metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.5D8, Dic5.8SD16, C20⋊Q8⋊2C2, C2.7(D5×D8), C4⋊C4.2D10, D4⋊C4⋊1D5, C10.20(C2×D8), C5⋊2(C4.4D8), D20⋊6C4⋊1C2, C2.8(D5×SD16), (C8×Dic5)⋊19C2, (C2×D4).17D10, C20.1(C4○D4), C20⋊D4.4C2, D4⋊Dic5⋊1C2, (C2×C8).199D10, D20⋊5C4⋊15C2, C4.20(C4○D20), C10.19(C2×SD16), C22.162(D4×D5), C4.46(D4⋊2D5), (C2×C40).178C22, (C2×C20).200C23, (C2×Dic5).131D4, (C2×D20).48C22, (D4×C10).21C22, C4⋊Dic5.60C22, C10.22(C4.4D4), (C4×Dic5).253C22, C2.12(Dic5.5D4), (C5×C4⋊C4).5C22, (C5×D4⋊C4)⋊16C2, (C2×C10).213(C2×D4), (C2×C4).307(C22×D5), (C2×C5⋊2C8).217C22, SmallGroup(320,387)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.5D8
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a5b, dcd=a5c-1 >
Subgroups: 590 in 118 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×C8, D4⋊C4, D4⋊C4, C4⋊1D4, C4⋊Q8, C5⋊2C8, C40, Dic10, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C4.4D8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×D20, C2×C5⋊D4, D4×C10, D20⋊6C4, C8×Dic5, D20⋊5C4, D4⋊Dic5, C5×D4⋊C4, C20⋊Q8, C20⋊D4, Dic5.5D8
Quotients: C1, C2, C22, D4, C23, D5, D8, SD16, C2×D4, C4○D4, D10, C4.4D4, C2×D8, C2×SD16, C22×D5, C4.4D8, C4○D20, D4×D5, D4⋊2D5, Dic5.5D4, D5×D8, D5×SD16, Dic5.5D8
►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 96 6 91)(2 95 7 100)(3 94 8 99)(4 93 9 98)(5 92 10 97)(11 61 16 66)(12 70 17 65)(13 69 18 64)(14 68 19 63)(15 67 20 62)(21 105 26 110)(22 104 27 109)(23 103 28 108)(24 102 29 107)(25 101 30 106)(31 123 36 128)(32 122 37 127)(33 121 38 126)(34 130 39 125)(35 129 40 124)(41 113 46 118)(42 112 47 117)(43 111 48 116)(44 120 49 115)(45 119 50 114)(51 156 56 151)(52 155 57 160)(53 154 58 159)(54 153 59 158)(55 152 60 157)(71 136 76 131)(72 135 77 140)(73 134 78 139)(74 133 79 138)(75 132 80 137)(81 150 86 145)(82 149 87 144)(83 148 88 143)(84 147 89 142)(85 146 90 141)
(1 151 31 148 28 19 43 136)(2 152 32 149 29 20 44 137)(3 153 33 150 30 11 45 138)(4 154 34 141 21 12 46 139)(5 155 35 142 22 13 47 140)(6 156 36 143 23 14 48 131)(7 157 37 144 24 15 49 132)(8 158 38 145 25 16 50 133)(9 159 39 146 26 17 41 134)(10 160 40 147 27 18 42 135)(51 123 88 108 63 111 76 96)(52 124 89 109 64 112 77 97)(53 125 90 110 65 113 78 98)(54 126 81 101 66 114 79 99)(55 127 82 102 67 115 80 100)(56 128 83 103 68 116 71 91)(57 129 84 104 69 117 72 92)(58 130 85 105 70 118 73 93)(59 121 86 106 61 119 74 94)(60 122 87 107 62 120 75 95)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 91)(7 92)(8 93)(9 94)(10 95)(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), (1,96,6,91)(2,95,7,100)(3,94,8,99)(4,93,9,98)(5,92,10,97)(11,61,16,66)(12,70,17,65)(13,69,18,64)(14,68,19,63)(15,67,20,62)(21,105,26,110)(22,104,27,109)(23,103,28,108)(24,102,29,107)(25,101,30,106)(31,123,36,128)(32,122,37,127)(33,121,38,126)(34,130,39,125)(35,129,40,124)(41,113,46,118)(42,112,47,117)(43,111,48,116)(44,120,49,115)(45,119,50,114)(51,156,56,151)(52,155,57,160)(53,154,58,159)(54,153,59,158)(55,152,60,157)(71,136,76,131)(72,135,77,140)(73,134,78,139)(74,133,79,138)(75,132,80,137)(81,150,86,145)(82,149,87,144)(83,148,88,143)(84,147,89,142)(85,146,90,141), (1,151,31,148,28,19,43,136)(2,152,32,149,29,20,44,137)(3,153,33,150,30,11,45,138)(4,154,34,141,21,12,46,139)(5,155,35,142,22,13,47,140)(6,156,36,143,23,14,48,131)(7,157,37,144,24,15,49,132)(8,158,38,145,25,16,50,133)(9,159,39,146,26,17,41,134)(10,160,40,147,27,18,42,135)(51,123,88,108,63,111,76,96)(52,124,89,109,64,112,77,97)(53,125,90,110,65,113,78,98)(54,126,81,101,66,114,79,99)(55,127,82,102,67,115,80,100)(56,128,83,103,68,116,71,91)(57,129,84,104,69,117,72,92)(58,130,85,105,70,118,73,93)(59,121,86,106,61,119,74,94)(60,122,87,107,62,120,75,95), (1,96)(2,97)(3,98)(4,99)(5,100)(6,91)(7,92)(8,93)(9,94)(10,95)(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), (1,96,6,91)(2,95,7,100)(3,94,8,99)(4,93,9,98)(5,92,10,97)(11,61,16,66)(12,70,17,65)(13,69,18,64)(14,68,19,63)(15,67,20,62)(21,105,26,110)(22,104,27,109)(23,103,28,108)(24,102,29,107)(25,101,30,106)(31,123,36,128)(32,122,37,127)(33,121,38,126)(34,130,39,125)(35,129,40,124)(41,113,46,118)(42,112,47,117)(43,111,48,116)(44,120,49,115)(45,119,50,114)(51,156,56,151)(52,155,57,160)(53,154,58,159)(54,153,59,158)(55,152,60,157)(71,136,76,131)(72,135,77,140)(73,134,78,139)(74,133,79,138)(75,132,80,137)(81,150,86,145)(82,149,87,144)(83,148,88,143)(84,147,89,142)(85,146,90,141), (1,151,31,148,28,19,43,136)(2,152,32,149,29,20,44,137)(3,153,33,150,30,11,45,138)(4,154,34,141,21,12,46,139)(5,155,35,142,22,13,47,140)(6,156,36,143,23,14,48,131)(7,157,37,144,24,15,49,132)(8,158,38,145,25,16,50,133)(9,159,39,146,26,17,41,134)(10,160,40,147,27,18,42,135)(51,123,88,108,63,111,76,96)(52,124,89,109,64,112,77,97)(53,125,90,110,65,113,78,98)(54,126,81,101,66,114,79,99)(55,127,82,102,67,115,80,100)(56,128,83,103,68,116,71,91)(57,129,84,104,69,117,72,92)(58,130,85,105,70,118,73,93)(59,121,86,106,61,119,74,94)(60,122,87,107,62,120,75,95), (1,96)(2,97)(3,98)(4,99)(5,100)(6,91)(7,92)(8,93)(9,94)(10,95)(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)], [(1,96,6,91),(2,95,7,100),(3,94,8,99),(4,93,9,98),(5,92,10,97),(11,61,16,66),(12,70,17,65),(13,69,18,64),(14,68,19,63),(15,67,20,62),(21,105,26,110),(22,104,27,109),(23,103,28,108),(24,102,29,107),(25,101,30,106),(31,123,36,128),(32,122,37,127),(33,121,38,126),(34,130,39,125),(35,129,40,124),(41,113,46,118),(42,112,47,117),(43,111,48,116),(44,120,49,115),(45,119,50,114),(51,156,56,151),(52,155,57,160),(53,154,58,159),(54,153,59,158),(55,152,60,157),(71,136,76,131),(72,135,77,140),(73,134,78,139),(74,133,79,138),(75,132,80,137),(81,150,86,145),(82,149,87,144),(83,148,88,143),(84,147,89,142),(85,146,90,141)], [(1,151,31,148,28,19,43,136),(2,152,32,149,29,20,44,137),(3,153,33,150,30,11,45,138),(4,154,34,141,21,12,46,139),(5,155,35,142,22,13,47,140),(6,156,36,143,23,14,48,131),(7,157,37,144,24,15,49,132),(8,158,38,145,25,16,50,133),(9,159,39,146,26,17,41,134),(10,160,40,147,27,18,42,135),(51,123,88,108,63,111,76,96),(52,124,89,109,64,112,77,97),(53,125,90,110,65,113,78,98),(54,126,81,101,66,114,79,99),(55,127,82,102,67,115,80,100),(56,128,83,103,68,116,71,91),(57,129,84,104,69,117,72,92),(58,130,85,105,70,118,73,93),(59,121,86,106,61,119,74,94),(60,122,87,107,62,120,75,95)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,91),(7,92),(8,93),(9,94),(10,95),(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)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 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 | 40 | 2 | 2 | 8 | 10 | 10 | 10 | 10 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D8 | SD16 | C4○D4 | D10 | D10 | D10 | C4○D20 | D4⋊2D5 | D4×D5 | D5×D8 | D5×SD16 |
| kernel | Dic5.5D8 | D20⋊6C4 | C8×Dic5 | D20⋊5C4 | D4⋊Dic5 | C5×D4⋊C4 | C20⋊Q8 | C20⋊D4 | C2×Dic5 | D4⋊C4 | Dic5 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5.5D8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 15 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 29 | 12 |
| 40 | 2 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,15,0,0,0,0,11,11,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,29,0,0,0,0,12,12],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
Dic5.5D8 in GAP, Magma, Sage, TeX
{\rm Dic}_5._5D_8 % in TeX
G:=Group("Dic5.5D8"); // GroupNames label
G:=SmallGroup(320,387);
// by ID
G=gap.SmallGroup(320,387);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,422,135,100,570,297,136,12550]);
// 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,b*c=c*b,d*b*d=a^5*b,d*c*d=a^5*c^-1>;
// generators/relations