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, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C10, C42, C2×C8, C2×C8, D8, C2×D4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×C8, C4⋊1D4, C2×D8, C2×D8, C5⋊2C8, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C8⋊4D4, D40, C2×C5⋊2C8, C4×Dic5, D4⋊D5, C2×C40, C5×D8, C2×D20, C2×C5⋊D4, D4×C10, C8×Dic5, C2×D40, C2×D4⋊D5, C20⋊D4, C10×D8, C40⋊5D4
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C4⋊1D4, C2×D8, C5⋊D4, C22×D5, C8⋊4D4, D4×D5, C2×C5⋊D4, D5×D8, 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 50 108 132)(2 59 109 141)(3 68 110 150)(4 77 111 159)(5 46 112 128)(6 55 113 137)(7 64 114 146)(8 73 115 155)(9 42 116 124)(10 51 117 133)(11 60 118 142)(12 69 119 151)(13 78 120 160)(14 47 81 129)(15 56 82 138)(16 65 83 147)(17 74 84 156)(18 43 85 125)(19 52 86 134)(20 61 87 143)(21 70 88 152)(22 79 89 121)(23 48 90 130)(24 57 91 139)(25 66 92 148)(26 75 93 157)(27 44 94 126)(28 53 95 135)(29 62 96 144)(30 71 97 153)(31 80 98 122)(32 49 99 131)(33 58 100 140)(34 67 101 149)(35 76 102 158)(36 45 103 127)(37 54 104 136)(38 63 105 145)(39 72 106 154)(40 41 107 123)
(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)(41 141)(42 140)(43 139)(44 138)(45 137)(46 136)(47 135)(48 134)(49 133)(50 132)(51 131)(52 130)(53 129)(54 128)(55 127)(56 126)(57 125)(58 124)(59 123)(60 122)(61 121)(62 160)(63 159)(64 158)(65 157)(66 156)(67 155)(68 154)(69 153)(70 152)(71 151)(72 150)(73 149)(74 148)(75 147)(76 146)(77 145)(78 144)(79 143)(80 142)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)
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), (1,50,108,132)(2,59,109,141)(3,68,110,150)(4,77,111,159)(5,46,112,128)(6,55,113,137)(7,64,114,146)(8,73,115,155)(9,42,116,124)(10,51,117,133)(11,60,118,142)(12,69,119,151)(13,78,120,160)(14,47,81,129)(15,56,82,138)(16,65,83,147)(17,74,84,156)(18,43,85,125)(19,52,86,134)(20,61,87,143)(21,70,88,152)(22,79,89,121)(23,48,90,130)(24,57,91,139)(25,66,92,148)(26,75,93,157)(27,44,94,126)(28,53,95,135)(29,62,96,144)(30,71,97,153)(31,80,98,122)(32,49,99,131)(33,58,100,140)(34,67,101,149)(35,76,102,158)(36,45,103,127)(37,54,104,136)(38,63,105,145)(39,72,106,154)(40,41,107,123), (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)(41,141)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,121)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)>;
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), (1,50,108,132)(2,59,109,141)(3,68,110,150)(4,77,111,159)(5,46,112,128)(6,55,113,137)(7,64,114,146)(8,73,115,155)(9,42,116,124)(10,51,117,133)(11,60,118,142)(12,69,119,151)(13,78,120,160)(14,47,81,129)(15,56,82,138)(16,65,83,147)(17,74,84,156)(18,43,85,125)(19,52,86,134)(20,61,87,143)(21,70,88,152)(22,79,89,121)(23,48,90,130)(24,57,91,139)(25,66,92,148)(26,75,93,157)(27,44,94,126)(28,53,95,135)(29,62,96,144)(30,71,97,153)(31,80,98,122)(32,49,99,131)(33,58,100,140)(34,67,101,149)(35,76,102,158)(36,45,103,127)(37,54,104,136)(38,63,105,145)(39,72,106,154)(40,41,107,123), (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)(41,141)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,121)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109) );
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)], [(1,50,108,132),(2,59,109,141),(3,68,110,150),(4,77,111,159),(5,46,112,128),(6,55,113,137),(7,64,114,146),(8,73,115,155),(9,42,116,124),(10,51,117,133),(11,60,118,142),(12,69,119,151),(13,78,120,160),(14,47,81,129),(15,56,82,138),(16,65,83,147),(17,74,84,156),(18,43,85,125),(19,52,86,134),(20,61,87,143),(21,70,88,152),(22,79,89,121),(23,48,90,130),(24,57,91,139),(25,66,92,148),(26,75,93,157),(27,44,94,126),(28,53,95,135),(29,62,96,144),(30,71,97,153),(31,80,98,122),(32,49,99,131),(33,58,100,140),(34,67,101,149),(35,76,102,158),(36,45,103,127),(37,54,104,136),(38,63,105,145),(39,72,106,154),(40,41,107,123)], [(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),(41,141),(42,140),(43,139),(44,138),(45,137),(46,136),(47,135),(48,134),(49,133),(50,132),(51,131),(52,130),(53,129),(54,128),(55,127),(56,126),(57,125),(58,124),(59,123),(60,122),(61,121),(62,160),(63,159),(64,158),(65,157),(66,156),(67,155),(68,154),(69,153),(70,152),(71,151),(72,150),(73,149),(74,148),(75,147),(76,146),(77,145),(78,144),(79,143),(80,142),(81,95),(82,94),(83,93),(84,92),(85,91),(86,90),(87,89),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109)]])
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