metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊5C8, C42.201D10, C5⋊6(C8×D4), C4⋊1(C8×D5), C20⋊6(C2×C8), C4⋊C8⋊18D5, D10⋊5(C2×C8), C5⋊2C8⋊30D4, (C4×D20).7C2, C10.76(C4×D4), C4.205(D4×D5), (C2×D20).24C4, C20.364(C2×D4), (C2×C8).216D10, C4⋊Dic5.30C4, D10⋊1C8⋊22C2, C10.52(C8○D4), (C4×C20).60C22, C10.35(C22×C8), D10⋊C4.21C4, C20.334(C4○D4), C2.2(D20⋊8C4), (C2×C20).831C23, (C2×C40).210C22, C4.53(Q8⋊2D5), C2.4(D20.2C4), (D5×C2×C8)⋊22C2, (C5×C4⋊C8)⋊15C2, (C4×C5⋊2C8)⋊4C2, C2.13(D5×C2×C8), (C2×C4).72(C4×D5), C22.48(C2×C4×D5), (C2×C20).331(C2×C4), (C2×C4×D5).347C22, (C2×Dic5).98(C2×C4), (C22×D5).74(C2×C4), (C2×C4).773(C22×D5), (C2×C10).187(C22×C4), (C2×C5⋊2C8).314C22, SmallGroup(320,461)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊5C8
G = < a,b,c | a20=b2=c8=1, bab=a-1, cac-1=a11, cbc-1=a10b >
Subgroups: 446 in 134 conjugacy classes, 61 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], C5, C8 [×5], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×6], C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×2], C20 [×2], C20, D10 [×4], D10 [×4], C2×C10, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C5⋊2C8 [×2], C5⋊2C8, C40 [×2], C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C8×D4, C8×D5 [×4], C2×C5⋊2C8 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20, C4×C5⋊2C8, D10⋊1C8 [×2], C5×C4⋊C8, C4×D20, D5×C2×C8 [×2], D20⋊5C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, D5, C2×C8 [×6], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C22×C8, C8○D4, C4×D5 [×2], C22×D5, C8×D4, C8×D5 [×2], C2×C4×D5, D4×D5, Q8⋊2D5, D20⋊8C4, D5×C2×C8, D20.2C4, D20⋊5C8
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
(1 72 102 46 123 154 21 87)(2 63 103 57 124 145 22 98)(3 74 104 48 125 156 23 89)(4 65 105 59 126 147 24 100)(5 76 106 50 127 158 25 91)(6 67 107 41 128 149 26 82)(7 78 108 52 129 160 27 93)(8 69 109 43 130 151 28 84)(9 80 110 54 131 142 29 95)(10 71 111 45 132 153 30 86)(11 62 112 56 133 144 31 97)(12 73 113 47 134 155 32 88)(13 64 114 58 135 146 33 99)(14 75 115 49 136 157 34 90)(15 66 116 60 137 148 35 81)(16 77 117 51 138 159 36 92)(17 68 118 42 139 150 37 83)(18 79 119 53 140 141 38 94)(19 70 120 44 121 152 39 85)(20 61 101 55 122 143 40 96)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157), (1,72,102,46,123,154,21,87)(2,63,103,57,124,145,22,98)(3,74,104,48,125,156,23,89)(4,65,105,59,126,147,24,100)(5,76,106,50,127,158,25,91)(6,67,107,41,128,149,26,82)(7,78,108,52,129,160,27,93)(8,69,109,43,130,151,28,84)(9,80,110,54,131,142,29,95)(10,71,111,45,132,153,30,86)(11,62,112,56,133,144,31,97)(12,73,113,47,134,155,32,88)(13,64,114,58,135,146,33,99)(14,75,115,49,136,157,34,90)(15,66,116,60,137,148,35,81)(16,77,117,51,138,159,36,92)(17,68,118,42,139,150,37,83)(18,79,119,53,140,141,38,94)(19,70,120,44,121,152,39,85)(20,61,101,55,122,143,40,96)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157), (1,72,102,46,123,154,21,87)(2,63,103,57,124,145,22,98)(3,74,104,48,125,156,23,89)(4,65,105,59,126,147,24,100)(5,76,106,50,127,158,25,91)(6,67,107,41,128,149,26,82)(7,78,108,52,129,160,27,93)(8,69,109,43,130,151,28,84)(9,80,110,54,131,142,29,95)(10,71,111,45,132,153,30,86)(11,62,112,56,133,144,31,97)(12,73,113,47,134,155,32,88)(13,64,114,58,135,146,33,99)(14,75,115,49,136,157,34,90)(15,66,116,60,137,148,35,81)(16,77,117,51,138,159,36,92)(17,68,118,42,139,150,37,83)(18,79,119,53,140,141,38,94)(19,70,120,44,121,152,39,85)(20,61,101,55,122,143,40,96) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(36,40),(37,39),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,60),(57,59),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(98,100),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(118,120),(121,139),(122,138),(123,137),(124,136),(125,135),(126,134),(127,133),(128,132),(129,131),(141,151),(142,150),(143,149),(144,148),(145,147),(152,160),(153,159),(154,158),(155,157)], [(1,72,102,46,123,154,21,87),(2,63,103,57,124,145,22,98),(3,74,104,48,125,156,23,89),(4,65,105,59,126,147,24,100),(5,76,106,50,127,158,25,91),(6,67,107,41,128,149,26,82),(7,78,108,52,129,160,27,93),(8,69,109,43,130,151,28,84),(9,80,110,54,131,142,29,95),(10,71,111,45,132,153,30,86),(11,62,112,56,133,144,31,97),(12,73,113,47,134,155,32,88),(13,64,114,58,135,146,33,99),(14,75,115,49,136,157,34,90),(15,66,116,60,137,148,35,81),(16,77,117,51,138,159,36,92),(17,68,118,42,139,150,37,83),(18,79,119,53,140,141,38,94),(19,70,120,44,121,152,39,85),(20,61,101,55,122,143,40,96)])
80 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 | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| 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 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D5 | C4○D4 | D10 | D10 | C8○D4 | C4×D5 | C8×D5 | D4×D5 | Q8⋊2D5 | D20.2C4 |
| kernel | D20⋊5C8 | C4×C5⋊2C8 | D10⋊1C8 | C5×C4⋊C8 | C4×D20 | D5×C2×C8 | C4⋊Dic5 | D10⋊C4 | C2×D20 | D20 | C5⋊2C8 | C4⋊C8 | C20 | C42 | C2×C8 | C10 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of D20⋊5C8 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 32 | 9 |
| 0 | 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 39 | 1 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 32 | 9 |
| 0 | 0 | 0 | 23 | 9 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,40,7,0,0,0,0,0,32,0,0,0,0,9,9],[40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,40,39,0,0,0,0,1],[3,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,23,0,0,0,9,9] >;
D20⋊5C8 in GAP, Magma, Sage, TeX
D_{20}\rtimes_5C_8 % in TeX
G:=Group("D20:5C8"); // GroupNames label
G:=SmallGroup(320,461);
// by ID
G=gap.SmallGroup(320,461);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,219,58,136,12550]);
// 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^10*b>;
// generators/relations