metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊5Q8, C20.18SD16, C42.79D10, C4⋊Q8⋊2D5, C4.11(Q8×D5), C4⋊C4.82D10, C5⋊5(D4⋊2Q8), C20⋊3C8⋊34C2, C20.38(C2×Q8), (C4×D20).18C2, (C2×C20).155D4, C4.10(Q8⋊D5), C20.81(C4○D4), C20.Q8⋊42C2, C10.76(C2×SD16), D20⋊6C4.14C2, C10.98(C8⋊C22), (C4×C20).131C22, (C2×C20).402C23, C4.34(Q8⋊2D5), C10.75(C22⋊Q8), C2.12(D10⋊3Q8), (C2×D20).256C22, C4⋊Dic5.347C22, C2.19(D4.D10), (C5×C4⋊Q8)⋊2C2, C2.14(C2×Q8⋊D5), (C2×C10).533(C2×D4), (C2×C4).188(C5⋊D4), (C5×C4⋊C4).129C22, (C2×C4).499(C22×D5), C22.205(C2×C5⋊D4), (C2×C5⋊2C8).136C22, SmallGroup(320,711)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊5Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 438 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, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C5⋊2C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4⋊2Q8, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C20⋊3C8, C20.Q8, D20⋊6C4, C4×D20, C5×C4⋊Q8, D20⋊5Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8⋊C22, C5⋊D4, C22×D5, D4⋊2Q8, Q8⋊D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, D4.D10, C2×Q8⋊D5, D10⋊3Q8, D20⋊5Q8
►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 97)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 81)(18 100)(19 99)(20 98)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(121 154)(122 153)(123 152)(124 151)(125 150)(126 149)(127 148)(128 147)(129 146)(130 145)(131 144)(132 143)(133 142)(134 141)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)
(1 49 88 108)(2 60 89 119)(3 51 90 110)(4 42 91 101)(5 53 92 112)(6 44 93 103)(7 55 94 114)(8 46 95 105)(9 57 96 116)(10 48 97 107)(11 59 98 118)(12 50 99 109)(13 41 100 120)(14 52 81 111)(15 43 82 102)(16 54 83 113)(17 45 84 104)(18 56 85 115)(19 47 86 106)(20 58 87 117)(21 127 75 146)(22 138 76 157)(23 129 77 148)(24 140 78 159)(25 131 79 150)(26 122 80 141)(27 133 61 152)(28 124 62 143)(29 135 63 154)(30 126 64 145)(31 137 65 156)(32 128 66 147)(33 139 67 158)(34 130 68 149)(35 121 69 160)(36 132 70 151)(37 123 71 142)(38 134 72 153)(39 125 73 144)(40 136 74 155)
(1 74 88 40)(2 75 89 21)(3 76 90 22)(4 77 91 23)(5 78 92 24)(6 79 93 25)(7 80 94 26)(8 61 95 27)(9 62 96 28)(10 63 97 29)(11 64 98 30)(12 65 99 31)(13 66 100 32)(14 67 81 33)(15 68 82 34)(16 69 83 35)(17 70 84 36)(18 71 85 37)(19 72 86 38)(20 73 87 39)(41 128 120 147)(42 129 101 148)(43 130 102 149)(44 131 103 150)(45 132 104 151)(46 133 105 152)(47 134 106 153)(48 135 107 154)(49 136 108 155)(50 137 109 156)(51 138 110 157)(52 139 111 158)(53 140 112 159)(54 121 113 160)(55 122 114 141)(56 123 115 142)(57 124 116 143)(58 125 117 144)(59 126 118 145)(60 127 119 146)
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), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,81)(18,100)(19,99)(20,98)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155), (1,49,88,108)(2,60,89,119)(3,51,90,110)(4,42,91,101)(5,53,92,112)(6,44,93,103)(7,55,94,114)(8,46,95,105)(9,57,96,116)(10,48,97,107)(11,59,98,118)(12,50,99,109)(13,41,100,120)(14,52,81,111)(15,43,82,102)(16,54,83,113)(17,45,84,104)(18,56,85,115)(19,47,86,106)(20,58,87,117)(21,127,75,146)(22,138,76,157)(23,129,77,148)(24,140,78,159)(25,131,79,150)(26,122,80,141)(27,133,61,152)(28,124,62,143)(29,135,63,154)(30,126,64,145)(31,137,65,156)(32,128,66,147)(33,139,67,158)(34,130,68,149)(35,121,69,160)(36,132,70,151)(37,123,71,142)(38,134,72,153)(39,125,73,144)(40,136,74,155), (1,74,88,40)(2,75,89,21)(3,76,90,22)(4,77,91,23)(5,78,92,24)(6,79,93,25)(7,80,94,26)(8,61,95,27)(9,62,96,28)(10,63,97,29)(11,64,98,30)(12,65,99,31)(13,66,100,32)(14,67,81,33)(15,68,82,34)(16,69,83,35)(17,70,84,36)(18,71,85,37)(19,72,86,38)(20,73,87,39)(41,128,120,147)(42,129,101,148)(43,130,102,149)(44,131,103,150)(45,132,104,151)(46,133,105,152)(47,134,106,153)(48,135,107,154)(49,136,108,155)(50,137,109,156)(51,138,110,157)(52,139,111,158)(53,140,112,159)(54,121,113,160)(55,122,114,141)(56,123,115,142)(57,124,116,143)(58,125,117,144)(59,126,118,145)(60,127,119,146)>;
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), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,81)(18,100)(19,99)(20,98)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155), (1,49,88,108)(2,60,89,119)(3,51,90,110)(4,42,91,101)(5,53,92,112)(6,44,93,103)(7,55,94,114)(8,46,95,105)(9,57,96,116)(10,48,97,107)(11,59,98,118)(12,50,99,109)(13,41,100,120)(14,52,81,111)(15,43,82,102)(16,54,83,113)(17,45,84,104)(18,56,85,115)(19,47,86,106)(20,58,87,117)(21,127,75,146)(22,138,76,157)(23,129,77,148)(24,140,78,159)(25,131,79,150)(26,122,80,141)(27,133,61,152)(28,124,62,143)(29,135,63,154)(30,126,64,145)(31,137,65,156)(32,128,66,147)(33,139,67,158)(34,130,68,149)(35,121,69,160)(36,132,70,151)(37,123,71,142)(38,134,72,153)(39,125,73,144)(40,136,74,155), (1,74,88,40)(2,75,89,21)(3,76,90,22)(4,77,91,23)(5,78,92,24)(6,79,93,25)(7,80,94,26)(8,61,95,27)(9,62,96,28)(10,63,97,29)(11,64,98,30)(12,65,99,31)(13,66,100,32)(14,67,81,33)(15,68,82,34)(16,69,83,35)(17,70,84,36)(18,71,85,37)(19,72,86,38)(20,73,87,39)(41,128,120,147)(42,129,101,148)(43,130,102,149)(44,131,103,150)(45,132,104,151)(46,133,105,152)(47,134,106,153)(48,135,107,154)(49,136,108,155)(50,137,109,156)(51,138,110,157)(52,139,111,158)(53,140,112,159)(54,121,113,160)(55,122,114,141)(56,123,115,142)(57,124,116,143)(58,125,117,144)(59,126,118,145)(60,127,119,146) );
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)], [(1,97),(2,96),(3,95),(4,94),(5,93),(6,92),(7,91),(8,90),(9,89),(10,88),(11,87),(12,86),(13,85),(14,84),(15,83),(16,82),(17,81),(18,100),(19,99),(20,98),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,80),(34,79),(35,78),(36,77),(37,76),(38,75),(39,74),(40,73),(41,110),(42,109),(43,108),(44,107),(45,106),(46,105),(47,104),(48,103),(49,102),(50,101),(51,120),(52,119),(53,118),(54,117),(55,116),(56,115),(57,114),(58,113),(59,112),(60,111),(121,154),(122,153),(123,152),(124,151),(125,150),(126,149),(127,148),(128,147),(129,146),(130,145),(131,144),(132,143),(133,142),(134,141),(135,160),(136,159),(137,158),(138,157),(139,156),(140,155)], [(1,49,88,108),(2,60,89,119),(3,51,90,110),(4,42,91,101),(5,53,92,112),(6,44,93,103),(7,55,94,114),(8,46,95,105),(9,57,96,116),(10,48,97,107),(11,59,98,118),(12,50,99,109),(13,41,100,120),(14,52,81,111),(15,43,82,102),(16,54,83,113),(17,45,84,104),(18,56,85,115),(19,47,86,106),(20,58,87,117),(21,127,75,146),(22,138,76,157),(23,129,77,148),(24,140,78,159),(25,131,79,150),(26,122,80,141),(27,133,61,152),(28,124,62,143),(29,135,63,154),(30,126,64,145),(31,137,65,156),(32,128,66,147),(33,139,67,158),(34,130,68,149),(35,121,69,160),(36,132,70,151),(37,123,71,142),(38,134,72,153),(39,125,73,144),(40,136,74,155)], [(1,74,88,40),(2,75,89,21),(3,76,90,22),(4,77,91,23),(5,78,92,24),(6,79,93,25),(7,80,94,26),(8,61,95,27),(9,62,96,28),(10,63,97,29),(11,64,98,30),(12,65,99,31),(13,66,100,32),(14,67,81,33),(15,68,82,34),(16,69,83,35),(17,70,84,36),(18,71,85,37),(19,72,86,38),(20,73,87,39),(41,128,120,147),(42,129,101,148),(43,130,102,149),(44,131,103,150),(45,132,104,151),(46,133,105,152),(47,134,106,153),(48,135,107,154),(49,136,108,155),(50,137,109,156),(51,138,110,157),(52,139,111,158),(53,140,112,159),(54,121,113,160),(55,122,114,141),(56,123,115,142),(57,124,116,143),(58,125,117,144),(59,126,118,145),(60,127,119,146)]])
47 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 | ··· | 20L | 20M | ··· | 20T |
| 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 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 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 | Q8 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | Q8⋊D5 | Q8×D5 | Q8⋊2D5 | D4.D10 |
| kernel | D20⋊5Q8 | C20⋊3C8 | C20.Q8 | D20⋊6C4 | C4×D20 | C5×C4⋊Q8 | D20 | C2×C20 | C4⋊Q8 | C20 | C20 | C42 | C4⋊C4 | C2×C4 | C10 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of D20⋊5Q8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 5 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 4 |
| 0 | 0 | 0 | 0 | 20 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 40 |
| 2 | 5 | 0 | 0 | 0 | 0 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 19 |
| 0 | 0 | 0 | 0 | 13 | 30 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 40 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 37 |
| 0 | 0 | 0 | 0 | 21 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,20,0,0,0,0,4,1],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,21,0,0,0,0,0,40],[2,40,0,0,0,0,5,39,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,13,0,0,0,0,19,30],[32,40,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,37,40] >;
D20⋊5Q8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_5Q_8 % in TeX
G:=Group("D20:5Q8"); // GroupNames label
G:=SmallGroup(320,711);
// by ID
G=gap.SmallGroup(320,711);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,268,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations