metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.4D4, C23.19D20, C40⋊6C4⋊5C2, (C2×C4).54D20, (C2×C8).79D10, C8.1(C5⋊D4), C5⋊4(C8.D4), (C2×C20).299D4, C20.423(C2×D4), (C2×Dic20)⋊12C2, (C2×C40).65C22, (C2×M4(2)).3D5, C20.232(C4○D4), C4.116(C4○D20), C20.44D4⋊42C2, C10.75(C4⋊D4), C2.23(C20⋊7D4), (C2×C20).777C23, (C22×C4).145D10, (C22×C10).105D4, C22.136(C2×D20), (C10×M4(2)).3C2, C4⋊Dic5.28C22, C20.48D4.17C2, C2.23(C8.D10), C10.23(C8.C22), (C22×C20).306C22, (C2×Dic10).22C22, C4.116(C2×C5⋊D4), (C2×C10).167(C2×D4), (C2×C4).726(C22×D5), SmallGroup(320,764)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.4D4
G = < a,b,c | a40=b4=1, c2=a20, bab-1=a19, cac-1=a-1, cbc-1=a20b-1 >
Subgroups: 406 in 110 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C40, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C8.D4, Dic20, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C2×Dic10, C22×C20, C20.44D4, C40⋊6C4, C2×Dic20, C20.48D4, C10×M4(2), C40.4D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8.C22, D20, C5⋊D4, C22×D5, C8.D4, C2×D20, C4○D20, C2×C5⋊D4, C8.D10, C20⋊7D4, C40.4D4
►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 79 120 134)(2 58 81 153)(3 77 82 132)(4 56 83 151)(5 75 84 130)(6 54 85 149)(7 73 86 128)(8 52 87 147)(9 71 88 126)(10 50 89 145)(11 69 90 124)(12 48 91 143)(13 67 92 122)(14 46 93 141)(15 65 94 160)(16 44 95 139)(17 63 96 158)(18 42 97 137)(19 61 98 156)(20 80 99 135)(21 59 100 154)(22 78 101 133)(23 57 102 152)(24 76 103 131)(25 55 104 150)(26 74 105 129)(27 53 106 148)(28 72 107 127)(29 51 108 146)(30 70 109 125)(31 49 110 144)(32 68 111 123)(33 47 112 142)(34 66 113 121)(35 45 114 140)(36 64 115 159)(37 43 116 138)(38 62 117 157)(39 41 118 136)(40 60 119 155)
(1 154 21 134)(2 153 22 133)(3 152 23 132)(4 151 24 131)(5 150 25 130)(6 149 26 129)(7 148 27 128)(8 147 28 127)(9 146 29 126)(10 145 30 125)(11 144 31 124)(12 143 32 123)(13 142 33 122)(14 141 34 121)(15 140 35 160)(16 139 36 159)(17 138 37 158)(18 137 38 157)(19 136 39 156)(20 135 40 155)(41 118 61 98)(42 117 62 97)(43 116 63 96)(44 115 64 95)(45 114 65 94)(46 113 66 93)(47 112 67 92)(48 111 68 91)(49 110 69 90)(50 109 70 89)(51 108 71 88)(52 107 72 87)(53 106 73 86)(54 105 74 85)(55 104 75 84)(56 103 76 83)(57 102 77 82)(58 101 78 81)(59 100 79 120)(60 99 80 119)
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), (1,79,120,134)(2,58,81,153)(3,77,82,132)(4,56,83,151)(5,75,84,130)(6,54,85,149)(7,73,86,128)(8,52,87,147)(9,71,88,126)(10,50,89,145)(11,69,90,124)(12,48,91,143)(13,67,92,122)(14,46,93,141)(15,65,94,160)(16,44,95,139)(17,63,96,158)(18,42,97,137)(19,61,98,156)(20,80,99,135)(21,59,100,154)(22,78,101,133)(23,57,102,152)(24,76,103,131)(25,55,104,150)(26,74,105,129)(27,53,106,148)(28,72,107,127)(29,51,108,146)(30,70,109,125)(31,49,110,144)(32,68,111,123)(33,47,112,142)(34,66,113,121)(35,45,114,140)(36,64,115,159)(37,43,116,138)(38,62,117,157)(39,41,118,136)(40,60,119,155), (1,154,21,134)(2,153,22,133)(3,152,23,132)(4,151,24,131)(5,150,25,130)(6,149,26,129)(7,148,27,128)(8,147,28,127)(9,146,29,126)(10,145,30,125)(11,144,31,124)(12,143,32,123)(13,142,33,122)(14,141,34,121)(15,140,35,160)(16,139,36,159)(17,138,37,158)(18,137,38,157)(19,136,39,156)(20,135,40,155)(41,118,61,98)(42,117,62,97)(43,116,63,96)(44,115,64,95)(45,114,65,94)(46,113,66,93)(47,112,67,92)(48,111,68,91)(49,110,69,90)(50,109,70,89)(51,108,71,88)(52,107,72,87)(53,106,73,86)(54,105,74,85)(55,104,75,84)(56,103,76,83)(57,102,77,82)(58,101,78,81)(59,100,79,120)(60,99,80,119)>;
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), (1,79,120,134)(2,58,81,153)(3,77,82,132)(4,56,83,151)(5,75,84,130)(6,54,85,149)(7,73,86,128)(8,52,87,147)(9,71,88,126)(10,50,89,145)(11,69,90,124)(12,48,91,143)(13,67,92,122)(14,46,93,141)(15,65,94,160)(16,44,95,139)(17,63,96,158)(18,42,97,137)(19,61,98,156)(20,80,99,135)(21,59,100,154)(22,78,101,133)(23,57,102,152)(24,76,103,131)(25,55,104,150)(26,74,105,129)(27,53,106,148)(28,72,107,127)(29,51,108,146)(30,70,109,125)(31,49,110,144)(32,68,111,123)(33,47,112,142)(34,66,113,121)(35,45,114,140)(36,64,115,159)(37,43,116,138)(38,62,117,157)(39,41,118,136)(40,60,119,155), (1,154,21,134)(2,153,22,133)(3,152,23,132)(4,151,24,131)(5,150,25,130)(6,149,26,129)(7,148,27,128)(8,147,28,127)(9,146,29,126)(10,145,30,125)(11,144,31,124)(12,143,32,123)(13,142,33,122)(14,141,34,121)(15,140,35,160)(16,139,36,159)(17,138,37,158)(18,137,38,157)(19,136,39,156)(20,135,40,155)(41,118,61,98)(42,117,62,97)(43,116,63,96)(44,115,64,95)(45,114,65,94)(46,113,66,93)(47,112,67,92)(48,111,68,91)(49,110,69,90)(50,109,70,89)(51,108,71,88)(52,107,72,87)(53,106,73,86)(54,105,74,85)(55,104,75,84)(56,103,76,83)(57,102,77,82)(58,101,78,81)(59,100,79,120)(60,99,80,119) );
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)], [(1,79,120,134),(2,58,81,153),(3,77,82,132),(4,56,83,151),(5,75,84,130),(6,54,85,149),(7,73,86,128),(8,52,87,147),(9,71,88,126),(10,50,89,145),(11,69,90,124),(12,48,91,143),(13,67,92,122),(14,46,93,141),(15,65,94,160),(16,44,95,139),(17,63,96,158),(18,42,97,137),(19,61,98,156),(20,80,99,135),(21,59,100,154),(22,78,101,133),(23,57,102,152),(24,76,103,131),(25,55,104,150),(26,74,105,129),(27,53,106,148),(28,72,107,127),(29,51,108,146),(30,70,109,125),(31,49,110,144),(32,68,111,123),(33,47,112,142),(34,66,113,121),(35,45,114,140),(36,64,115,159),(37,43,116,138),(38,62,117,157),(39,41,118,136),(40,60,119,155)], [(1,154,21,134),(2,153,22,133),(3,152,23,132),(4,151,24,131),(5,150,25,130),(6,149,26,129),(7,148,27,128),(8,147,28,127),(9,146,29,126),(10,145,30,125),(11,144,31,124),(12,143,32,123),(13,142,33,122),(14,141,34,121),(15,140,35,160),(16,139,36,159),(17,138,37,158),(18,137,38,157),(19,136,39,156),(20,135,40,155),(41,118,61,98),(42,117,62,97),(43,116,63,96),(44,115,64,95),(45,114,65,94),(46,113,66,93),(47,112,67,92),(48,111,68,91),(49,110,69,90),(50,109,70,89),(51,108,71,88),(52,107,72,87),(53,106,73,86),(54,105,74,85),(55,104,75,84),(56,103,76,83),(57,102,77,82),(58,101,78,81),(59,100,79,120),(60,99,80,119)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 40 | 40 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | D20 | C4○D20 | C8.C22 | C8.D10 |
| kernel | C40.4D4 | C20.44D4 | C40⋊6C4 | C2×Dic20 | C20.48D4 | C10×M4(2) | C40 | C2×C20 | C22×C10 | C2×M4(2) | C20 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 8 | 2 | 8 |
Matrix representation of C40.4D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 33 | 0 | 0 |
| 0 | 0 | 20 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 16 |
| 0 | 0 | 0 | 0 | 1 | 33 |
| 19 | 39 | 0 | 0 | 0 | 0 |
| 17 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 19 | 39 | 0 | 0 | 0 | 0 |
| 16 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,20,0,0,0,0,33,4,0,0,0,0,0,0,8,1,0,0,0,0,16,33],[19,17,0,0,0,0,39,22,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,1,40,0,0,0,0,0,40,0,0],[19,16,0,0,0,0,39,22,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;
C40.4D4 in GAP, Magma, Sage, TeX
C_{40}._4D_4 % in TeX
G:=Group("C40.4D4"); // GroupNames label
G:=SmallGroup(320,764);
// by ID
G=gap.SmallGroup(320,764);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,344,254,387,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=a^20*b^-1>;
// generators/relations