metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.16D8, C20.15SD16, C42.217D10, C4.5(D4⋊D5), C10.56(C2×D8), C4⋊1D4.3D5, C5⋊3(C4.4D8), C20⋊2Q8⋊19C2, (C2×D4).53D10, (C2×C20).146D4, C4.3(D4.D5), C20.74(C4○D4), D4⋊Dic5⋊20C2, C10.57(C2×SD16), C4.22(D4⋊2D5), (C4×C20).118C22, (C2×C20).388C23, (D4×C10).69C22, C10.43(C4.4D4), C4⋊Dic5.154C22, C2.10(C20.17D4), (C4×C5⋊2C8)⋊14C2, C2.11(C2×D4⋊D5), (C5×C4⋊1D4).2C2, C2.11(C2×D4.D5), (C2×C10).519(C2×D4), (C2×C4).130(C5⋊D4), (C2×C4).486(C22×D5), C22.192(C2×C5⋊D4), (C2×C5⋊2C8).264C22, SmallGroup(320,697)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.16D8
G = < a,b,c | a20=b8=1, c2=a10, bab-1=a9, cac-1=a-1, cbc-1=a10b-1 >
Subgroups: 414 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C4⋊C4, C2×C8, C2×D4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C5×D4, C22×C10, C4.4D8, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C4×C20, C2×Dic10, D4×C10, D4×C10, C4×C5⋊2C8, D4⋊Dic5, C20⋊2Q8, C5×C4⋊1D4, C20.16D8
Quotients: C1, C2, C22, D4, C23, D5, D8, SD16, C2×D4, C4○D4, D10, C4.4D4, C2×D8, C2×SD16, C5⋊D4, C22×D5, C4.4D8, D4⋊D5, D4.D5, D4⋊2D5, C2×C5⋊D4, C2×D4⋊D5, C2×D4.D5, C20.17D4, C20.16D8
►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 24 123 141 64 85 47 104)(2 33 124 150 65 94 48 113)(3 22 125 159 66 83 49 102)(4 31 126 148 67 92 50 111)(5 40 127 157 68 81 51 120)(6 29 128 146 69 90 52 109)(7 38 129 155 70 99 53 118)(8 27 130 144 71 88 54 107)(9 36 131 153 72 97 55 116)(10 25 132 142 73 86 56 105)(11 34 133 151 74 95 57 114)(12 23 134 160 75 84 58 103)(13 32 135 149 76 93 59 112)(14 21 136 158 77 82 60 101)(15 30 137 147 78 91 41 110)(16 39 138 156 79 100 42 119)(17 28 139 145 80 89 43 108)(18 37 140 154 61 98 44 117)(19 26 121 143 62 87 45 106)(20 35 122 152 63 96 46 115)
(1 119 11 109)(2 118 12 108)(3 117 13 107)(4 116 14 106)(5 115 15 105)(6 114 16 104)(7 113 17 103)(8 112 18 102)(9 111 19 101)(10 110 20 120)(21 55 31 45)(22 54 32 44)(23 53 33 43)(24 52 34 42)(25 51 35 41)(26 50 36 60)(27 49 37 59)(28 48 38 58)(29 47 39 57)(30 46 40 56)(61 159 71 149)(62 158 72 148)(63 157 73 147)(64 156 74 146)(65 155 75 145)(66 154 76 144)(67 153 77 143)(68 152 78 142)(69 151 79 141)(70 150 80 160)(81 132 91 122)(82 131 92 121)(83 130 93 140)(84 129 94 139)(85 128 95 138)(86 127 96 137)(87 126 97 136)(88 125 98 135)(89 124 99 134)(90 123 100 133)
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), (1,24,123,141,64,85,47,104)(2,33,124,150,65,94,48,113)(3,22,125,159,66,83,49,102)(4,31,126,148,67,92,50,111)(5,40,127,157,68,81,51,120)(6,29,128,146,69,90,52,109)(7,38,129,155,70,99,53,118)(8,27,130,144,71,88,54,107)(9,36,131,153,72,97,55,116)(10,25,132,142,73,86,56,105)(11,34,133,151,74,95,57,114)(12,23,134,160,75,84,58,103)(13,32,135,149,76,93,59,112)(14,21,136,158,77,82,60,101)(15,30,137,147,78,91,41,110)(16,39,138,156,79,100,42,119)(17,28,139,145,80,89,43,108)(18,37,140,154,61,98,44,117)(19,26,121,143,62,87,45,106)(20,35,122,152,63,96,46,115), (1,119,11,109)(2,118,12,108)(3,117,13,107)(4,116,14,106)(5,115,15,105)(6,114,16,104)(7,113,17,103)(8,112,18,102)(9,111,19,101)(10,110,20,120)(21,55,31,45)(22,54,32,44)(23,53,33,43)(24,52,34,42)(25,51,35,41)(26,50,36,60)(27,49,37,59)(28,48,38,58)(29,47,39,57)(30,46,40,56)(61,159,71,149)(62,158,72,148)(63,157,73,147)(64,156,74,146)(65,155,75,145)(66,154,76,144)(67,153,77,143)(68,152,78,142)(69,151,79,141)(70,150,80,160)(81,132,91,122)(82,131,92,121)(83,130,93,140)(84,129,94,139)(85,128,95,138)(86,127,96,137)(87,126,97,136)(88,125,98,135)(89,124,99,134)(90,123,100,133)>;
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), (1,24,123,141,64,85,47,104)(2,33,124,150,65,94,48,113)(3,22,125,159,66,83,49,102)(4,31,126,148,67,92,50,111)(5,40,127,157,68,81,51,120)(6,29,128,146,69,90,52,109)(7,38,129,155,70,99,53,118)(8,27,130,144,71,88,54,107)(9,36,131,153,72,97,55,116)(10,25,132,142,73,86,56,105)(11,34,133,151,74,95,57,114)(12,23,134,160,75,84,58,103)(13,32,135,149,76,93,59,112)(14,21,136,158,77,82,60,101)(15,30,137,147,78,91,41,110)(16,39,138,156,79,100,42,119)(17,28,139,145,80,89,43,108)(18,37,140,154,61,98,44,117)(19,26,121,143,62,87,45,106)(20,35,122,152,63,96,46,115), (1,119,11,109)(2,118,12,108)(3,117,13,107)(4,116,14,106)(5,115,15,105)(6,114,16,104)(7,113,17,103)(8,112,18,102)(9,111,19,101)(10,110,20,120)(21,55,31,45)(22,54,32,44)(23,53,33,43)(24,52,34,42)(25,51,35,41)(26,50,36,60)(27,49,37,59)(28,48,38,58)(29,47,39,57)(30,46,40,56)(61,159,71,149)(62,158,72,148)(63,157,73,147)(64,156,74,146)(65,155,75,145)(66,154,76,144)(67,153,77,143)(68,152,78,142)(69,151,79,141)(70,150,80,160)(81,132,91,122)(82,131,92,121)(83,130,93,140)(84,129,94,139)(85,128,95,138)(86,127,96,137)(87,126,97,136)(88,125,98,135)(89,124,99,134)(90,123,100,133) );
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)], [(1,24,123,141,64,85,47,104),(2,33,124,150,65,94,48,113),(3,22,125,159,66,83,49,102),(4,31,126,148,67,92,50,111),(5,40,127,157,68,81,51,120),(6,29,128,146,69,90,52,109),(7,38,129,155,70,99,53,118),(8,27,130,144,71,88,54,107),(9,36,131,153,72,97,55,116),(10,25,132,142,73,86,56,105),(11,34,133,151,74,95,57,114),(12,23,134,160,75,84,58,103),(13,32,135,149,76,93,59,112),(14,21,136,158,77,82,60,101),(15,30,137,147,78,91,41,110),(16,39,138,156,79,100,42,119),(17,28,139,145,80,89,43,108),(18,37,140,154,61,98,44,117),(19,26,121,143,62,87,45,106),(20,35,122,152,63,96,46,115)], [(1,119,11,109),(2,118,12,108),(3,117,13,107),(4,116,14,106),(5,115,15,105),(6,114,16,104),(7,113,17,103),(8,112,18,102),(9,111,19,101),(10,110,20,120),(21,55,31,45),(22,54,32,44),(23,53,33,43),(24,52,34,42),(25,51,35,41),(26,50,36,60),(27,49,37,59),(28,48,38,58),(29,47,39,57),(30,46,40,56),(61,159,71,149),(62,158,72,148),(63,157,73,147),(64,156,74,146),(65,155,75,145),(66,154,76,144),(67,153,77,143),(68,152,78,142),(69,151,79,141),(70,150,80,160),(81,132,91,122),(82,131,92,121),(83,130,93,140),(84,129,94,139),(85,128,95,138),(86,127,96,137),(87,126,97,136),(88,125,98,135),(89,124,99,134),(90,123,100,133)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | ··· | 2 | 40 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
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 | D4.D5 | D4⋊2D5 |
| kernel | C20.16D8 | C4×C5⋊2C8 | D4⋊Dic5 | C20⋊2Q8 | C5×C4⋊1D4 | C2×C20 | C4⋊1D4 | C20 | C20 | C20 | C42 | C2×D4 | 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 C20.16D8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 2 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 29 | 29 | 0 | 0 | 0 | 0 |
| 12 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 37 | 0 | 0 |
| 0 | 0 | 14 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 15 |
| 0 | 0 | 0 | 0 | 26 | 15 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 12 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 4 | 0 | 0 |
| 0 | 0 | 27 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 26 |
| 0 | 0 | 0 | 0 | 26 | 26 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,25,2,0,0,0,0,0,23,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[29,12,0,0,0,0,29,29,0,0,0,0,0,0,4,14,0,0,0,0,37,37,0,0,0,0,0,0,15,26,0,0,0,0,15,15],[12,12,0,0,0,0,12,29,0,0,0,0,0,0,37,27,0,0,0,0,4,4,0,0,0,0,0,0,15,26,0,0,0,0,26,26] >;
C20.16D8 in GAP, Magma, Sage, TeX
C_{20}._{16}D_8 % in TeX
G:=Group("C20.16D8"); // GroupNames label
G:=SmallGroup(320,697);
// by ID
G=gap.SmallGroup(320,697);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,64,590,135,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;
// generators/relations