direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C20, (C4×C8)⋊7C10, C8⋊4(C2×C20), C40⋊36(C2×C4), (C4×C40)⋊23C2, (C4×D4)⋊1C10, D4⋊1(C2×C20), C2.3(C10×D8), (D4×C20)⋊30C2, (C2×D8).7C10, C2.D8⋊14C10, C10.75(C2×D8), C2.12(D4×C20), (C10×D8).14C2, C10.144(C4×D4), (C2×C20).360D4, C4.9(C22×C20), D4⋊C4⋊21C10, C42.70(C2×C10), C22.51(D4×C10), C20.256(C4○D4), C10.116(C4○D8), (C4×C20).355C22, C20.213(C22×C4), (C2×C40).421C22, (C2×C20).904C23, (D4×C10).290C22, C2.3(C5×C4○D8), C4.1(C5×C4○D4), (C5×D4)⋊24(C2×C4), (C5×C2.D8)⋊29C2, (C2×C4).50(C5×D4), C4⋊C4.45(C2×C10), (C2×C8).65(C2×C10), (C5×D4⋊C4)⋊44C2, (C2×D4).48(C2×C10), (C2×C10).627(C2×D4), (C5×C4⋊C4).366C22, (C2×C4).79(C22×C10), SmallGroup(320,938)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C20
G = < a,b,c | a20=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 250 in 134 conjugacy classes, 74 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C20, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C4×D8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×D8, C22×C20, D4×C10, C4×C40, C5×D4⋊C4, C5×C2.D8, D4×C20, C10×D8, D8×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, D8, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C2×D8, C4○D8, C2×C20, C5×D4, C22×C10, C4×D8, C5×D8, C22×C20, D4×C10, C5×C4○D4, D4×C20, C10×D8, C5×C4○D8, D8×C20
►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 67 141 126 44 87 30 113)(2 68 142 127 45 88 31 114)(3 69 143 128 46 89 32 115)(4 70 144 129 47 90 33 116)(5 71 145 130 48 91 34 117)(6 72 146 131 49 92 35 118)(7 73 147 132 50 93 36 119)(8 74 148 133 51 94 37 120)(9 75 149 134 52 95 38 101)(10 76 150 135 53 96 39 102)(11 77 151 136 54 97 40 103)(12 78 152 137 55 98 21 104)(13 79 153 138 56 99 22 105)(14 80 154 139 57 100 23 106)(15 61 155 140 58 81 24 107)(16 62 156 121 59 82 25 108)(17 63 157 122 60 83 26 109)(18 64 158 123 41 84 27 110)(19 65 159 124 42 85 28 111)(20 66 160 125 43 86 29 112)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 78)(22 79)(23 80)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 121)(60 122)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)
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), (1,67,141,126,44,87,30,113)(2,68,142,127,45,88,31,114)(3,69,143,128,46,89,32,115)(4,70,144,129,47,90,33,116)(5,71,145,130,48,91,34,117)(6,72,146,131,49,92,35,118)(7,73,147,132,50,93,36,119)(8,74,148,133,51,94,37,120)(9,75,149,134,52,95,38,101)(10,76,150,135,53,96,39,102)(11,77,151,136,54,97,40,103)(12,78,152,137,55,98,21,104)(13,79,153,138,56,99,22,105)(14,80,154,139,57,100,23,106)(15,61,155,140,58,81,24,107)(16,62,156,121,59,82,25,108)(17,63,157,122,60,83,26,109)(18,64,158,123,41,84,27,110)(19,65,159,124,42,85,28,111)(20,66,160,125,43,86,29,112), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,78)(22,79)(23,80)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,121)(60,122)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)>;
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), (1,67,141,126,44,87,30,113)(2,68,142,127,45,88,31,114)(3,69,143,128,46,89,32,115)(4,70,144,129,47,90,33,116)(5,71,145,130,48,91,34,117)(6,72,146,131,49,92,35,118)(7,73,147,132,50,93,36,119)(8,74,148,133,51,94,37,120)(9,75,149,134,52,95,38,101)(10,76,150,135,53,96,39,102)(11,77,151,136,54,97,40,103)(12,78,152,137,55,98,21,104)(13,79,153,138,56,99,22,105)(14,80,154,139,57,100,23,106)(15,61,155,140,58,81,24,107)(16,62,156,121,59,82,25,108)(17,63,157,122,60,83,26,109)(18,64,158,123,41,84,27,110)(19,65,159,124,42,85,28,111)(20,66,160,125,43,86,29,112), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,78)(22,79)(23,80)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,121)(60,122)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154) );
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)], [(1,67,141,126,44,87,30,113),(2,68,142,127,45,88,31,114),(3,69,143,128,46,89,32,115),(4,70,144,129,47,90,33,116),(5,71,145,130,48,91,34,117),(6,72,146,131,49,92,35,118),(7,73,147,132,50,93,36,119),(8,74,148,133,51,94,37,120),(9,75,149,134,52,95,38,101),(10,76,150,135,53,96,39,102),(11,77,151,136,54,97,40,103),(12,78,152,137,55,98,21,104),(13,79,153,138,56,99,22,105),(14,80,154,139,57,100,23,106),(15,61,155,140,58,81,24,107),(16,62,156,121,59,82,25,108),(17,63,157,122,60,83,26,109),(18,64,158,123,41,84,27,110),(19,65,159,124,42,85,28,111),(20,66,160,125,43,86,29,112)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,78),(22,79),(23,80),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77),(41,123),(42,124),(43,125),(44,126),(45,127),(46,128),(47,129),(48,130),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,137),(56,138),(57,139),(58,140),(59,121),(60,122),(81,155),(82,156),(83,157),(84,158),(85,159),(86,160),(87,141),(88,142),(89,143),(90,144),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(97,151),(98,152),(99,153),(100,154)]])
140 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 | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AV | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | D8 | C4○D4 | C4○D8 | C5×D4 | C5×D8 | C5×C4○D4 | C5×C4○D8 |
| kernel | D8×C20 | C4×C40 | C5×D4⋊C4 | C5×C2.D8 | D4×C20 | C10×D8 | C5×D8 | C4×D8 | C4×C8 | D4⋊C4 | C2.D8 | C4×D4 | C2×D8 | D8 | C2×C20 | C20 | C20 | C10 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 8 | 4 | 32 | 2 | 4 | 2 | 4 | 8 | 16 | 8 | 16 |
Matrix representation of D8×C20 ►in GL3(𝔽41) generated by
| 32 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 40 | 0 | 0 |
| 0 | 29 | 12 |
| 0 | 29 | 29 |
| 1 | 0 | 0 |
| 0 | 29 | 12 |
| 0 | 12 | 12 |
G:=sub<GL(3,GF(41))| [32,0,0,0,4,0,0,0,4],[40,0,0,0,29,29,0,12,29],[1,0,0,0,29,12,0,12,12] >;
D8×C20 in GAP, Magma, Sage, TeX
D_8\times C_{20} % in TeX
G:=Group("D8xC20"); // GroupNames label
G:=SmallGroup(320,938);
// by ID
G=gap.SmallGroup(320,938);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,436,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations