metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊4C8, C20.54D8, C20.49SD16, C42.192D10, C20.17M4(2), C4⋊C8⋊1D5, C5⋊4(D4⋊C8), C4.1(C8×D5), C10.22C4≀C2, C20.27(C2×C8), (C4×D20).6C2, C4.27(D4⋊D5), (C2×D20).23C4, (C2×C20).225D4, (C2×C4).109D20, C4.1(C8⋊D5), C4.15(Q8⋊D5), C4⋊Dic5.26C4, (C4×C20).41C22, C2.1(D20⋊6C4), C2.7(D10⋊1C8), C2.1(D20⋊7C4), C10.20(C22⋊C8), C10.18(D4⋊C4), C22.34(D10⋊C4), (C5×C4⋊C8)⋊1C2, (C4×C5⋊2C8)⋊1C2, (C2×C4).65(C4×D5), (C2×C20).221(C2×C4), (C2×C4).265(C5⋊D4), (C2×C10).109(C22⋊C4), SmallGroup(320,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊4C8
G = < a,b,c | a20=b2=c8=1, bab=a-1, cac-1=a11, cbc-1=a15b >
Subgroups: 374 in 82 conjugacy classes, 35 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C4×C8, C4⋊C8, C4×D4, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D4⋊C8, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C4×C5⋊2C8, C5×C4⋊C8, C4×D20, D20⋊4C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), D8, SD16, D10, C22⋊C8, D4⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, D4⋊C8, C8×D5, C8⋊D5, D10⋊C4, D4⋊D5, Q8⋊D5, D20⋊6C4, D10⋊1C8, D20⋊7C4, D20⋊4C8
►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)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(61 143)(62 142)(63 141)(64 160)(65 159)(66 158)(67 157)(68 156)(69 155)(70 154)(71 153)(72 152)(73 151)(74 150)(75 149)(76 148)(77 147)(78 146)(79 145)(80 144)(81 131)(82 130)(83 129)(84 128)(85 127)(86 126)(87 125)(88 124)(89 123)(90 122)(91 121)(92 140)(93 139)(94 138)(95 137)(96 136)(97 135)(98 134)(99 133)(100 132)
(1 144 27 133 111 76 41 95)(2 155 28 124 112 67 42 86)(3 146 29 135 113 78 43 97)(4 157 30 126 114 69 44 88)(5 148 31 137 115 80 45 99)(6 159 32 128 116 71 46 90)(7 150 33 139 117 62 47 81)(8 141 34 130 118 73 48 92)(9 152 35 121 119 64 49 83)(10 143 36 132 120 75 50 94)(11 154 37 123 101 66 51 85)(12 145 38 134 102 77 52 96)(13 156 39 125 103 68 53 87)(14 147 40 136 104 79 54 98)(15 158 21 127 105 70 55 89)(16 149 22 138 106 61 56 100)(17 160 23 129 107 72 57 91)(18 151 24 140 108 63 58 82)(19 142 25 131 109 74 59 93)(20 153 26 122 110 65 60 84)
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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(61,143)(62,142)(63,141)(64,160)(65,159)(66,158)(67,157)(68,156)(69,155)(70,154)(71,153)(72,152)(73,151)(74,150)(75,149)(76,148)(77,147)(78,146)(79,145)(80,144)(81,131)(82,130)(83,129)(84,128)(85,127)(86,126)(87,125)(88,124)(89,123)(90,122)(91,121)(92,140)(93,139)(94,138)(95,137)(96,136)(97,135)(98,134)(99,133)(100,132), (1,144,27,133,111,76,41,95)(2,155,28,124,112,67,42,86)(3,146,29,135,113,78,43,97)(4,157,30,126,114,69,44,88)(5,148,31,137,115,80,45,99)(6,159,32,128,116,71,46,90)(7,150,33,139,117,62,47,81)(8,141,34,130,118,73,48,92)(9,152,35,121,119,64,49,83)(10,143,36,132,120,75,50,94)(11,154,37,123,101,66,51,85)(12,145,38,134,102,77,52,96)(13,156,39,125,103,68,53,87)(14,147,40,136,104,79,54,98)(15,158,21,127,105,70,55,89)(16,149,22,138,106,61,56,100)(17,160,23,129,107,72,57,91)(18,151,24,140,108,63,58,82)(19,142,25,131,109,74,59,93)(20,153,26,122,110,65,60,84)>;
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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(61,143)(62,142)(63,141)(64,160)(65,159)(66,158)(67,157)(68,156)(69,155)(70,154)(71,153)(72,152)(73,151)(74,150)(75,149)(76,148)(77,147)(78,146)(79,145)(80,144)(81,131)(82,130)(83,129)(84,128)(85,127)(86,126)(87,125)(88,124)(89,123)(90,122)(91,121)(92,140)(93,139)(94,138)(95,137)(96,136)(97,135)(98,134)(99,133)(100,132), (1,144,27,133,111,76,41,95)(2,155,28,124,112,67,42,86)(3,146,29,135,113,78,43,97)(4,157,30,126,114,69,44,88)(5,148,31,137,115,80,45,99)(6,159,32,128,116,71,46,90)(7,150,33,139,117,62,47,81)(8,141,34,130,118,73,48,92)(9,152,35,121,119,64,49,83)(10,143,36,132,120,75,50,94)(11,154,37,123,101,66,51,85)(12,145,38,134,102,77,52,96)(13,156,39,125,103,68,53,87)(14,147,40,136,104,79,54,98)(15,158,21,127,105,70,55,89)(16,149,22,138,106,61,56,100)(17,160,23,129,107,72,57,91)(18,151,24,140,108,63,58,82)(19,142,25,131,109,74,59,93)(20,153,26,122,110,65,60,84) );
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)], [(1,110),(2,109),(3,108),(4,107),(5,106),(6,105),(7,104),(8,103),(9,102),(10,101),(11,120),(12,119),(13,118),(14,117),(15,116),(16,115),(17,114),(18,113),(19,112),(20,111),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,60),(28,59),(29,58),(30,57),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(61,143),(62,142),(63,141),(64,160),(65,159),(66,158),(67,157),(68,156),(69,155),(70,154),(71,153),(72,152),(73,151),(74,150),(75,149),(76,148),(77,147),(78,146),(79,145),(80,144),(81,131),(82,130),(83,129),(84,128),(85,127),(86,126),(87,125),(88,124),(89,123),(90,122),(91,121),(92,140),(93,139),(94,138),(95,137),(96,136),(97,135),(98,134),(99,133),(100,132)], [(1,144,27,133,111,76,41,95),(2,155,28,124,112,67,42,86),(3,146,29,135,113,78,43,97),(4,157,30,126,114,69,44,88),(5,148,31,137,115,80,45,99),(6,159,32,128,116,71,46,90),(7,150,33,139,117,62,47,81),(8,141,34,130,118,73,48,92),(9,152,35,121,119,64,49,83),(10,143,36,132,120,75,50,94),(11,154,37,123,101,66,51,85),(12,145,38,134,102,77,52,96),(13,156,39,125,103,68,53,87),(14,147,40,136,104,79,54,98),(15,158,21,127,105,70,55,89),(16,149,22,138,106,61,56,100),(17,160,23,129,107,72,57,91),(18,151,24,140,108,63,58,82),(19,142,25,131,109,74,59,93),(20,153,26,122,110,65,60,84)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D5 | M4(2) | D8 | SD16 | D10 | C4≀C2 | C4×D5 | D20 | C5⋊D4 | C8×D5 | C8⋊D5 | D4⋊D5 | Q8⋊D5 | D20⋊7C4 |
| kernel | D20⋊4C8 | C4×C5⋊2C8 | C5×C4⋊C8 | C4×D20 | C4⋊Dic5 | C2×D20 | D20 | C2×C20 | C4⋊C8 | C20 | C20 | C20 | C42 | C10 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 4 |
Matrix representation of D20⋊4C8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 21 | 39 | 0 | 0 | 0 | 0 |
| 20 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 28 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 26 |
| 0 | 0 | 0 | 0 | 26 | 15 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[21,20,0,0,0,0,39,20,0,0,0,0,0,0,39,13,0,0,0,0,28,2,0,0,0,0,0,0,26,26,0,0,0,0,26,15] >;
D20⋊4C8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_4C_8 % in TeX
G:=Group("D20:4C8"); // GroupNames label
G:=SmallGroup(320,41);
// by ID
G=gap.SmallGroup(320,41);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,100,1123,570,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations