metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.19D8, C20.19SD16, C42.224D10, C4⋊Q8⋊6D5, C4⋊C4.84D10, C4.7(D4⋊D5), C10.61(C2×D8), C5⋊4(C4.4D8), C4.5(Q8⋊D5), (C2×C20).158D4, D20⋊6C4⋊42C2, C20.83(C4○D4), C20⋊4D4.10C2, C10.78(C2×SD16), (C4×C20).135C22, (C2×C20).406C23, C4.16(Q8⋊2D5), C10.58(C4.4D4), (C2×D20).113C22, C2.11(C20.23D4), (C5×C4⋊Q8)⋊6C2, (C4×C5⋊2C8)⋊18C2, C2.16(C2×D4⋊D5), C2.16(C2×Q8⋊D5), (C2×C10).537(C2×D4), (C2×C4).137(C5⋊D4), (C5×C4⋊C4).131C22, (C2×C4).503(C22×D5), C22.209(C2×C5⋊D4), (C2×C5⋊2C8).271C22, SmallGroup(320,715)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.D8
G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a-1, cbc=a10b-1 >
Subgroups: 606 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C42, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×D4 [×4], C2×Q8, C20 [×6], C20 [×2], D10 [×6], C2×C10, C4×C8, D4⋊C4 [×4], C4⋊1D4, C4⋊Q8, C5⋊2C8 [×2], D20 [×8], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C4.4D8, C2×C5⋊2C8 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×D20 [×2], C2×D20 [×2], Q8×C10, C4×C5⋊2C8, D20⋊6C4 [×4], C20⋊4D4, C5×C4⋊Q8, C20.D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C2×D8, C2×SD16, C5⋊D4 [×2], C22×D5, C4.4D8, D4⋊D5 [×2], Q8⋊D5 [×2], Q8⋊2D5 [×2], C2×C5⋊D4, C2×D4⋊D5, C2×Q8⋊D5, C20.23D4, C20.D8
►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 81 31 77 121 59 111 151)(2 90 32 66 122 48 112 160)(3 99 33 75 123 57 113 149)(4 88 34 64 124 46 114 158)(5 97 35 73 125 55 115 147)(6 86 36 62 126 44 116 156)(7 95 37 71 127 53 117 145)(8 84 38 80 128 42 118 154)(9 93 39 69 129 51 119 143)(10 82 40 78 130 60 120 152)(11 91 21 67 131 49 101 141)(12 100 22 76 132 58 102 150)(13 89 23 65 133 47 103 159)(14 98 24 74 134 56 104 148)(15 87 25 63 135 45 105 157)(16 96 26 72 136 54 106 146)(17 85 27 61 137 43 107 155)(18 94 28 70 138 52 108 144)(19 83 29 79 139 41 109 153)(20 92 30 68 140 50 110 142)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 101)(22 120)(23 119)(24 118)(25 117)(26 116)(27 115)(28 114)(29 113)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 104)(39 103)(40 102)(41 65)(42 64)(43 63)(44 62)(45 61)(46 80)(47 79)(48 78)(49 77)(50 76)(51 75)(52 74)(53 73)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(81 141)(82 160)(83 159)(84 158)(85 157)(86 156)(87 155)(88 154)(89 153)(90 152)(91 151)(92 150)(93 149)(94 148)(95 147)(96 146)(97 145)(98 144)(99 143)(100 142)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)
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), (1,81,31,77,121,59,111,151)(2,90,32,66,122,48,112,160)(3,99,33,75,123,57,113,149)(4,88,34,64,124,46,114,158)(5,97,35,73,125,55,115,147)(6,86,36,62,126,44,116,156)(7,95,37,71,127,53,117,145)(8,84,38,80,128,42,118,154)(9,93,39,69,129,51,119,143)(10,82,40,78,130,60,120,152)(11,91,21,67,131,49,101,141)(12,100,22,76,132,58,102,150)(13,89,23,65,133,47,103,159)(14,98,24,74,134,56,104,148)(15,87,25,63,135,45,105,157)(16,96,26,72,136,54,106,146)(17,85,27,61,137,43,107,155)(18,94,28,70,138,52,108,144)(19,83,29,79,139,41,109,153)(20,92,30,68,140,50,110,142), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,101)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,65)(42,64)(43,63)(44,62)(45,61)(46,80)(47,79)(48,78)(49,77)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(81,141)(82,160)(83,159)(84,158)(85,157)(86,156)(87,155)(88,154)(89,153)(90,152)(91,151)(92,150)(93,149)(94,148)(95,147)(96,146)(97,145)(98,144)(99,143)(100,142)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)>;
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), (1,81,31,77,121,59,111,151)(2,90,32,66,122,48,112,160)(3,99,33,75,123,57,113,149)(4,88,34,64,124,46,114,158)(5,97,35,73,125,55,115,147)(6,86,36,62,126,44,116,156)(7,95,37,71,127,53,117,145)(8,84,38,80,128,42,118,154)(9,93,39,69,129,51,119,143)(10,82,40,78,130,60,120,152)(11,91,21,67,131,49,101,141)(12,100,22,76,132,58,102,150)(13,89,23,65,133,47,103,159)(14,98,24,74,134,56,104,148)(15,87,25,63,135,45,105,157)(16,96,26,72,136,54,106,146)(17,85,27,61,137,43,107,155)(18,94,28,70,138,52,108,144)(19,83,29,79,139,41,109,153)(20,92,30,68,140,50,110,142), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,101)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,65)(42,64)(43,63)(44,62)(45,61)(46,80)(47,79)(48,78)(49,77)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(81,141)(82,160)(83,159)(84,158)(85,157)(86,156)(87,155)(88,154)(89,153)(90,152)(91,151)(92,150)(93,149)(94,148)(95,147)(96,146)(97,145)(98,144)(99,143)(100,142)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132) );
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)], [(1,81,31,77,121,59,111,151),(2,90,32,66,122,48,112,160),(3,99,33,75,123,57,113,149),(4,88,34,64,124,46,114,158),(5,97,35,73,125,55,115,147),(6,86,36,62,126,44,116,156),(7,95,37,71,127,53,117,145),(8,84,38,80,128,42,118,154),(9,93,39,69,129,51,119,143),(10,82,40,78,130,60,120,152),(11,91,21,67,131,49,101,141),(12,100,22,76,132,58,102,150),(13,89,23,65,133,47,103,159),(14,98,24,74,134,56,104,148),(15,87,25,63,135,45,105,157),(16,96,26,72,136,54,106,146),(17,85,27,61,137,43,107,155),(18,94,28,70,138,52,108,144),(19,83,29,79,139,41,109,153),(20,92,30,68,140,50,110,142)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,101),(22,120),(23,119),(24,118),(25,117),(26,116),(27,115),(28,114),(29,113),(30,112),(31,111),(32,110),(33,109),(34,108),(35,107),(36,106),(37,105),(38,104),(39,103),(40,102),(41,65),(42,64),(43,63),(44,62),(45,61),(46,80),(47,79),(48,78),(49,77),(50,76),(51,75),(52,74),(53,73),(54,72),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(81,141),(82,160),(83,159),(84,158),(85,157),(86,156),(87,155),(88,154),(89,153),(90,152),(91,151),(92,150),(93,149),(94,148),(95,147),(96,146),(97,145),(98,144),(99,143),(100,142),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132)])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | ··· | 2 | 8 | 8 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | D8 | SD16 | C4○D4 | D10 | D10 | C5⋊D4 | D4⋊D5 | Q8⋊D5 | Q8⋊2D5 |
| kernel | C20.D8 | C4×C5⋊2C8 | D20⋊6C4 | C20⋊4D4 | C5×C4⋊Q8 | C2×C20 | C4⋊Q8 | C20 | C20 | C20 | C42 | C4⋊C4 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 4 |
Matrix representation of C20.D8 ►in GL6(𝔽41)
| 20 | 9 | 0 | 0 | 0 | 0 |
| 1 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 8 |
| 0 | 0 | 0 | 0 | 6 | 19 |
| 28 | 12 | 0 | 0 | 0 | 0 |
| 15 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 28 | 34 |
| 20 | 9 | 0 | 0 | 0 | 0 |
| 24 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 15 | 40 |
G:=sub<GL(6,GF(41))| [20,1,0,0,0,0,9,21,0,0,0,0,0,0,6,5,0,0,0,0,1,1,0,0,0,0,0,0,22,6,0,0,0,0,8,19],[28,15,0,0,0,0,12,2,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,7,28,0,0,0,0,10,34],[20,24,0,0,0,0,9,21,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,15,0,0,0,0,0,40] >;
C20.D8 in GAP, Magma, Sage, TeX
C_{20}.D_8 % in TeX
G:=Group("C20.D8"); // GroupNames label
G:=SmallGroup(320,715);
// by ID
G=gap.SmallGroup(320,715);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,219,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=a^10*b^-1>;
// generators/relations