metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊4D8, C5⋊3(C4×D8), D4⋊D5⋊5C4, D4⋊1(C4×D5), C2.1(D5×D8), D20⋊11(C2×C4), (D4×Dic5)⋊1C2, C10.60(C4×D4), C10.18(C2×D8), D20⋊8C4⋊1C2, D4⋊C4⋊21D5, C4⋊C4.128D10, (C8×Dic5)⋊17C2, (C2×C8).197D10, C10.D8⋊1C2, D20⋊5C4⋊14C2, C22.65(D4×D5), (C2×D4).123D10, C10.36(C4○D8), C20.36(C22×C4), C20.142(C4○D4), C4.43(D4⋊2D5), (C2×C40).175C22, (C2×C20).196C23, (C2×Dic5).267D4, (D4×C10).17C22, (C2×D20).47C22, C4⋊Dic5.56C22, C2.1(SD16⋊3D5), C2.14(Dic5⋊4D4), (C4×Dic5).251C22, C4.1(C2×C4×D5), C5⋊2C8⋊17(C2×C4), (C5×D4)⋊11(C2×C4), (C2×D4⋊D5).1C2, (C5×C4⋊C4).1C22, (C5×D4⋊C4)⋊15C2, (C2×C10).209(C2×D4), (C2×C4).303(C22×D5), (C2×C5⋊2C8).215C22, SmallGroup(320,383)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊4D8
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 566 in 134 conjugacy classes, 51 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, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×C8, D4⋊C4, D4⋊C4, C2.D8, C4×D4, C2×D8, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C4×D8, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D10⋊C4, D4⋊D5, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C22×Dic5, D4×C10, C10.D8, C8×Dic5, D20⋊5C4, C5×D4⋊C4, D20⋊8C4, C2×D4⋊D5, D4×Dic5, Dic5⋊4D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, D8, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×D8, C4○D8, C4×D5, C22×D5, C4×D8, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, D5×D8, SD16⋊3D5, Dic5⋊4D8
►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 68 6 63)(2 67 7 62)(3 66 8 61)(4 65 9 70)(5 64 10 69)(11 126 16 121)(12 125 17 130)(13 124 18 129)(14 123 19 128)(15 122 20 127)(21 51 26 56)(22 60 27 55)(23 59 28 54)(24 58 29 53)(25 57 30 52)(31 77 36 72)(32 76 37 71)(33 75 38 80)(34 74 39 79)(35 73 40 78)(41 85 46 90)(42 84 47 89)(43 83 48 88)(44 82 49 87)(45 81 50 86)(91 141 96 146)(92 150 97 145)(93 149 98 144)(94 148 99 143)(95 147 100 142)(101 131 106 136)(102 140 107 135)(103 139 108 134)(104 138 109 133)(105 137 110 132)(111 157 116 152)(112 156 117 151)(113 155 118 160)(114 154 119 159)(115 153 120 158)
(1 114 34 94 28 128 48 108)(2 113 35 93 29 127 49 107)(3 112 36 92 30 126 50 106)(4 111 37 91 21 125 41 105)(5 120 38 100 22 124 42 104)(6 119 39 99 23 123 43 103)(7 118 40 98 24 122 44 102)(8 117 31 97 25 121 45 101)(9 116 32 96 26 130 46 110)(10 115 33 95 27 129 47 109)(11 81 131 61 151 77 145 57)(12 90 132 70 152 76 146 56)(13 89 133 69 153 75 147 55)(14 88 134 68 154 74 148 54)(15 87 135 67 155 73 149 53)(16 86 136 66 156 72 150 52)(17 85 137 65 157 71 141 51)(18 84 138 64 158 80 142 60)(19 83 139 63 159 79 143 59)(20 82 140 62 160 78 144 58)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 91)(9 92)(10 93)(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), (1,68,6,63)(2,67,7,62)(3,66,8,61)(4,65,9,70)(5,64,10,69)(11,126,16,121)(12,125,17,130)(13,124,18,129)(14,123,19,128)(15,122,20,127)(21,51,26,56)(22,60,27,55)(23,59,28,54)(24,58,29,53)(25,57,30,52)(31,77,36,72)(32,76,37,71)(33,75,38,80)(34,74,39,79)(35,73,40,78)(41,85,46,90)(42,84,47,89)(43,83,48,88)(44,82,49,87)(45,81,50,86)(91,141,96,146)(92,150,97,145)(93,149,98,144)(94,148,99,143)(95,147,100,142)(101,131,106,136)(102,140,107,135)(103,139,108,134)(104,138,109,133)(105,137,110,132)(111,157,116,152)(112,156,117,151)(113,155,118,160)(114,154,119,159)(115,153,120,158), (1,114,34,94,28,128,48,108)(2,113,35,93,29,127,49,107)(3,112,36,92,30,126,50,106)(4,111,37,91,21,125,41,105)(5,120,38,100,22,124,42,104)(6,119,39,99,23,123,43,103)(7,118,40,98,24,122,44,102)(8,117,31,97,25,121,45,101)(9,116,32,96,26,130,46,110)(10,115,33,95,27,129,47,109)(11,81,131,61,151,77,145,57)(12,90,132,70,152,76,146,56)(13,89,133,69,153,75,147,55)(14,88,134,68,154,74,148,54)(15,87,135,67,155,73,149,53)(16,86,136,66,156,72,150,52)(17,85,137,65,157,71,141,51)(18,84,138,64,158,80,142,60)(19,83,139,63,159,79,143,59)(20,82,140,62,160,78,144,58), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(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), (1,68,6,63)(2,67,7,62)(3,66,8,61)(4,65,9,70)(5,64,10,69)(11,126,16,121)(12,125,17,130)(13,124,18,129)(14,123,19,128)(15,122,20,127)(21,51,26,56)(22,60,27,55)(23,59,28,54)(24,58,29,53)(25,57,30,52)(31,77,36,72)(32,76,37,71)(33,75,38,80)(34,74,39,79)(35,73,40,78)(41,85,46,90)(42,84,47,89)(43,83,48,88)(44,82,49,87)(45,81,50,86)(91,141,96,146)(92,150,97,145)(93,149,98,144)(94,148,99,143)(95,147,100,142)(101,131,106,136)(102,140,107,135)(103,139,108,134)(104,138,109,133)(105,137,110,132)(111,157,116,152)(112,156,117,151)(113,155,118,160)(114,154,119,159)(115,153,120,158), (1,114,34,94,28,128,48,108)(2,113,35,93,29,127,49,107)(3,112,36,92,30,126,50,106)(4,111,37,91,21,125,41,105)(5,120,38,100,22,124,42,104)(6,119,39,99,23,123,43,103)(7,118,40,98,24,122,44,102)(8,117,31,97,25,121,45,101)(9,116,32,96,26,130,46,110)(10,115,33,95,27,129,47,109)(11,81,131,61,151,77,145,57)(12,90,132,70,152,76,146,56)(13,89,133,69,153,75,147,55)(14,88,134,68,154,74,148,54)(15,87,135,67,155,73,149,53)(16,86,136,66,156,72,150,52)(17,85,137,65,157,71,141,51)(18,84,138,64,158,80,142,60)(19,83,139,63,159,79,143,59)(20,82,140,62,160,78,144,58), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(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)], [(1,68,6,63),(2,67,7,62),(3,66,8,61),(4,65,9,70),(5,64,10,69),(11,126,16,121),(12,125,17,130),(13,124,18,129),(14,123,19,128),(15,122,20,127),(21,51,26,56),(22,60,27,55),(23,59,28,54),(24,58,29,53),(25,57,30,52),(31,77,36,72),(32,76,37,71),(33,75,38,80),(34,74,39,79),(35,73,40,78),(41,85,46,90),(42,84,47,89),(43,83,48,88),(44,82,49,87),(45,81,50,86),(91,141,96,146),(92,150,97,145),(93,149,98,144),(94,148,99,143),(95,147,100,142),(101,131,106,136),(102,140,107,135),(103,139,108,134),(104,138,109,133),(105,137,110,132),(111,157,116,152),(112,156,117,151),(113,155,118,160),(114,154,119,159),(115,153,120,158)], [(1,114,34,94,28,128,48,108),(2,113,35,93,29,127,49,107),(3,112,36,92,30,126,50,106),(4,111,37,91,21,125,41,105),(5,120,38,100,22,124,42,104),(6,119,39,99,23,123,43,103),(7,118,40,98,24,122,44,102),(8,117,31,97,25,121,45,101),(9,116,32,96,26,130,46,110),(10,115,33,95,27,129,47,109),(11,81,131,61,151,77,145,57),(12,90,132,70,152,76,146,56),(13,89,133,69,153,75,147,55),(14,88,134,68,154,74,148,54),(15,87,135,67,155,73,149,53),(16,86,136,66,156,72,150,52),(17,85,137,65,157,71,141,51),(18,84,138,64,158,80,142,60),(19,83,139,63,159,79,143,59),(20,82,140,62,160,78,144,58)], [(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,91),(9,92),(10,93),(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 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 |
56 irreducible representations
| dim | 1 | 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 | C4 | D4 | D5 | D8 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4×D5 | D4⋊2D5 | D4×D5 | D5×D8 | SD16⋊3D5 |
| kernel | Dic5⋊4D8 | C10.D8 | C8×Dic5 | D20⋊5C4 | C5×D4⋊C4 | D20⋊8C4 | C2×D4⋊D5 | D4×Dic5 | D4⋊D5 | C2×Dic5 | D4⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C10 | D4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5⋊4D8 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 7 | 40 |
| 0 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 22 | 19 |
| 0 | 0 | 9 | 19 |
| 0 | 12 | 0 | 0 |
| 17 | 17 | 0 | 0 |
| 0 | 0 | 7 | 34 |
| 0 | 0 | 1 | 34 |
| 0 | 12 | 0 | 0 |
| 24 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,7,1,0,0,40,0],[9,0,0,0,0,9,0,0,0,0,22,9,0,0,19,19],[0,17,0,0,12,17,0,0,0,0,7,1,0,0,34,34],[0,24,0,0,12,0,0,0,0,0,40,0,0,0,0,40] >;
Dic5⋊4D8 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_4D_8 % in TeX
G:=Group("Dic5:4D8"); // GroupNames label
G:=SmallGroup(320,383);
// by ID
G=gap.SmallGroup(320,383);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,135,100,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations