metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.18D4, C8.20D20, D20.20D4, Dic10.20D4, M4(2).9D10, (C2×C8).70D10, C4.56(C2×D20), C8.C4⋊5D5, C4.135(D4×D5), C20.136(C2×D4), C5⋊2(D4.5D4), (C2×Dic20)⋊21C2, C20.47D4⋊3C2, C8.D10.2C2, C2.22(C4⋊2D20), C10.49(C4⋊D4), (C2×C40).102C22, (C2×C20).312C23, 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 C40.18D4
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 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4 [×2], Q8 [×5], D5, C10, C10, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], SD16 [×2], Q16 [×4], C2×Q8 [×2], C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22 [×2], C5⋊2C8, C40 [×2], C40 [×2], Dic10, Dic10 [×4], C4×D5, D20, C2×Dic5 [×2], C5⋊D4, C2×C20, D4.5D4, C8×D5, C8⋊D5, C40⋊C2 [×2], Dic20 [×4], C4.Dic5, C2×C40, C5×M4(2) [×2], C2×Dic10 [×2], C4○D20, C20.47D4 [×2], C5×C8.C4, D20.3C4, C2×Dic20, C8.D10 [×2], C40.18D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C22×D5, D4.5D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊2D20, C40.18D4
►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)
(1 153 90 57 21 133 110 77)(2 144 91 48 22 124 111 68)(3 135 92 79 23 155 112 59)(4 126 93 70 24 146 113 50)(5 157 94 61 25 137 114 41)(6 148 95 52 26 128 115 72)(7 139 96 43 27 159 116 63)(8 130 97 74 28 150 117 54)(9 121 98 65 29 141 118 45)(10 152 99 56 30 132 119 76)(11 143 100 47 31 123 120 67)(12 134 101 78 32 154 81 58)(13 125 102 69 33 145 82 49)(14 156 103 60 34 136 83 80)(15 147 104 51 35 127 84 71)(16 138 105 42 36 158 85 62)(17 129 106 73 37 149 86 53)(18 160 107 64 38 140 87 44)(19 151 108 55 39 131 88 75)(20 142 109 46 40 122 89 66)
(1 57 21 77)(2 56 22 76)(3 55 23 75)(4 54 24 74)(5 53 25 73)(6 52 26 72)(7 51 27 71)(8 50 28 70)(9 49 29 69)(10 48 30 68)(11 47 31 67)(12 46 32 66)(13 45 33 65)(14 44 34 64)(15 43 35 63)(16 42 36 62)(17 41 37 61)(18 80 38 60)(19 79 39 59)(20 78 40 58)(81 122 101 142)(82 121 102 141)(83 160 103 140)(84 159 104 139)(85 158 105 138)(86 157 106 137)(87 156 107 136)(88 155 108 135)(89 154 109 134)(90 153 110 133)(91 152 111 132)(92 151 112 131)(93 150 113 130)(94 149 114 129)(95 148 115 128)(96 147 116 127)(97 146 117 126)(98 145 118 125)(99 144 119 124)(100 143 120 123)
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), (1,153,90,57,21,133,110,77)(2,144,91,48,22,124,111,68)(3,135,92,79,23,155,112,59)(4,126,93,70,24,146,113,50)(5,157,94,61,25,137,114,41)(6,148,95,52,26,128,115,72)(7,139,96,43,27,159,116,63)(8,130,97,74,28,150,117,54)(9,121,98,65,29,141,118,45)(10,152,99,56,30,132,119,76)(11,143,100,47,31,123,120,67)(12,134,101,78,32,154,81,58)(13,125,102,69,33,145,82,49)(14,156,103,60,34,136,83,80)(15,147,104,51,35,127,84,71)(16,138,105,42,36,158,85,62)(17,129,106,73,37,149,86,53)(18,160,107,64,38,140,87,44)(19,151,108,55,39,131,88,75)(20,142,109,46,40,122,89,66), (1,57,21,77)(2,56,22,76)(3,55,23,75)(4,54,24,74)(5,53,25,73)(6,52,26,72)(7,51,27,71)(8,50,28,70)(9,49,29,69)(10,48,30,68)(11,47,31,67)(12,46,32,66)(13,45,33,65)(14,44,34,64)(15,43,35,63)(16,42,36,62)(17,41,37,61)(18,80,38,60)(19,79,39,59)(20,78,40,58)(81,122,101,142)(82,121,102,141)(83,160,103,140)(84,159,104,139)(85,158,105,138)(86,157,106,137)(87,156,107,136)(88,155,108,135)(89,154,109,134)(90,153,110,133)(91,152,111,132)(92,151,112,131)(93,150,113,130)(94,149,114,129)(95,148,115,128)(96,147,116,127)(97,146,117,126)(98,145,118,125)(99,144,119,124)(100,143,120,123)>;
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), (1,153,90,57,21,133,110,77)(2,144,91,48,22,124,111,68)(3,135,92,79,23,155,112,59)(4,126,93,70,24,146,113,50)(5,157,94,61,25,137,114,41)(6,148,95,52,26,128,115,72)(7,139,96,43,27,159,116,63)(8,130,97,74,28,150,117,54)(9,121,98,65,29,141,118,45)(10,152,99,56,30,132,119,76)(11,143,100,47,31,123,120,67)(12,134,101,78,32,154,81,58)(13,125,102,69,33,145,82,49)(14,156,103,60,34,136,83,80)(15,147,104,51,35,127,84,71)(16,138,105,42,36,158,85,62)(17,129,106,73,37,149,86,53)(18,160,107,64,38,140,87,44)(19,151,108,55,39,131,88,75)(20,142,109,46,40,122,89,66), (1,57,21,77)(2,56,22,76)(3,55,23,75)(4,54,24,74)(5,53,25,73)(6,52,26,72)(7,51,27,71)(8,50,28,70)(9,49,29,69)(10,48,30,68)(11,47,31,67)(12,46,32,66)(13,45,33,65)(14,44,34,64)(15,43,35,63)(16,42,36,62)(17,41,37,61)(18,80,38,60)(19,79,39,59)(20,78,40,58)(81,122,101,142)(82,121,102,141)(83,160,103,140)(84,159,104,139)(85,158,105,138)(86,157,106,137)(87,156,107,136)(88,155,108,135)(89,154,109,134)(90,153,110,133)(91,152,111,132)(92,151,112,131)(93,150,113,130)(94,149,114,129)(95,148,115,128)(96,147,116,127)(97,146,117,126)(98,145,118,125)(99,144,119,124)(100,143,120,123) );
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)], [(1,153,90,57,21,133,110,77),(2,144,91,48,22,124,111,68),(3,135,92,79,23,155,112,59),(4,126,93,70,24,146,113,50),(5,157,94,61,25,137,114,41),(6,148,95,52,26,128,115,72),(7,139,96,43,27,159,116,63),(8,130,97,74,28,150,117,54),(9,121,98,65,29,141,118,45),(10,152,99,56,30,132,119,76),(11,143,100,47,31,123,120,67),(12,134,101,78,32,154,81,58),(13,125,102,69,33,145,82,49),(14,156,103,60,34,136,83,80),(15,147,104,51,35,127,84,71),(16,138,105,42,36,158,85,62),(17,129,106,73,37,149,86,53),(18,160,107,64,38,140,87,44),(19,151,108,55,39,131,88,75),(20,142,109,46,40,122,89,66)], [(1,57,21,77),(2,56,22,76),(3,55,23,75),(4,54,24,74),(5,53,25,73),(6,52,26,72),(7,51,27,71),(8,50,28,70),(9,49,29,69),(10,48,30,68),(11,47,31,67),(12,46,32,66),(13,45,33,65),(14,44,34,64),(15,43,35,63),(16,42,36,62),(17,41,37,61),(18,80,38,60),(19,79,39,59),(20,78,40,58),(81,122,101,142),(82,121,102,141),(83,160,103,140),(84,159,104,139),(85,158,105,138),(86,157,106,137),(87,156,107,136),(88,155,108,135),(89,154,109,134),(90,153,110,133),(91,152,111,132),(92,151,112,131),(93,150,113,130),(94,149,114,129),(95,148,115,128),(96,147,116,127),(97,146,117,126),(98,145,118,125),(99,144,119,124),(100,143,120,123)])
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 | C40.18D4 |
| kernel | C40.18D4 | C20.47D4 | 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 C40.18D4 ►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] >;
C40.18D4 in GAP, Magma, Sage, TeX
C_{40}._{18}D_4 % in TeX
G:=Group("C40.18D4"); // 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