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, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, C20, D10, D10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C5⋊2C8, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C8×D4, C8×D5, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C4×C5⋊2C8, D10⋊1C8, C5×C4⋊C8, C4×D20, D5×C2×C8, D20⋊5C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D5, C2×C8, C22×C4, C2×D4, C4○D4, D10, C4×D4, C22×C8, C8○D4, C4×D5, C22×D5, C8×D4, C8×D5, 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 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)
(1 159 69 28 115 122 42 87)(2 150 70 39 116 133 43 98)(3 141 71 30 117 124 44 89)(4 152 72 21 118 135 45 100)(5 143 73 32 119 126 46 91)(6 154 74 23 120 137 47 82)(7 145 75 34 101 128 48 93)(8 156 76 25 102 139 49 84)(9 147 77 36 103 130 50 95)(10 158 78 27 104 121 51 86)(11 149 79 38 105 132 52 97)(12 160 80 29 106 123 53 88)(13 151 61 40 107 134 54 99)(14 142 62 31 108 125 55 90)(15 153 63 22 109 136 56 81)(16 144 64 33 110 127 57 92)(17 155 65 24 111 138 58 83)(18 146 66 35 112 129 59 94)(19 157 67 26 113 140 60 85)(20 148 68 37 114 131 41 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)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152), (1,159,69,28,115,122,42,87)(2,150,70,39,116,133,43,98)(3,141,71,30,117,124,44,89)(4,152,72,21,118,135,45,100)(5,143,73,32,119,126,46,91)(6,154,74,23,120,137,47,82)(7,145,75,34,101,128,48,93)(8,156,76,25,102,139,49,84)(9,147,77,36,103,130,50,95)(10,158,78,27,104,121,51,86)(11,149,79,38,105,132,52,97)(12,160,80,29,106,123,53,88)(13,151,61,40,107,134,54,99)(14,142,62,31,108,125,55,90)(15,153,63,22,109,136,56,81)(16,144,64,33,110,127,57,92)(17,155,65,24,111,138,58,83)(18,146,66,35,112,129,59,94)(19,157,67,26,113,140,60,85)(20,148,68,37,114,131,41,96)>;
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)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,127)(122,126)(123,125)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152), (1,159,69,28,115,122,42,87)(2,150,70,39,116,133,43,98)(3,141,71,30,117,124,44,89)(4,152,72,21,118,135,45,100)(5,143,73,32,119,126,46,91)(6,154,74,23,120,137,47,82)(7,145,75,34,101,128,48,93)(8,156,76,25,102,139,49,84)(9,147,77,36,103,130,50,95)(10,158,78,27,104,121,51,86)(11,149,79,38,105,132,52,97)(12,160,80,29,106,123,53,88)(13,151,61,40,107,134,54,99)(14,142,62,31,108,125,55,90)(15,153,63,22,109,136,56,81)(16,144,64,33,110,127,57,92)(17,155,65,24,111,138,58,83)(18,146,66,35,112,129,59,94)(19,157,67,26,113,140,60,85)(20,148,68,37,114,131,41,96) );
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),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,60),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(98,100),(101,103),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(121,127),(122,126),(123,125),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,154),(149,153),(150,152)], [(1,159,69,28,115,122,42,87),(2,150,70,39,116,133,43,98),(3,141,71,30,117,124,44,89),(4,152,72,21,118,135,45,100),(5,143,73,32,119,126,46,91),(6,154,74,23,120,137,47,82),(7,145,75,34,101,128,48,93),(8,156,76,25,102,139,49,84),(9,147,77,36,103,130,50,95),(10,158,78,27,104,121,51,86),(11,149,79,38,105,132,52,97),(12,160,80,29,106,123,53,88),(13,151,61,40,107,134,54,99),(14,142,62,31,108,125,55,90),(15,153,63,22,109,136,56,81),(16,144,64,33,110,127,57,92),(17,155,65,24,111,138,58,83),(18,146,66,35,112,129,59,94),(19,157,67,26,113,140,60,85),(20,148,68,37,114,131,41,96)]])
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