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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], C5, C8 [×2], C8, C2×C4 [×3], C2×C4 [×6], D4 [×4], D4 [×2], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], C20 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×8], C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C40 [×2], C40, C2×C20 [×3], C2×C20 [×6], C5×D4 [×4], C5×D4 [×2], C22×C10 [×2], C4×D8, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40 [×2], C5×D8 [×4], C22×C20 [×2], D4×C10 [×2], C4×C40, C5×D4⋊C4 [×2], C5×C2.D8, D4×C20 [×2], C10×D8, D8×C20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], D8 [×2], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C2×D8, C4○D8, C2×C20 [×6], C5×D4 [×2], C22×C10, C4×D8, C5×D8 [×2], 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 82 44 155 25 73 102 121)(2 83 45 156 26 74 103 122)(3 84 46 157 27 75 104 123)(4 85 47 158 28 76 105 124)(5 86 48 159 29 77 106 125)(6 87 49 160 30 78 107 126)(7 88 50 141 31 79 108 127)(8 89 51 142 32 80 109 128)(9 90 52 143 33 61 110 129)(10 91 53 144 34 62 111 130)(11 92 54 145 35 63 112 131)(12 93 55 146 36 64 113 132)(13 94 56 147 37 65 114 133)(14 95 57 148 38 66 115 134)(15 96 58 149 39 67 116 135)(16 97 59 150 40 68 117 136)(17 98 60 151 21 69 118 137)(18 99 41 152 22 70 119 138)(19 100 42 153 23 71 120 139)(20 81 43 154 24 72 101 140)
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 151)(22 152)(23 153)(24 154)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)
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), (1,82,44,155,25,73,102,121)(2,83,45,156,26,74,103,122)(3,84,46,157,27,75,104,123)(4,85,47,158,28,76,105,124)(5,86,48,159,29,77,106,125)(6,87,49,160,30,78,107,126)(7,88,50,141,31,79,108,127)(8,89,51,142,32,80,109,128)(9,90,52,143,33,61,110,129)(10,91,53,144,34,62,111,130)(11,92,54,145,35,63,112,131)(12,93,55,146,36,64,113,132)(13,94,56,147,37,65,114,133)(14,95,57,148,38,66,115,134)(15,96,58,149,39,67,116,135)(16,97,59,150,40,68,117,136)(17,98,60,151,21,69,118,137)(18,99,41,152,22,70,119,138)(19,100,42,153,23,71,120,139)(20,81,43,154,24,72,101,140), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)>;
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), (1,82,44,155,25,73,102,121)(2,83,45,156,26,74,103,122)(3,84,46,157,27,75,104,123)(4,85,47,158,28,76,105,124)(5,86,48,159,29,77,106,125)(6,87,49,160,30,78,107,126)(7,88,50,141,31,79,108,127)(8,89,51,142,32,80,109,128)(9,90,52,143,33,61,110,129)(10,91,53,144,34,62,111,130)(11,92,54,145,35,63,112,131)(12,93,55,146,36,64,113,132)(13,94,56,147,37,65,114,133)(14,95,57,148,38,66,115,134)(15,96,58,149,39,67,116,135)(16,97,59,150,40,68,117,136)(17,98,60,151,21,69,118,137)(18,99,41,152,22,70,119,138)(19,100,42,153,23,71,120,139)(20,81,43,154,24,72,101,140), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120) );
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)], [(1,82,44,155,25,73,102,121),(2,83,45,156,26,74,103,122),(3,84,46,157,27,75,104,123),(4,85,47,158,28,76,105,124),(5,86,48,159,29,77,106,125),(6,87,49,160,30,78,107,126),(7,88,50,141,31,79,108,127),(8,89,51,142,32,80,109,128),(9,90,52,143,33,61,110,129),(10,91,53,144,34,62,111,130),(11,92,54,145,35,63,112,131),(12,93,55,146,36,64,113,132),(13,94,56,147,37,65,114,133),(14,95,57,148,38,66,115,134),(15,96,58,149,39,67,116,135),(16,97,59,150,40,68,117,136),(17,98,60,151,21,69,118,137),(18,99,41,152,22,70,119,138),(19,100,42,153,23,71,120,139),(20,81,43,154,24,72,101,140)], [(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,151),(22,152),(23,153),(24,154),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,141),(32,142),(33,143),(34,144),(35,145),(36,146),(37,147),(38,148),(39,149),(40,150),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120)])
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