metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.31D4, C40⋊8C4⋊5C2, C4.23(D4×D5), C5⋊2C8.9D4, (C2×C8).89D10, C5⋊4(C8.2D4), C8.3(C5⋊D4), (C2×D4).70D10, C20.174(C2×D4), (C2×Q8).52D10, (C2×SD16).2D5, (C2×Dic20)⋊25C2, Dic5⋊Q8⋊4C2, (C2×Dic5).78D4, (C10×SD16).2C2, C22.264(D4×D5), C2.20(C20⋊D4), C10.29(C4⋊1D4), (C2×C40).114C22, (C2×C20).444C23, C20.17D4.8C2, (D4×C10).93C22, (Q8×C10).74C22, C2.28(SD16⋊D5), C10.48(C8.C22), (C4×Dic5).57C22, (C2×Dic10).129C22, C4.7(C2×C5⋊D4), (C2×C5⋊Q16)⋊17C2, (C2×D4.D5).9C2, (C2×C10).356(C2×D4), (C2×C4).533(C22×D5), (C2×C5⋊2C8).156C22, SmallGroup(320,794)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.31D4
G = < a,b,c | a40=b4=1, c2=a20, bab-1=a29, cac-1=a-1, cbc-1=b-1 >
Subgroups: 462 in 124 conjugacy classes, 43 normal (31 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, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×SD16, C2×Q16, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C8.2D4, Dic20, C2×C5⋊2C8, C4×Dic5, C10.D4, D4.D5, C5⋊Q16, C23.D5, C2×C40, C5×SD16, C2×Dic10, D4×C10, Q8×C10, C40⋊8C4, C2×Dic20, C2×D4.D5, C20.17D4, C2×C5⋊Q16, Dic5⋊Q8, C10×SD16, C40.31D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C8.C22, C5⋊D4, C22×D5, C8.2D4, D4×D5, C2×C5⋊D4, SD16⋊D5, C20⋊D4, C40.31D4
►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 99 74 136)(2 88 75 125)(3 117 76 154)(4 106 77 143)(5 95 78 132)(6 84 79 121)(7 113 80 150)(8 102 41 139)(9 91 42 128)(10 120 43 157)(11 109 44 146)(12 98 45 135)(13 87 46 124)(14 116 47 153)(15 105 48 142)(16 94 49 131)(17 83 50 160)(18 112 51 149)(19 101 52 138)(20 90 53 127)(21 119 54 156)(22 108 55 145)(23 97 56 134)(24 86 57 123)(25 115 58 152)(26 104 59 141)(27 93 60 130)(28 82 61 159)(29 111 62 148)(30 100 63 137)(31 89 64 126)(32 118 65 155)(33 107 66 144)(34 96 67 133)(35 85 68 122)(36 114 69 151)(37 103 70 140)(38 92 71 129)(39 81 72 158)(40 110 73 147)
(1 151 21 131)(2 150 22 130)(3 149 23 129)(4 148 24 128)(5 147 25 127)(6 146 26 126)(7 145 27 125)(8 144 28 124)(9 143 29 123)(10 142 30 122)(11 141 31 121)(12 140 32 160)(13 139 33 159)(14 138 34 158)(15 137 35 157)(16 136 36 156)(17 135 37 155)(18 134 38 154)(19 133 39 153)(20 132 40 152)(41 107 61 87)(42 106 62 86)(43 105 63 85)(44 104 64 84)(45 103 65 83)(46 102 66 82)(47 101 67 81)(48 100 68 120)(49 99 69 119)(50 98 70 118)(51 97 71 117)(52 96 72 116)(53 95 73 115)(54 94 74 114)(55 93 75 113)(56 92 76 112)(57 91 77 111)(58 90 78 110)(59 89 79 109)(60 88 80 108)
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), (1,99,74,136)(2,88,75,125)(3,117,76,154)(4,106,77,143)(5,95,78,132)(6,84,79,121)(7,113,80,150)(8,102,41,139)(9,91,42,128)(10,120,43,157)(11,109,44,146)(12,98,45,135)(13,87,46,124)(14,116,47,153)(15,105,48,142)(16,94,49,131)(17,83,50,160)(18,112,51,149)(19,101,52,138)(20,90,53,127)(21,119,54,156)(22,108,55,145)(23,97,56,134)(24,86,57,123)(25,115,58,152)(26,104,59,141)(27,93,60,130)(28,82,61,159)(29,111,62,148)(30,100,63,137)(31,89,64,126)(32,118,65,155)(33,107,66,144)(34,96,67,133)(35,85,68,122)(36,114,69,151)(37,103,70,140)(38,92,71,129)(39,81,72,158)(40,110,73,147), (1,151,21,131)(2,150,22,130)(3,149,23,129)(4,148,24,128)(5,147,25,127)(6,146,26,126)(7,145,27,125)(8,144,28,124)(9,143,29,123)(10,142,30,122)(11,141,31,121)(12,140,32,160)(13,139,33,159)(14,138,34,158)(15,137,35,157)(16,136,36,156)(17,135,37,155)(18,134,38,154)(19,133,39,153)(20,132,40,152)(41,107,61,87)(42,106,62,86)(43,105,63,85)(44,104,64,84)(45,103,65,83)(46,102,66,82)(47,101,67,81)(48,100,68,120)(49,99,69,119)(50,98,70,118)(51,97,71,117)(52,96,72,116)(53,95,73,115)(54,94,74,114)(55,93,75,113)(56,92,76,112)(57,91,77,111)(58,90,78,110)(59,89,79,109)(60,88,80,108)>;
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), (1,99,74,136)(2,88,75,125)(3,117,76,154)(4,106,77,143)(5,95,78,132)(6,84,79,121)(7,113,80,150)(8,102,41,139)(9,91,42,128)(10,120,43,157)(11,109,44,146)(12,98,45,135)(13,87,46,124)(14,116,47,153)(15,105,48,142)(16,94,49,131)(17,83,50,160)(18,112,51,149)(19,101,52,138)(20,90,53,127)(21,119,54,156)(22,108,55,145)(23,97,56,134)(24,86,57,123)(25,115,58,152)(26,104,59,141)(27,93,60,130)(28,82,61,159)(29,111,62,148)(30,100,63,137)(31,89,64,126)(32,118,65,155)(33,107,66,144)(34,96,67,133)(35,85,68,122)(36,114,69,151)(37,103,70,140)(38,92,71,129)(39,81,72,158)(40,110,73,147), (1,151,21,131)(2,150,22,130)(3,149,23,129)(4,148,24,128)(5,147,25,127)(6,146,26,126)(7,145,27,125)(8,144,28,124)(9,143,29,123)(10,142,30,122)(11,141,31,121)(12,140,32,160)(13,139,33,159)(14,138,34,158)(15,137,35,157)(16,136,36,156)(17,135,37,155)(18,134,38,154)(19,133,39,153)(20,132,40,152)(41,107,61,87)(42,106,62,86)(43,105,63,85)(44,104,64,84)(45,103,65,83)(46,102,66,82)(47,101,67,81)(48,100,68,120)(49,99,69,119)(50,98,70,118)(51,97,71,117)(52,96,72,116)(53,95,73,115)(54,94,74,114)(55,93,75,113)(56,92,76,112)(57,91,77,111)(58,90,78,110)(59,89,79,109)(60,88,80,108) );
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)], [(1,99,74,136),(2,88,75,125),(3,117,76,154),(4,106,77,143),(5,95,78,132),(6,84,79,121),(7,113,80,150),(8,102,41,139),(9,91,42,128),(10,120,43,157),(11,109,44,146),(12,98,45,135),(13,87,46,124),(14,116,47,153),(15,105,48,142),(16,94,49,131),(17,83,50,160),(18,112,51,149),(19,101,52,138),(20,90,53,127),(21,119,54,156),(22,108,55,145),(23,97,56,134),(24,86,57,123),(25,115,58,152),(26,104,59,141),(27,93,60,130),(28,82,61,159),(29,111,62,148),(30,100,63,137),(31,89,64,126),(32,118,65,155),(33,107,66,144),(34,96,67,133),(35,85,68,122),(36,114,69,151),(37,103,70,140),(38,92,71,129),(39,81,72,158),(40,110,73,147)], [(1,151,21,131),(2,150,22,130),(3,149,23,129),(4,148,24,128),(5,147,25,127),(6,146,26,126),(7,145,27,125),(8,144,28,124),(9,143,29,123),(10,142,30,122),(11,141,31,121),(12,140,32,160),(13,139,33,159),(14,138,34,158),(15,137,35,157),(16,136,36,156),(17,135,37,155),(18,134,38,154),(19,133,39,153),(20,132,40,152),(41,107,61,87),(42,106,62,86),(43,105,63,85),(44,104,64,84),(45,103,65,83),(46,102,66,82),(47,101,67,81),(48,100,68,120),(49,99,69,119),(50,98,70,118),(51,97,71,117),(52,96,72,116),(53,95,73,115),(54,94,74,114),(55,93,75,113),(56,92,76,112),(57,91,77,111),(58,90,78,110),(59,89,79,109),(60,88,80,108)]])
44 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 | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| 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 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 8 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | C8.C22 | D4×D5 | D4×D5 | SD16⋊D5 |
| kernel | C40.31D4 | C40⋊8C4 | C2×Dic20 | C2×D4.D5 | C20.17D4 | C2×C5⋊Q16 | Dic5⋊Q8 | C10×SD16 | C5⋊2C8 | C40 | C2×Dic5 | C2×SD16 | C2×C8 | C2×D4 | C2×Q8 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40.31D4 ►in GL6(𝔽41)
| 16 | 9 | 0 | 0 | 0 | 0 |
| 17 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 5 | 0 | 36 | 36 |
| 0 | 0 | 26 | 33 | 23 | 8 |
| 0 | 0 | 0 | 0 | 28 | 8 |
| 16 | 9 | 0 | 0 | 0 | 0 |
| 17 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 17 | 14 | 0 | 0 |
| 0 | 0 | 29 | 0 | 27 | 12 |
| 0 | 0 | 13 | 27 | 28 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 2 | 33 |
| 0 | 0 | 10 | 0 | 21 | 31 |
| 0 | 0 | 25 | 21 | 0 | 0 |
| 0 | 0 | 40 | 36 | 2 | 33 |
G:=sub<GL(6,GF(41))| [16,17,0,0,0,0,9,25,0,0,0,0,0,0,0,5,26,0,0,0,0,0,33,0,0,0,36,36,23,28,0,0,0,36,8,8],[16,17,0,0,0,0,9,25,0,0,0,0,0,0,27,17,29,13,0,0,27,14,0,27,0,0,0,0,27,28,0,0,0,0,12,14],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,8,10,25,40,0,0,0,0,21,36,0,0,2,21,0,2,0,0,33,31,0,33] >;
C40.31D4 in GAP, Magma, Sage, TeX
C_{40}._{31}D_4 % in TeX
G:=Group("C40.31D4"); // GroupNames label
G:=SmallGroup(320,794);
// by ID
G=gap.SmallGroup(320,794);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,232,1094,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=a^29,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations