metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.22D4, (C2×D8).4D5, C4.22(D4×D5), (C8×Dic5)⋊6C2, (C10×D8).5C2, C5⋊2C8.31D4, (C2×D4).63D10, (C2×C8).237D10, C20.165(C2×D4), C5⋊3(C8.12D4), C8.15(C5⋊D4), (C2×Dic20)⋊18C2, C10.33(C4○D8), C20.17D4⋊5C2, (C2×C40).89C22, C22.255(D4×D5), C2.17(D8⋊3D5), C2.19(C20⋊D4), C10.28(C4⋊1D4), (C2×C20).432C23, (C2×Dic5).156D4, (D4×C10).82C22, (C4×Dic5).270C22, (C2×Dic10).125C22, C4.6(C2×C5⋊D4), (C2×D4.D5)⋊18C2, (C2×C10).345(C2×D4), (C2×C4).522(C22×D5), (C2×C5⋊2C8).278C22, SmallGroup(320,782)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.22D4
G = < a,b,c | a40=b4=1, c2=a20, bab-1=a9, cac-1=a-1, cbc-1=a20b-1 >
Subgroups: 494 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C8.12D4, Dic20, C2×C5⋊2C8, C4×Dic5, D4.D5, C23.D5, C2×C40, C5×D8, C2×Dic10, D4×C10, C8×Dic5, C2×Dic20, C2×D4.D5, C20.17D4, C10×D8, C40.22D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C4○D8, C5⋊D4, C22×D5, C8.12D4, D4×D5, C2×C5⋊D4, D8⋊3D5, C20⋊D4, C40.22D4
►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 64 151 97)(2 73 152 106)(3 42 153 115)(4 51 154 84)(5 60 155 93)(6 69 156 102)(7 78 157 111)(8 47 158 120)(9 56 159 89)(10 65 160 98)(11 74 121 107)(12 43 122 116)(13 52 123 85)(14 61 124 94)(15 70 125 103)(16 79 126 112)(17 48 127 81)(18 57 128 90)(19 66 129 99)(20 75 130 108)(21 44 131 117)(22 53 132 86)(23 62 133 95)(24 71 134 104)(25 80 135 113)(26 49 136 82)(27 58 137 91)(28 67 138 100)(29 76 139 109)(30 45 140 118)(31 54 141 87)(32 63 142 96)(33 72 143 105)(34 41 144 114)(35 50 145 83)(36 59 146 92)(37 68 147 101)(38 77 148 110)(39 46 149 119)(40 55 150 88)
(1 117 21 97)(2 116 22 96)(3 115 23 95)(4 114 24 94)(5 113 25 93)(6 112 26 92)(7 111 27 91)(8 110 28 90)(9 109 29 89)(10 108 30 88)(11 107 31 87)(12 106 32 86)(13 105 33 85)(14 104 34 84)(15 103 35 83)(16 102 36 82)(17 101 37 81)(18 100 38 120)(19 99 39 119)(20 98 40 118)(41 134 61 154)(42 133 62 153)(43 132 63 152)(44 131 64 151)(45 130 65 150)(46 129 66 149)(47 128 67 148)(48 127 68 147)(49 126 69 146)(50 125 70 145)(51 124 71 144)(52 123 72 143)(53 122 73 142)(54 121 74 141)(55 160 75 140)(56 159 76 139)(57 158 77 138)(58 157 78 137)(59 156 79 136)(60 155 80 135)
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), (1,64,151,97)(2,73,152,106)(3,42,153,115)(4,51,154,84)(5,60,155,93)(6,69,156,102)(7,78,157,111)(8,47,158,120)(9,56,159,89)(10,65,160,98)(11,74,121,107)(12,43,122,116)(13,52,123,85)(14,61,124,94)(15,70,125,103)(16,79,126,112)(17,48,127,81)(18,57,128,90)(19,66,129,99)(20,75,130,108)(21,44,131,117)(22,53,132,86)(23,62,133,95)(24,71,134,104)(25,80,135,113)(26,49,136,82)(27,58,137,91)(28,67,138,100)(29,76,139,109)(30,45,140,118)(31,54,141,87)(32,63,142,96)(33,72,143,105)(34,41,144,114)(35,50,145,83)(36,59,146,92)(37,68,147,101)(38,77,148,110)(39,46,149,119)(40,55,150,88), (1,117,21,97)(2,116,22,96)(3,115,23,95)(4,114,24,94)(5,113,25,93)(6,112,26,92)(7,111,27,91)(8,110,28,90)(9,109,29,89)(10,108,30,88)(11,107,31,87)(12,106,32,86)(13,105,33,85)(14,104,34,84)(15,103,35,83)(16,102,36,82)(17,101,37,81)(18,100,38,120)(19,99,39,119)(20,98,40,118)(41,134,61,154)(42,133,62,153)(43,132,63,152)(44,131,64,151)(45,130,65,150)(46,129,66,149)(47,128,67,148)(48,127,68,147)(49,126,69,146)(50,125,70,145)(51,124,71,144)(52,123,72,143)(53,122,73,142)(54,121,74,141)(55,160,75,140)(56,159,76,139)(57,158,77,138)(58,157,78,137)(59,156,79,136)(60,155,80,135)>;
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), (1,64,151,97)(2,73,152,106)(3,42,153,115)(4,51,154,84)(5,60,155,93)(6,69,156,102)(7,78,157,111)(8,47,158,120)(9,56,159,89)(10,65,160,98)(11,74,121,107)(12,43,122,116)(13,52,123,85)(14,61,124,94)(15,70,125,103)(16,79,126,112)(17,48,127,81)(18,57,128,90)(19,66,129,99)(20,75,130,108)(21,44,131,117)(22,53,132,86)(23,62,133,95)(24,71,134,104)(25,80,135,113)(26,49,136,82)(27,58,137,91)(28,67,138,100)(29,76,139,109)(30,45,140,118)(31,54,141,87)(32,63,142,96)(33,72,143,105)(34,41,144,114)(35,50,145,83)(36,59,146,92)(37,68,147,101)(38,77,148,110)(39,46,149,119)(40,55,150,88), (1,117,21,97)(2,116,22,96)(3,115,23,95)(4,114,24,94)(5,113,25,93)(6,112,26,92)(7,111,27,91)(8,110,28,90)(9,109,29,89)(10,108,30,88)(11,107,31,87)(12,106,32,86)(13,105,33,85)(14,104,34,84)(15,103,35,83)(16,102,36,82)(17,101,37,81)(18,100,38,120)(19,99,39,119)(20,98,40,118)(41,134,61,154)(42,133,62,153)(43,132,63,152)(44,131,64,151)(45,130,65,150)(46,129,66,149)(47,128,67,148)(48,127,68,147)(49,126,69,146)(50,125,70,145)(51,124,71,144)(52,123,72,143)(53,122,73,142)(54,121,74,141)(55,160,75,140)(56,159,76,139)(57,158,77,138)(58,157,78,137)(59,156,79,136)(60,155,80,135) );
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)], [(1,64,151,97),(2,73,152,106),(3,42,153,115),(4,51,154,84),(5,60,155,93),(6,69,156,102),(7,78,157,111),(8,47,158,120),(9,56,159,89),(10,65,160,98),(11,74,121,107),(12,43,122,116),(13,52,123,85),(14,61,124,94),(15,70,125,103),(16,79,126,112),(17,48,127,81),(18,57,128,90),(19,66,129,99),(20,75,130,108),(21,44,131,117),(22,53,132,86),(23,62,133,95),(24,71,134,104),(25,80,135,113),(26,49,136,82),(27,58,137,91),(28,67,138,100),(29,76,139,109),(30,45,140,118),(31,54,141,87),(32,63,142,96),(33,72,143,105),(34,41,144,114),(35,50,145,83),(36,59,146,92),(37,68,147,101),(38,77,148,110),(39,46,149,119),(40,55,150,88)], [(1,117,21,97),(2,116,22,96),(3,115,23,95),(4,114,24,94),(5,113,25,93),(6,112,26,92),(7,111,27,91),(8,110,28,90),(9,109,29,89),(10,108,30,88),(11,107,31,87),(12,106,32,86),(13,105,33,85),(14,104,34,84),(15,103,35,83),(16,102,36,82),(17,101,37,81),(18,100,38,120),(19,99,39,119),(20,98,40,118),(41,134,61,154),(42,133,62,153),(43,132,63,152),(44,131,64,151),(45,130,65,150),(46,129,66,149),(47,128,67,148),(48,127,68,147),(49,126,69,146),(50,125,70,145),(51,124,71,144),(52,123,72,143),(53,122,73,142),(54,121,74,141),(55,160,75,140),(56,159,76,139),(57,158,77,138),(58,157,78,137),(59,156,79,136),(60,155,80,135)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 10 | 10 | 10 | 10 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | C4○D8 | C5⋊D4 | D4×D5 | D4×D5 | D8⋊3D5 |
| kernel | C40.22D4 | C8×Dic5 | C2×Dic20 | C2×D4.D5 | C20.17D4 | C10×D8 | C5⋊2C8 | C40 | C2×Dic5 | C2×D8 | C2×C8 | C2×D4 | C10 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 8 |
Matrix representation of C40.22D4 ►in GL4(𝔽41) generated by
| 37 | 8 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 29 | 29 |
| 9 | 0 | 0 | 0 |
| 26 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 32 | 22 | 0 | 0 |
| 15 | 9 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [37,0,0,0,8,10,0,0,0,0,29,29,0,0,12,29],[9,26,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[32,15,0,0,22,9,0,0,0,0,32,0,0,0,0,9] >;
C40.22D4 in GAP, Magma, Sage, TeX
C_{40}._{22}D_4 % in TeX
G:=Group("C40.22D4"); // GroupNames label
G:=SmallGroup(320,782);
// by ID
G=gap.SmallGroup(320,782);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,1094,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=a^20*b^-1>;
// generators/relations