metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.20D20, C40.18D4, D20.20D4, Dic10.20D4, M4(2).9D10, C4.135(D4×D5), C8.C4⋊5D5, C4.56(C2×D20), (C2×C8).70D10, C20.136(C2×D4), C5⋊2(D4.5D4), (C2×Dic20)⋊21C2, C4.12D20⋊3C2, C8.D10.2C2, C10.49(C4⋊D4), C2.22(C4⋊D20), (C2×C20).312C23, (C2×C40).102C22, D20.3C4.2C2, C4○D20.39C22, C22.6(Q8⋊2D5), (C5×M4(2)).6C22, C4.Dic5.37C22, (C2×Dic10).98C22, (C5×C8.C4)⋊6C2, (C2×C10).3(C4○D4), (C2×C4).113(C22×D5), SmallGroup(320,523)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.20D20
G = < a,b,c | a40=1, b4=c2=a20, bab-1=a31, cac-1=a-1, cbc-1=b3 >
Subgroups: 414 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C5⋊2C8, C40, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, D4.5D4, C8×D5, C8⋊D5, C40⋊C2, Dic20, C4.Dic5, C2×C40, C5×M4(2), C2×Dic10, C4○D20, C4.12D20, C5×C8.C4, D20.3C4, C2×Dic20, C8.D10, C8.20D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C22×D5, D4.5D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, C8.20D20
►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 78 141 110 21 58 121 90)(2 69 142 101 22 49 122 81)(3 60 143 92 23 80 123 112)(4 51 144 83 24 71 124 103)(5 42 145 114 25 62 125 94)(6 73 146 105 26 53 126 85)(7 64 147 96 27 44 127 116)(8 55 148 87 28 75 128 107)(9 46 149 118 29 66 129 98)(10 77 150 109 30 57 130 89)(11 68 151 100 31 48 131 120)(12 59 152 91 32 79 132 111)(13 50 153 82 33 70 133 102)(14 41 154 113 34 61 134 93)(15 72 155 104 35 52 135 84)(16 63 156 95 36 43 136 115)(17 54 157 86 37 74 137 106)(18 45 158 117 38 65 138 97)(19 76 159 108 39 56 139 88)(20 67 160 99 40 47 140 119)
(1 110 21 90)(2 109 22 89)(3 108 23 88)(4 107 24 87)(5 106 25 86)(6 105 26 85)(7 104 27 84)(8 103 28 83)(9 102 29 82)(10 101 30 81)(11 100 31 120)(12 99 32 119)(13 98 33 118)(14 97 34 117)(15 96 35 116)(16 95 36 115)(17 94 37 114)(18 93 38 113)(19 92 39 112)(20 91 40 111)(41 158 61 138)(42 157 62 137)(43 156 63 136)(44 155 64 135)(45 154 65 134)(46 153 66 133)(47 152 67 132)(48 151 68 131)(49 150 69 130)(50 149 70 129)(51 148 71 128)(52 147 72 127)(53 146 73 126)(54 145 74 125)(55 144 75 124)(56 143 76 123)(57 142 77 122)(58 141 78 121)(59 140 79 160)(60 139 80 159)
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), (1,78,141,110,21,58,121,90)(2,69,142,101,22,49,122,81)(3,60,143,92,23,80,123,112)(4,51,144,83,24,71,124,103)(5,42,145,114,25,62,125,94)(6,73,146,105,26,53,126,85)(7,64,147,96,27,44,127,116)(8,55,148,87,28,75,128,107)(9,46,149,118,29,66,129,98)(10,77,150,109,30,57,130,89)(11,68,151,100,31,48,131,120)(12,59,152,91,32,79,132,111)(13,50,153,82,33,70,133,102)(14,41,154,113,34,61,134,93)(15,72,155,104,35,52,135,84)(16,63,156,95,36,43,136,115)(17,54,157,86,37,74,137,106)(18,45,158,117,38,65,138,97)(19,76,159,108,39,56,139,88)(20,67,160,99,40,47,140,119), (1,110,21,90)(2,109,22,89)(3,108,23,88)(4,107,24,87)(5,106,25,86)(6,105,26,85)(7,104,27,84)(8,103,28,83)(9,102,29,82)(10,101,30,81)(11,100,31,120)(12,99,32,119)(13,98,33,118)(14,97,34,117)(15,96,35,116)(16,95,36,115)(17,94,37,114)(18,93,38,113)(19,92,39,112)(20,91,40,111)(41,158,61,138)(42,157,62,137)(43,156,63,136)(44,155,64,135)(45,154,65,134)(46,153,66,133)(47,152,67,132)(48,151,68,131)(49,150,69,130)(50,149,70,129)(51,148,71,128)(52,147,72,127)(53,146,73,126)(54,145,74,125)(55,144,75,124)(56,143,76,123)(57,142,77,122)(58,141,78,121)(59,140,79,160)(60,139,80,159)>;
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), (1,78,141,110,21,58,121,90)(2,69,142,101,22,49,122,81)(3,60,143,92,23,80,123,112)(4,51,144,83,24,71,124,103)(5,42,145,114,25,62,125,94)(6,73,146,105,26,53,126,85)(7,64,147,96,27,44,127,116)(8,55,148,87,28,75,128,107)(9,46,149,118,29,66,129,98)(10,77,150,109,30,57,130,89)(11,68,151,100,31,48,131,120)(12,59,152,91,32,79,132,111)(13,50,153,82,33,70,133,102)(14,41,154,113,34,61,134,93)(15,72,155,104,35,52,135,84)(16,63,156,95,36,43,136,115)(17,54,157,86,37,74,137,106)(18,45,158,117,38,65,138,97)(19,76,159,108,39,56,139,88)(20,67,160,99,40,47,140,119), (1,110,21,90)(2,109,22,89)(3,108,23,88)(4,107,24,87)(5,106,25,86)(6,105,26,85)(7,104,27,84)(8,103,28,83)(9,102,29,82)(10,101,30,81)(11,100,31,120)(12,99,32,119)(13,98,33,118)(14,97,34,117)(15,96,35,116)(16,95,36,115)(17,94,37,114)(18,93,38,113)(19,92,39,112)(20,91,40,111)(41,158,61,138)(42,157,62,137)(43,156,63,136)(44,155,64,135)(45,154,65,134)(46,153,66,133)(47,152,67,132)(48,151,68,131)(49,150,69,130)(50,149,70,129)(51,148,71,128)(52,147,72,127)(53,146,73,126)(54,145,74,125)(55,144,75,124)(56,143,76,123)(57,142,77,122)(58,141,78,121)(59,140,79,160)(60,139,80,159) );
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)], [(1,78,141,110,21,58,121,90),(2,69,142,101,22,49,122,81),(3,60,143,92,23,80,123,112),(4,51,144,83,24,71,124,103),(5,42,145,114,25,62,125,94),(6,73,146,105,26,53,126,85),(7,64,147,96,27,44,127,116),(8,55,148,87,28,75,128,107),(9,46,149,118,29,66,129,98),(10,77,150,109,30,57,130,89),(11,68,151,100,31,48,131,120),(12,59,152,91,32,79,132,111),(13,50,153,82,33,70,133,102),(14,41,154,113,34,61,134,93),(15,72,155,104,35,52,135,84),(16,63,156,95,36,43,136,115),(17,54,157,86,37,74,137,106),(18,45,158,117,38,65,138,97),(19,76,159,108,39,56,139,88),(20,67,160,99,40,47,140,119)], [(1,110,21,90),(2,109,22,89),(3,108,23,88),(4,107,24,87),(5,106,25,86),(6,105,26,85),(7,104,27,84),(8,103,28,83),(9,102,29,82),(10,101,30,81),(11,100,31,120),(12,99,32,119),(13,98,33,118),(14,97,34,117),(15,96,35,116),(16,95,36,115),(17,94,37,114),(18,93,38,113),(19,92,39,112),(20,91,40,111),(41,158,61,138),(42,157,62,137),(43,156,63,136),(44,155,64,135),(45,154,65,134),(46,153,66,133),(47,152,67,132),(48,151,68,131),(49,150,69,130),(50,149,70,129),(51,148,71,128),(52,147,72,127),(53,146,73,126),(54,145,74,125),(55,144,75,124),(56,143,76,123),(57,142,77,122),(58,141,78,121),(59,140,79,160),(60,139,80,159)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 2 | 2 | 20 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D20 | D4.5D4 | D4×D5 | Q8⋊2D5 | C8.20D20 |
| kernel | C8.20D20 | C4.12D20 | C5×C8.C4 | D20.3C4 | C2×Dic20 | C8.D10 | C40 | Dic10 | D20 | C8.C4 | C2×C10 | C2×C8 | M4(2) | C8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C8.20D20 ►in GL4(𝔽41) generated by
| 24 | 4 | 27 | 37 |
| 10 | 0 | 31 | 10 |
| 36 | 35 | 0 | 0 |
| 40 | 35 | 27 | 37 |
| 4 | 25 | 35 | 18 |
| 34 | 20 | 17 | 9 |
| 34 | 1 | 35 | 16 |
| 36 | 28 | 19 | 23 |
| 17 | 6 | 22 | 35 |
| 25 | 30 | 26 | 13 |
| 34 | 1 | 35 | 16 |
| 34 | 37 | 39 | 0 |
G:=sub<GL(4,GF(41))| [24,10,36,40,4,0,35,35,27,31,0,27,37,10,0,37],[4,34,34,36,25,20,1,28,35,17,35,19,18,9,16,23],[17,25,34,34,6,30,1,37,22,26,35,39,35,13,16,0] >;
C8.20D20 in GAP, Magma, Sage, TeX
C_8._{20}D_{20} % in TeX
G:=Group("C8.20D20"); // GroupNames label
G:=SmallGroup(320,523);
// by ID
G=gap.SmallGroup(320,523);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,58,1123,136,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=1,b^4=c^2=a^20,b*a*b^-1=a^31,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations