metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊5D4, Dic5⋊1D8, (C2×D8)⋊3D5, (C10×D8)⋊5C2, C5⋊2C8⋊14D4, C8⋊7(C5⋊D4), C5⋊3(C8⋊4D4), C2.27(D5×D8), C4.20(D4×D5), (C2×D40)⋊18C2, C20⋊D4⋊4C2, (C8×Dic5)⋊5C2, C20.45(C2×D4), C10.44(C2×D8), (C2×D4).60D10, (C2×C8).236D10, (C2×C40).88C22, C22.252(D4×D5), C10.26(C4⋊1D4), C2.17(C20⋊D4), (C2×C20).428C23, (C2×Dic5).155D4, (D4×C10).78C22, (C2×D20).118C22, (C4×Dic5).269C22, C4.4(C2×C5⋊D4), (C2×D4⋊D5)⋊17C2, (C2×C10).341(C2×D4), (C2×C4).518(C22×D5), (C2×C5⋊2C8).277C22, SmallGroup(320,778)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊5D4
G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a-1, cbc=b-1 >
Subgroups: 878 in 162 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×12], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×16], C23 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C42, C2×C8, C2×C8, D8 [×8], C2×D4 [×2], C2×D4 [×6], Dic5 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×6], C4×C8, C4⋊1D4 [×2], C2×D8, C2×D8 [×3], C5⋊2C8 [×2], C40 [×2], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5 [×2], C22×C10 [×2], C8⋊4D4, D40 [×2], C2×C5⋊2C8, C4×Dic5, D4⋊D5 [×4], C2×C40, C5×D8 [×2], C2×D20 [×2], C2×C5⋊D4 [×4], D4×C10 [×2], C8×Dic5, C2×D40, C2×D4⋊D5 [×2], C20⋊D4 [×2], C10×D8, C40⋊5D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, D8 [×4], C2×D4 [×3], D10 [×3], C4⋊1D4, C2×D8 [×2], C5⋊D4 [×2], C22×D5, C8⋊4D4, D4×D5 [×2], C2×C5⋊D4, D5×D8 [×2], C20⋊D4, C40⋊5D4
►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)
(1 115 44 124)(2 84 45 133)(3 93 46 142)(4 102 47 151)(5 111 48 160)(6 120 49 129)(7 89 50 138)(8 98 51 147)(9 107 52 156)(10 116 53 125)(11 85 54 134)(12 94 55 143)(13 103 56 152)(14 112 57 121)(15 81 58 130)(16 90 59 139)(17 99 60 148)(18 108 61 157)(19 117 62 126)(20 86 63 135)(21 95 64 144)(22 104 65 153)(23 113 66 122)(24 82 67 131)(25 91 68 140)(26 100 69 149)(27 109 70 158)(28 118 71 127)(29 87 72 136)(30 96 73 145)(31 105 74 154)(32 114 75 123)(33 83 76 132)(34 92 77 141)(35 101 78 150)(36 110 79 159)(37 119 80 128)(38 88 41 137)(39 97 42 146)(40 106 43 155)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 47)(42 46)(43 45)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(81 158)(82 157)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 144)(96 143)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)(109 130)(110 129)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 160)(120 159)
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), (1,115,44,124)(2,84,45,133)(3,93,46,142)(4,102,47,151)(5,111,48,160)(6,120,49,129)(7,89,50,138)(8,98,51,147)(9,107,52,156)(10,116,53,125)(11,85,54,134)(12,94,55,143)(13,103,56,152)(14,112,57,121)(15,81,58,130)(16,90,59,139)(17,99,60,148)(18,108,61,157)(19,117,62,126)(20,86,63,135)(21,95,64,144)(22,104,65,153)(23,113,66,122)(24,82,67,131)(25,91,68,140)(26,100,69,149)(27,109,70,158)(28,118,71,127)(29,87,72,136)(30,96,73,145)(31,105,74,154)(32,114,75,123)(33,83,76,132)(34,92,77,141)(35,101,78,150)(36,110,79,159)(37,119,80,128)(38,88,41,137)(39,97,42,146)(40,106,43,155), (2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,47)(42,46)(43,45)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(81,158)(82,157)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)(109,130)(110,129)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,160)(120,159)>;
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), (1,115,44,124)(2,84,45,133)(3,93,46,142)(4,102,47,151)(5,111,48,160)(6,120,49,129)(7,89,50,138)(8,98,51,147)(9,107,52,156)(10,116,53,125)(11,85,54,134)(12,94,55,143)(13,103,56,152)(14,112,57,121)(15,81,58,130)(16,90,59,139)(17,99,60,148)(18,108,61,157)(19,117,62,126)(20,86,63,135)(21,95,64,144)(22,104,65,153)(23,113,66,122)(24,82,67,131)(25,91,68,140)(26,100,69,149)(27,109,70,158)(28,118,71,127)(29,87,72,136)(30,96,73,145)(31,105,74,154)(32,114,75,123)(33,83,76,132)(34,92,77,141)(35,101,78,150)(36,110,79,159)(37,119,80,128)(38,88,41,137)(39,97,42,146)(40,106,43,155), (2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,47)(42,46)(43,45)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(81,158)(82,157)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)(109,130)(110,129)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,160)(120,159) );
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)], [(1,115,44,124),(2,84,45,133),(3,93,46,142),(4,102,47,151),(5,111,48,160),(6,120,49,129),(7,89,50,138),(8,98,51,147),(9,107,52,156),(10,116,53,125),(11,85,54,134),(12,94,55,143),(13,103,56,152),(14,112,57,121),(15,81,58,130),(16,90,59,139),(17,99,60,148),(18,108,61,157),(19,117,62,126),(20,86,63,135),(21,95,64,144),(22,104,65,153),(23,113,66,122),(24,82,67,131),(25,91,68,140),(26,100,69,149),(27,109,70,158),(28,118,71,127),(29,87,72,136),(30,96,73,145),(31,105,74,154),(32,114,75,123),(33,83,76,132),(34,92,77,141),(35,101,78,150),(36,110,79,159),(37,119,80,128),(38,88,41,137),(39,97,42,146),(40,106,43,155)], [(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(41,47),(42,46),(43,45),(48,80),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(81,158),(82,157),(83,156),(84,155),(85,154),(86,153),(87,152),(88,151),(89,150),(90,149),(91,148),(92,147),(93,146),(94,145),(95,144),(96,143),(97,142),(98,141),(99,140),(100,139),(101,138),(102,137),(103,136),(104,135),(105,134),(106,133),(107,132),(108,131),(109,130),(110,129),(111,128),(112,127),(113,126),(114,125),(115,124),(116,123),(117,122),(118,121),(119,160),(120,159)])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 2 | 2 | 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 | 40 | 40 | 2 | 2 | 10 | 10 | 10 | 10 | 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 | D8 | D10 | D10 | C5⋊D4 | D4×D5 | D4×D5 | D5×D8 |
| kernel | C40⋊5D4 | C8×Dic5 | C2×D40 | C2×D4⋊D5 | C20⋊D4 | C10×D8 | C5⋊2C8 | C40 | C2×Dic5 | C2×D8 | Dic5 | C2×C8 | C2×D4 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 8 | 2 | 2 | 8 |
Matrix representation of C40⋊5D4 ►in GL4(𝔽41) generated by
| 0 | 35 | 0 | 0 |
| 7 | 6 | 0 | 0 |
| 0 | 0 | 17 | 15 |
| 0 | 0 | 30 | 0 |
| 21 | 15 | 0 | 0 |
| 17 | 20 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 6 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 18 | 40 |
G:=sub<GL(4,GF(41))| [0,7,0,0,35,6,0,0,0,0,17,30,0,0,15,0],[21,17,0,0,15,20,0,0,0,0,1,0,0,0,0,1],[6,1,0,0,6,35,0,0,0,0,1,18,0,0,0,40] >;
C40⋊5D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_5D_4 % in TeX
G:=Group("C40:5D4"); // GroupNames label
G:=SmallGroup(320,778);
// by ID
G=gap.SmallGroup(320,778);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,422,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations