,113),(12,117,17,112),(13,116,18,111),(14,115,19,120),(15,114,20,119),(21,65,26,70),(22,64,27,69),(23,63,28,68),(24,62,29,67),(25,61,30,66),(31,87,36,82),(32,86,37,81),(33,85,38,90),(34,84,39,89),(35,83,40,88),(41,77,46,72),(42,76,47,71),(43,75,48,80),(44,74,49,79),(45,73,50,78),(91,139,96,134),(92,138,97,133),(93,137,98,132),(94,136,99,131),(95,135,100,140),(101,145,106,150),(102,144,107,149),(103,143,108,148),(104,142,109,147),(105,141,110,146),(121,157,126,152),(122,156,127,151),(123,155,128,160),(124,154,129,159),(125,153,130,158)], [(1,160,40,148,28,14,48,140),(2,151,31,149,29,15,49,131),(3,152,32,150,30,16,50,132),(4,153,33,141,21,17,41,133),(5,154,34,142,22,18,42,134),(6,155,35,143,23,19,43,135),(7,156,36,144,24,20,44,136),(8,157,37,145,25,11,45,137),(9,158,38,146,26,12,46,138),(10,159,39,147,27,13,47,139),(51,129,89,109,69,111,71,91),(52,130,90,110,70,112,72,92),(53,121,81,101,61,113,73,93),(54,122,82,102,62,114,74,94),(55,123,83,103,63,115,75,95),(56,124,84,104,64,116,76,96),(57,125,85,105,65,117,77,97),(58,126,86,106,66,118,78,98),(59,127,87,107,67,119,79,99),(60,128,88,108,68,120,80,100)], [(1,100),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,81),(18,82),(19,83),(20,84),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 40 | 2 | 2 | 10 | 10 | 20 | 20 | 20 | 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 | D8 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4⋊2D5 | D4×D5 | D5×D8 | D8⋊D5 |
| kernel | Dic5⋊D8 | C20.8Q8 | D20⋊5C4 | D4⋊Dic5 | C2×D4⋊D5 | D4×Dic5 | C20⋊D4 | C10×D8 | C2×Dic5 | C5×D4 | C2×D8 | Dic5 | C20 | C2×C8 | C2×D4 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5⋊D8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 33 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 15 | 4 | 0 | 0 | 0 | 0 |
| 5 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 35 | 0 | 0 |
| 0 | 0 | 8 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 26 | 37 | 0 | 0 | 0 | 0 |
| 15 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 22 |
| 0 | 0 | 0 | 0 | 3 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 22 |
| 0 | 0 | 0 | 0 | 4 | 35 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,5,0,0,0,0,4,26,0,0,0,0,0,0,34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,15,0,0,0,0,37,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,3,0,0,0,0,22,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,4,0,0,0,0,22,35] >;
Dic5⋊D8 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes D_8 % in TeX
G:=Group("Dic5:D8"); // GroupNames label
G:=SmallGroup(320,777);
// by ID
G=gap.SmallGroup(320,777);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations