metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊4Q8, C20.13D8, C4.13D40, C42.37D10, C4⋊C8⋊4D5, C10.8(C2×D8), C4.44(Q8×D5), C40⋊5C4⋊12C2, C5⋊2(D4⋊Q8), C2.10(C2×D40), (C2×C8).22D10, C20⋊2Q8⋊13C2, (C4×D20).12C2, (C2×C4).134D20, (C2×C20).123D4, C20.103(C2×Q8), D20⋊5C4.3C2, (C2×C40).25C22, (C4×C20).72C22, C20.287(C4○D4), (C2×C20).756C23, C22.119(C2×D20), C10.31(C22⋊Q8), C4⋊Dic5.19C22, C4.111(D4⋊2D5), C2.12(D10⋊2Q8), C2.19(C8.D10), (C2×D20).202C22, C10.16(C8.C22), (C5×C4⋊C8)⋊6C2, (C2×C10).139(C2×D4), (C2×C4).701(C22×D5), SmallGroup(320,473)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊4Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 518 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D4⋊Q8, C4⋊Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C40⋊5C4, D20⋊5C4, C5×C4⋊C8, C20⋊2Q8, C4×D20, D20⋊4Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, D8, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×D8, C8.C22, D20, C22×D5, D4⋊Q8, D40, C2×D20, D4⋊2D5, Q8×D5, D10⋊2Q8, C2×D40, C8.D10, D20⋊4Q8
►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 64)(2 63)(3 62)(4 61)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 160)(22 159)(23 158)(24 157)(25 156)(26 155)(27 154)(28 153)(29 152)(30 151)(31 150)(32 149)(33 148)(34 147)(35 146)(36 145)(37 144)(38 143)(39 142)(40 141)(41 102)(42 101)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 108)(56 107)(57 106)(58 105)(59 104)(60 103)(81 128)(82 127)(83 126)(84 125)(85 124)(86 123)(87 122)(88 121)(89 140)(90 139)(91 138)(92 137)(93 136)(94 135)(95 134)(96 133)(97 132)(98 131)(99 130)(100 129)
(1 83 65 122)(2 82 66 121)(3 81 67 140)(4 100 68 139)(5 99 69 138)(6 98 70 137)(7 97 71 136)(8 96 72 135)(9 95 73 134)(10 94 74 133)(11 93 75 132)(12 92 76 131)(13 91 77 130)(14 90 78 129)(15 89 79 128)(16 88 80 127)(17 87 61 126)(18 86 62 125)(19 85 63 124)(20 84 64 123)(21 58 151 111)(22 57 152 110)(23 56 153 109)(24 55 154 108)(25 54 155 107)(26 53 156 106)(27 52 157 105)(28 51 158 104)(29 50 159 103)(30 49 160 102)(31 48 141 101)(32 47 142 120)(33 46 143 119)(34 45 144 118)(35 44 145 117)(36 43 146 116)(37 42 147 115)(38 41 148 114)(39 60 149 113)(40 59 150 112)
(1 156 65 26)(2 157 66 27)(3 158 67 28)(4 159 68 29)(5 160 69 30)(6 141 70 31)(7 142 71 32)(8 143 72 33)(9 144 73 34)(10 145 74 35)(11 146 75 36)(12 147 76 37)(13 148 77 38)(14 149 78 39)(15 150 79 40)(16 151 80 21)(17 152 61 22)(18 153 62 23)(19 154 63 24)(20 155 64 25)(41 130 114 91)(42 131 115 92)(43 132 116 93)(44 133 117 94)(45 134 118 95)(46 135 119 96)(47 136 120 97)(48 137 101 98)(49 138 102 99)(50 139 103 100)(51 140 104 81)(52 121 105 82)(53 122 106 83)(54 123 107 84)(55 124 108 85)(56 125 109 86)(57 126 110 87)(58 127 111 88)(59 128 112 89)(60 129 113 90)
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), (1,64)(2,63)(3,62)(4,61)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,160)(22,159)(23,158)(24,157)(25,156)(26,155)(27,154)(28,153)(29,152)(30,151)(31,150)(32,149)(33,148)(34,147)(35,146)(36,145)(37,144)(38,143)(39,142)(40,141)(41,102)(42,101)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129), (1,83,65,122)(2,82,66,121)(3,81,67,140)(4,100,68,139)(5,99,69,138)(6,98,70,137)(7,97,71,136)(8,96,72,135)(9,95,73,134)(10,94,74,133)(11,93,75,132)(12,92,76,131)(13,91,77,130)(14,90,78,129)(15,89,79,128)(16,88,80,127)(17,87,61,126)(18,86,62,125)(19,85,63,124)(20,84,64,123)(21,58,151,111)(22,57,152,110)(23,56,153,109)(24,55,154,108)(25,54,155,107)(26,53,156,106)(27,52,157,105)(28,51,158,104)(29,50,159,103)(30,49,160,102)(31,48,141,101)(32,47,142,120)(33,46,143,119)(34,45,144,118)(35,44,145,117)(36,43,146,116)(37,42,147,115)(38,41,148,114)(39,60,149,113)(40,59,150,112), (1,156,65,26)(2,157,66,27)(3,158,67,28)(4,159,68,29)(5,160,69,30)(6,141,70,31)(7,142,71,32)(8,143,72,33)(9,144,73,34)(10,145,74,35)(11,146,75,36)(12,147,76,37)(13,148,77,38)(14,149,78,39)(15,150,79,40)(16,151,80,21)(17,152,61,22)(18,153,62,23)(19,154,63,24)(20,155,64,25)(41,130,114,91)(42,131,115,92)(43,132,116,93)(44,133,117,94)(45,134,118,95)(46,135,119,96)(47,136,120,97)(48,137,101,98)(49,138,102,99)(50,139,103,100)(51,140,104,81)(52,121,105,82)(53,122,106,83)(54,123,107,84)(55,124,108,85)(56,125,109,86)(57,126,110,87)(58,127,111,88)(59,128,112,89)(60,129,113,90)>;
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), (1,64)(2,63)(3,62)(4,61)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,160)(22,159)(23,158)(24,157)(25,156)(26,155)(27,154)(28,153)(29,152)(30,151)(31,150)(32,149)(33,148)(34,147)(35,146)(36,145)(37,144)(38,143)(39,142)(40,141)(41,102)(42,101)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129), (1,83,65,122)(2,82,66,121)(3,81,67,140)(4,100,68,139)(5,99,69,138)(6,98,70,137)(7,97,71,136)(8,96,72,135)(9,95,73,134)(10,94,74,133)(11,93,75,132)(12,92,76,131)(13,91,77,130)(14,90,78,129)(15,89,79,128)(16,88,80,127)(17,87,61,126)(18,86,62,125)(19,85,63,124)(20,84,64,123)(21,58,151,111)(22,57,152,110)(23,56,153,109)(24,55,154,108)(25,54,155,107)(26,53,156,106)(27,52,157,105)(28,51,158,104)(29,50,159,103)(30,49,160,102)(31,48,141,101)(32,47,142,120)(33,46,143,119)(34,45,144,118)(35,44,145,117)(36,43,146,116)(37,42,147,115)(38,41,148,114)(39,60,149,113)(40,59,150,112), (1,156,65,26)(2,157,66,27)(3,158,67,28)(4,159,68,29)(5,160,69,30)(6,141,70,31)(7,142,71,32)(8,143,72,33)(9,144,73,34)(10,145,74,35)(11,146,75,36)(12,147,76,37)(13,148,77,38)(14,149,78,39)(15,150,79,40)(16,151,80,21)(17,152,61,22)(18,153,62,23)(19,154,63,24)(20,155,64,25)(41,130,114,91)(42,131,115,92)(43,132,116,93)(44,133,117,94)(45,134,118,95)(46,135,119,96)(47,136,120,97)(48,137,101,98)(49,138,102,99)(50,139,103,100)(51,140,104,81)(52,121,105,82)(53,122,106,83)(54,123,107,84)(55,124,108,85)(56,125,109,86)(57,126,110,87)(58,127,111,88)(59,128,112,89)(60,129,113,90) );
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)], [(1,64),(2,63),(3,62),(4,61),(5,80),(6,79),(7,78),(8,77),(9,76),(10,75),(11,74),(12,73),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,160),(22,159),(23,158),(24,157),(25,156),(26,155),(27,154),(28,153),(29,152),(30,151),(31,150),(32,149),(33,148),(34,147),(35,146),(36,145),(37,144),(38,143),(39,142),(40,141),(41,102),(42,101),(43,120),(44,119),(45,118),(46,117),(47,116),(48,115),(49,114),(50,113),(51,112),(52,111),(53,110),(54,109),(55,108),(56,107),(57,106),(58,105),(59,104),(60,103),(81,128),(82,127),(83,126),(84,125),(85,124),(86,123),(87,122),(88,121),(89,140),(90,139),(91,138),(92,137),(93,136),(94,135),(95,134),(96,133),(97,132),(98,131),(99,130),(100,129)], [(1,83,65,122),(2,82,66,121),(3,81,67,140),(4,100,68,139),(5,99,69,138),(6,98,70,137),(7,97,71,136),(8,96,72,135),(9,95,73,134),(10,94,74,133),(11,93,75,132),(12,92,76,131),(13,91,77,130),(14,90,78,129),(15,89,79,128),(16,88,80,127),(17,87,61,126),(18,86,62,125),(19,85,63,124),(20,84,64,123),(21,58,151,111),(22,57,152,110),(23,56,153,109),(24,55,154,108),(25,54,155,107),(26,53,156,106),(27,52,157,105),(28,51,158,104),(29,50,159,103),(30,49,160,102),(31,48,141,101),(32,47,142,120),(33,46,143,119),(34,45,144,118),(35,44,145,117),(36,43,146,116),(37,42,147,115),(38,41,148,114),(39,60,149,113),(40,59,150,112)], [(1,156,65,26),(2,157,66,27),(3,158,67,28),(4,159,68,29),(5,160,69,30),(6,141,70,31),(7,142,71,32),(8,143,72,33),(9,144,73,34),(10,145,74,35),(11,146,75,36),(12,147,76,37),(13,148,77,38),(14,149,78,39),(15,150,79,40),(16,151,80,21),(17,152,61,22),(18,153,62,23),(19,154,63,24),(20,155,64,25),(41,130,114,91),(42,131,115,92),(43,132,116,93),(44,133,117,94),(45,134,118,95),(46,135,119,96),(47,136,120,97),(48,137,101,98),(49,138,102,99),(50,139,103,100),(51,140,104,81),(52,121,105,82),(53,122,106,83),(54,123,107,84),(55,124,108,85),(56,125,109,86),(57,126,110,87),(58,127,111,88),(59,128,112,89),(60,129,113,90)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 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 | Q8 | D4 | D5 | D8 | C4○D4 | D10 | D10 | D20 | D40 | C8.C22 | D4⋊2D5 | Q8×D5 | C8.D10 |
| kernel | D20⋊4Q8 | C40⋊5C4 | D20⋊5C4 | C5×C4⋊C8 | C20⋊2Q8 | C4×D20 | D20 | C2×C20 | C4⋊C8 | C20 | C20 | C42 | C2×C8 | C2×C4 | C4 | C10 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of D20⋊4Q8 ►in GL4(𝔽41) generated by
| 32 | 30 | 0 | 0 |
| 11 | 27 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 32 | 30 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 20 | 15 | 0 | 0 |
| 39 | 21 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [32,11,0,0,30,27,0,0,0,0,40,0,0,0,0,40],[32,11,0,0,30,9,0,0,0,0,1,0,0,0,0,40],[20,39,0,0,15,21,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,9] >;
D20⋊4Q8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_4Q_8 % in TeX
G:=Group("D20:4Q8"); // GroupNames label
G:=SmallGroup(320,473);
// by ID
G=gap.SmallGroup(320,473);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,310,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations