metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.1D4, C8.1D20, C42.21D10, C8⋊C4⋊4D5, C20⋊2Q8⋊4C2, (C2×C20).39D4, (C2×C4).28D20, C4.35(C2×D20), (C2×C8).56D10, C5⋊1(C8.2D4), C20.278(C2×D4), (C4×C20).6C22, (C2×Dic20)⋊10C2, C10.8(C4⋊1D4), (C2×C40).57C22, C4.D20.5C2, C2.10(C20⋊4D4), (C2×C20).736C23, C2.9(C8.D10), (C2×D20).11C22, C22.100(C2×D20), C10.5(C8.C22), (C2×Dic10).11C22, (C5×C8⋊C4)⋊5C2, (C2×C40⋊C2).2C2, (C2×C10).119(C2×D4), (C2×C4).680(C22×D5), SmallGroup(320,342)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.D20
G = < a,b,c | a8=b20=1, c2=a4, bab-1=a5, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 590 in 124 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C40, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C8.2D4, C40⋊C2, Dic20, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×Dic10, C2×D20, C5×C8⋊C4, C20⋊2Q8, C4.D20, C2×C40⋊C2, C2×Dic20, C8.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C8.C22, D20, C22×D5, C8.2D4, C2×D20, C20⋊4D4, C8.D10, C8.D20
►On 160 points
Generators in S160
(1 38 128 93 61 144 46 101)(2 145 129 102 62 39 47 94)(3 40 130 95 63 146 48 103)(4 147 131 104 64 21 49 96)(5 22 132 97 65 148 50 105)(6 149 133 106 66 23 51 98)(7 24 134 99 67 150 52 107)(8 151 135 108 68 25 53 100)(9 26 136 81 69 152 54 109)(10 153 137 110 70 27 55 82)(11 28 138 83 71 154 56 111)(12 155 139 112 72 29 57 84)(13 30 140 85 73 156 58 113)(14 157 121 114 74 31 59 86)(15 32 122 87 75 158 60 115)(16 159 123 116 76 33 41 88)(17 34 124 89 77 160 42 117)(18 141 125 118 78 35 43 90)(19 36 126 91 79 142 44 119)(20 143 127 120 80 37 45 92)
(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 80 61 20)(2 19 62 79)(3 78 63 18)(4 17 64 77)(5 76 65 16)(6 15 66 75)(7 74 67 14)(8 13 68 73)(9 72 69 12)(10 11 70 71)(21 89 147 117)(22 116 148 88)(23 87 149 115)(24 114 150 86)(25 85 151 113)(26 112 152 84)(27 83 153 111)(28 110 154 82)(29 81 155 109)(30 108 156 100)(31 99 157 107)(32 106 158 98)(33 97 159 105)(34 104 160 96)(35 95 141 103)(36 102 142 94)(37 93 143 101)(38 120 144 92)(39 91 145 119)(40 118 146 90)(41 132 123 50)(42 49 124 131)(43 130 125 48)(44 47 126 129)(45 128 127 46)(51 122 133 60)(52 59 134 121)(53 140 135 58)(54 57 136 139)(55 138 137 56)
G:=sub<Sym(160)| (1,38,128,93,61,144,46,101)(2,145,129,102,62,39,47,94)(3,40,130,95,63,146,48,103)(4,147,131,104,64,21,49,96)(5,22,132,97,65,148,50,105)(6,149,133,106,66,23,51,98)(7,24,134,99,67,150,52,107)(8,151,135,108,68,25,53,100)(9,26,136,81,69,152,54,109)(10,153,137,110,70,27,55,82)(11,28,138,83,71,154,56,111)(12,155,139,112,72,29,57,84)(13,30,140,85,73,156,58,113)(14,157,121,114,74,31,59,86)(15,32,122,87,75,158,60,115)(16,159,123,116,76,33,41,88)(17,34,124,89,77,160,42,117)(18,141,125,118,78,35,43,90)(19,36,126,91,79,142,44,119)(20,143,127,120,80,37,45,92), (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), (1,80,61,20)(2,19,62,79)(3,78,63,18)(4,17,64,77)(5,76,65,16)(6,15,66,75)(7,74,67,14)(8,13,68,73)(9,72,69,12)(10,11,70,71)(21,89,147,117)(22,116,148,88)(23,87,149,115)(24,114,150,86)(25,85,151,113)(26,112,152,84)(27,83,153,111)(28,110,154,82)(29,81,155,109)(30,108,156,100)(31,99,157,107)(32,106,158,98)(33,97,159,105)(34,104,160,96)(35,95,141,103)(36,102,142,94)(37,93,143,101)(38,120,144,92)(39,91,145,119)(40,118,146,90)(41,132,123,50)(42,49,124,131)(43,130,125,48)(44,47,126,129)(45,128,127,46)(51,122,133,60)(52,59,134,121)(53,140,135,58)(54,57,136,139)(55,138,137,56)>;
G:=Group( (1,38,128,93,61,144,46,101)(2,145,129,102,62,39,47,94)(3,40,130,95,63,146,48,103)(4,147,131,104,64,21,49,96)(5,22,132,97,65,148,50,105)(6,149,133,106,66,23,51,98)(7,24,134,99,67,150,52,107)(8,151,135,108,68,25,53,100)(9,26,136,81,69,152,54,109)(10,153,137,110,70,27,55,82)(11,28,138,83,71,154,56,111)(12,155,139,112,72,29,57,84)(13,30,140,85,73,156,58,113)(14,157,121,114,74,31,59,86)(15,32,122,87,75,158,60,115)(16,159,123,116,76,33,41,88)(17,34,124,89,77,160,42,117)(18,141,125,118,78,35,43,90)(19,36,126,91,79,142,44,119)(20,143,127,120,80,37,45,92), (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), (1,80,61,20)(2,19,62,79)(3,78,63,18)(4,17,64,77)(5,76,65,16)(6,15,66,75)(7,74,67,14)(8,13,68,73)(9,72,69,12)(10,11,70,71)(21,89,147,117)(22,116,148,88)(23,87,149,115)(24,114,150,86)(25,85,151,113)(26,112,152,84)(27,83,153,111)(28,110,154,82)(29,81,155,109)(30,108,156,100)(31,99,157,107)(32,106,158,98)(33,97,159,105)(34,104,160,96)(35,95,141,103)(36,102,142,94)(37,93,143,101)(38,120,144,92)(39,91,145,119)(40,118,146,90)(41,132,123,50)(42,49,124,131)(43,130,125,48)(44,47,126,129)(45,128,127,46)(51,122,133,60)(52,59,134,121)(53,140,135,58)(54,57,136,139)(55,138,137,56) );
G=PermutationGroup([[(1,38,128,93,61,144,46,101),(2,145,129,102,62,39,47,94),(3,40,130,95,63,146,48,103),(4,147,131,104,64,21,49,96),(5,22,132,97,65,148,50,105),(6,149,133,106,66,23,51,98),(7,24,134,99,67,150,52,107),(8,151,135,108,68,25,53,100),(9,26,136,81,69,152,54,109),(10,153,137,110,70,27,55,82),(11,28,138,83,71,154,56,111),(12,155,139,112,72,29,57,84),(13,30,140,85,73,156,58,113),(14,157,121,114,74,31,59,86),(15,32,122,87,75,158,60,115),(16,159,123,116,76,33,41,88),(17,34,124,89,77,160,42,117),(18,141,125,118,78,35,43,90),(19,36,126,91,79,142,44,119),(20,143,127,120,80,37,45,92)], [(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)], [(1,80,61,20),(2,19,62,79),(3,78,63,18),(4,17,64,77),(5,76,65,16),(6,15,66,75),(7,74,67,14),(8,13,68,73),(9,72,69,12),(10,11,70,71),(21,89,147,117),(22,116,148,88),(23,87,149,115),(24,114,150,86),(25,85,151,113),(26,112,152,84),(27,83,153,111),(28,110,154,82),(29,81,155,109),(30,108,156,100),(31,99,157,107),(32,106,158,98),(33,97,159,105),(34,104,160,96),(35,95,141,103),(36,102,142,94),(37,93,143,101),(38,120,144,92),(39,91,145,119),(40,118,146,90),(41,132,123,50),(42,49,124,131),(43,130,125,48),(44,47,126,129),(45,128,127,46),(51,122,133,60),(52,59,134,121),(53,140,135,58),(54,57,136,139),(55,138,137,56)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 4 | 4 | 40 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D20 | D20 | C8.C22 | C8.D10 |
| kernel | C8.D20 | C5×C8⋊C4 | C20⋊2Q8 | C4.D20 | C2×C40⋊C2 | C2×Dic20 | C40 | C2×C20 | C8⋊C4 | C42 | C2×C8 | C8 | C2×C4 | C10 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 16 | 8 | 2 | 8 |
Matrix representation of C8.D20 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 21 | 40 | 29 |
| 0 | 0 | 20 | 13 | 12 | 1 |
| 0 | 0 | 15 | 0 | 4 | 20 |
| 0 | 0 | 0 | 15 | 21 | 28 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 21 | 37 |
| 0 | 0 | 40 | 5 | 4 | 8 |
| 0 | 0 | 21 | 28 | 29 | 40 |
| 0 | 0 | 13 | 30 | 1 | 36 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 40 | 20 | 4 |
| 0 | 0 | 1 | 12 | 28 | 21 |
| 0 | 0 | 20 | 4 | 0 | 26 |
| 0 | 0 | 28 | 21 | 26 | 0 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,37,20,15,0,0,0,21,13,0,15,0,0,40,12,4,21,0,0,29,1,20,28],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,12,40,21,13,0,0,1,5,28,30,0,0,21,4,29,1,0,0,37,8,40,36],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,29,1,20,28,0,0,40,12,4,21,0,0,20,28,0,26,0,0,4,21,26,0] >;
C8.D20 in GAP, Magma, Sage, TeX
C_8.D_{20} % in TeX
G:=Group("C8.D20"); // GroupNames label
G:=SmallGroup(320,342);
// by ID
G=gap.SmallGroup(320,342);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=1,c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations