metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊1C8, Dic5.20D8, C20.1M4(2), Dic5.15SD16, C5⋊1(D4⋊C8), C10.8C4≀C2, C4⋊C4.3F5, C20.1(C2×C8), C20⋊C8⋊1C2, C4.1(D5⋊C8), C4.1(C4.F5), (C2×D20).10C4, C10.1(C22⋊C8), C2.1(D20⋊C4), C2.1(Q8⋊2F5), (C2×Dic5).95D4, D20⋊8C4.15C2, C10.2(D4⋊C4), C2.3(D10⋊C8), C22.31(C22⋊F5), (C4×Dic5).186C22, (C4×C5⋊C8)⋊1C2, (C5×C4⋊C4).3C4, (C2×C4).64(C2×F5), (C2×C20).30(C2×C4), (C2×C10).21(C22⋊C4), SmallGroup(320,209)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊C8
G = < a,b,c | a20=b2=c8=1, bab=a-1, cac-1=a3, cbc-1=a17b >
Subgroups: 402 in 82 conjugacy classes, 30 normal (28 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, Dic5, C20, C20, D10, C2×C10, C4×C8, C4⋊C8, C4×D4, C5⋊C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, D4⋊C8, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C5⋊C8, C2×C4×D5, C2×D20, C4×C5⋊C8, C20⋊C8, D20⋊8C4, D20⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), D8, SD16, F5, C22⋊C8, D4⋊C4, C4≀C2, C2×F5, D4⋊C8, D5⋊C8, C4.F5, C22⋊F5, D10⋊C8, D20⋊C4, Q8⋊2F5, D20⋊C8
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)(121 124)(122 123)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 160)(155 159)(156 158)
(1 150 28 102 57 73 133 85)(2 157 37 105 58 80 122 88)(3 144 26 108 59 67 131 91)(4 151 35 111 60 74 140 94)(5 158 24 114 41 61 129 97)(6 145 33 117 42 68 138 100)(7 152 22 120 43 75 127 83)(8 159 31 103 44 62 136 86)(9 146 40 106 45 69 125 89)(10 153 29 109 46 76 134 92)(11 160 38 112 47 63 123 95)(12 147 27 115 48 70 132 98)(13 154 36 118 49 77 121 81)(14 141 25 101 50 64 130 84)(15 148 34 104 51 71 139 87)(16 155 23 107 52 78 128 90)(17 142 32 110 53 65 137 93)(18 149 21 113 54 72 126 96)(19 156 30 116 55 79 135 99)(20 143 39 119 56 66 124 82)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,160)(155,159)(156,158), (1,150,28,102,57,73,133,85)(2,157,37,105,58,80,122,88)(3,144,26,108,59,67,131,91)(4,151,35,111,60,74,140,94)(5,158,24,114,41,61,129,97)(6,145,33,117,42,68,138,100)(7,152,22,120,43,75,127,83)(8,159,31,103,44,62,136,86)(9,146,40,106,45,69,125,89)(10,153,29,109,46,76,134,92)(11,160,38,112,47,63,123,95)(12,147,27,115,48,70,132,98)(13,154,36,118,49,77,121,81)(14,141,25,101,50,64,130,84)(15,148,34,104,51,71,139,87)(16,155,23,107,52,78,128,90)(17,142,32,110,53,65,137,93)(18,149,21,113,54,72,126,96)(19,156,30,116,55,79,135,99)(20,143,39,119,56,66,124,82)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,160)(155,159)(156,158), (1,150,28,102,57,73,133,85)(2,157,37,105,58,80,122,88)(3,144,26,108,59,67,131,91)(4,151,35,111,60,74,140,94)(5,158,24,114,41,61,129,97)(6,145,33,117,42,68,138,100)(7,152,22,120,43,75,127,83)(8,159,31,103,44,62,136,86)(9,146,40,106,45,69,125,89)(10,153,29,109,46,76,134,92)(11,160,38,112,47,63,123,95)(12,147,27,115,48,70,132,98)(13,154,36,118,49,77,121,81)(14,141,25,101,50,64,130,84)(15,148,34,104,51,71,139,87)(16,155,23,107,52,78,128,90)(17,142,32,110,53,65,137,93)(18,149,21,113,54,72,126,96)(19,156,30,116,55,79,135,99)(20,143,39,119,56,66,124,82) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(81,83),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(118,120),(121,124),(122,123),(125,140),(126,139),(127,138),(128,137),(129,136),(130,135),(131,134),(132,133),(141,153),(142,152),(143,151),(144,150),(145,149),(146,148),(154,160),(155,159),(156,158)], [(1,150,28,102,57,73,133,85),(2,157,37,105,58,80,122,88),(3,144,26,108,59,67,131,91),(4,151,35,111,60,74,140,94),(5,158,24,114,41,61,129,97),(6,145,33,117,42,68,138,100),(7,152,22,120,43,75,127,83),(8,159,31,103,44,62,136,86),(9,146,40,106,45,69,125,89),(10,153,29,109,46,76,134,92),(11,160,38,112,47,63,123,95),(12,147,27,115,48,70,132,98),(13,154,36,118,49,77,121,81),(14,141,25,101,50,64,130,84),(15,148,34,104,51,71,139,87),(16,155,23,107,52,78,128,90),(17,142,32,110,53,65,137,93),(18,149,21,113,54,72,126,96),(19,156,30,116,55,79,135,99),(20,143,39,119,56,66,124,82)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D8 | SD16 | M4(2) | C4≀C2 | F5 | C2×F5 | D5⋊C8 | C4.F5 | C22⋊F5 | D20⋊C4 | Q8⋊2F5 |
| kernel | D20⋊C8 | C4×C5⋊C8 | C20⋊C8 | D20⋊8C4 | C5×C4⋊C4 | C2×D20 | D20 | C2×Dic5 | Dic5 | Dic5 | C20 | C10 | C4⋊C4 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 1 | 1 |
Matrix representation of D20⋊C8 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 | 0 | 0 |
| 21 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 32 | 23 | 0 |
| 0 | 0 | 0 | 0 | 32 | 32 | 0 | 9 |
| 0 | 0 | 0 | 0 | 9 | 0 | 32 | 32 |
| 0 | 0 | 0 | 0 | 0 | 23 | 32 | 9 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[40,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[21,7,0,0,0,0,0,0,32,20,0,0,0,0,0,0,0,0,26,15,0,0,0,0,0,0,15,15,0,0,0,0,0,0,0,0,9,32,9,0,0,0,0,0,32,32,0,23,0,0,0,0,23,0,32,32,0,0,0,0,0,9,32,9] >;
D20⋊C8 in GAP, Magma, Sage, TeX
D_{20}\rtimes C_8 % in TeX
G:=Group("D20:C8"); // GroupNames label
G:=SmallGroup(320,209);
// by ID
G=gap.SmallGroup(320,209);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,100,1123,570,136,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^17*b>;
// generators/relations