metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊9C4, C42.17D10, C8⋊2(C4×D5), C40⋊12(C2×C4), (C4×D20)⋊2C2, C8⋊C4⋊2D5, C40⋊6C4⋊2C2, D20⋊19(C2×C4), C5⋊2(D8⋊C4), (C2×D40).7C2, C10.42(C4×D4), C2.15(C4×D20), (C2×C8).54D10, (C2×C4).115D20, (C2×C20).237D4, D20⋊5C4⋊37C2, C2.2(C8⋊D10), C10.3(C8⋊C22), (C2×C40).55C22, (C4×C20).15C22, C22.31(C2×D20), C20.223(C4○D4), C4.107(C4○D20), C20.165(C22×C4), (C2×C20).732C23, (C2×D20).196C22, C4⋊Dic5.266C22, C4.64(C2×C4×D5), (C5×C8⋊C4)⋊3C2, (C2×C10).115(C2×D4), (C2×C4).676(C22×D5), SmallGroup(320,338)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊9C4
G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a21, cbc-1=a20b >
Subgroups: 662 in 132 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C40, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, D8⋊C4, D40, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C40⋊6C4, D20⋊5C4, C5×C8⋊C4, C4×D20, C2×D40, D40⋊9C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8⋊C22, C4×D5, D20, C22×D5, D8⋊C4, C2×C4×D5, C2×D20, C4○D20, C4×D20, C8⋊D10, D40⋊9C4
►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)
(1 159)(2 158)(3 157)(4 156)(5 155)(6 154)(7 153)(8 152)(9 151)(10 150)(11 149)(12 148)(13 147)(14 146)(15 145)(16 144)(17 143)(18 142)(19 141)(20 140)(21 139)(22 138)(23 137)(24 136)(25 135)(26 134)(27 133)(28 132)(29 131)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 123)(38 122)(39 121)(40 160)(41 83)(42 82)(43 81)(44 120)(45 119)(46 118)(47 117)(48 116)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 108)(57 107)(58 106)(59 105)(60 104)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)
(1 51 145 119)(2 72 146 100)(3 53 147 81)(4 74 148 102)(5 55 149 83)(6 76 150 104)(7 57 151 85)(8 78 152 106)(9 59 153 87)(10 80 154 108)(11 61 155 89)(12 42 156 110)(13 63 157 91)(14 44 158 112)(15 65 159 93)(16 46 160 114)(17 67 121 95)(18 48 122 116)(19 69 123 97)(20 50 124 118)(21 71 125 99)(22 52 126 120)(23 73 127 101)(24 54 128 82)(25 75 129 103)(26 56 130 84)(27 77 131 105)(28 58 132 86)(29 79 133 107)(30 60 134 88)(31 41 135 109)(32 62 136 90)(33 43 137 111)(34 64 138 92)(35 45 139 113)(36 66 140 94)(37 47 141 115)(38 68 142 96)(39 49 143 117)(40 70 144 98)
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), (1,159)(2,158)(3,157)(4,156)(5,155)(6,154)(7,153)(8,152)(9,151)(10,150)(11,149)(12,148)(13,147)(14,146)(15,145)(16,144)(17,143)(18,142)(19,141)(20,140)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,160)(41,83)(42,82)(43,81)(44,120)(45,119)(46,118)(47,117)(48,116)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,108)(57,107)(58,106)(59,105)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84), (1,51,145,119)(2,72,146,100)(3,53,147,81)(4,74,148,102)(5,55,149,83)(6,76,150,104)(7,57,151,85)(8,78,152,106)(9,59,153,87)(10,80,154,108)(11,61,155,89)(12,42,156,110)(13,63,157,91)(14,44,158,112)(15,65,159,93)(16,46,160,114)(17,67,121,95)(18,48,122,116)(19,69,123,97)(20,50,124,118)(21,71,125,99)(22,52,126,120)(23,73,127,101)(24,54,128,82)(25,75,129,103)(26,56,130,84)(27,77,131,105)(28,58,132,86)(29,79,133,107)(30,60,134,88)(31,41,135,109)(32,62,136,90)(33,43,137,111)(34,64,138,92)(35,45,139,113)(36,66,140,94)(37,47,141,115)(38,68,142,96)(39,49,143,117)(40,70,144,98)>;
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), (1,159)(2,158)(3,157)(4,156)(5,155)(6,154)(7,153)(8,152)(9,151)(10,150)(11,149)(12,148)(13,147)(14,146)(15,145)(16,144)(17,143)(18,142)(19,141)(20,140)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,160)(41,83)(42,82)(43,81)(44,120)(45,119)(46,118)(47,117)(48,116)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,108)(57,107)(58,106)(59,105)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84), (1,51,145,119)(2,72,146,100)(3,53,147,81)(4,74,148,102)(5,55,149,83)(6,76,150,104)(7,57,151,85)(8,78,152,106)(9,59,153,87)(10,80,154,108)(11,61,155,89)(12,42,156,110)(13,63,157,91)(14,44,158,112)(15,65,159,93)(16,46,160,114)(17,67,121,95)(18,48,122,116)(19,69,123,97)(20,50,124,118)(21,71,125,99)(22,52,126,120)(23,73,127,101)(24,54,128,82)(25,75,129,103)(26,56,130,84)(27,77,131,105)(28,58,132,86)(29,79,133,107)(30,60,134,88)(31,41,135,109)(32,62,136,90)(33,43,137,111)(34,64,138,92)(35,45,139,113)(36,66,140,94)(37,47,141,115)(38,68,142,96)(39,49,143,117)(40,70,144,98) );
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)], [(1,159),(2,158),(3,157),(4,156),(5,155),(6,154),(7,153),(8,152),(9,151),(10,150),(11,149),(12,148),(13,147),(14,146),(15,145),(16,144),(17,143),(18,142),(19,141),(20,140),(21,139),(22,138),(23,137),(24,136),(25,135),(26,134),(27,133),(28,132),(29,131),(30,130),(31,129),(32,128),(33,127),(34,126),(35,125),(36,124),(37,123),(38,122),(39,121),(40,160),(41,83),(42,82),(43,81),(44,120),(45,119),(46,118),(47,117),(48,116),(49,115),(50,114),(51,113),(52,112),(53,111),(54,110),(55,109),(56,108),(57,107),(58,106),(59,105),(60,104),(61,103),(62,102),(63,101),(64,100),(65,99),(66,98),(67,97),(68,96),(69,95),(70,94),(71,93),(72,92),(73,91),(74,90),(75,89),(76,88),(77,87),(78,86),(79,85),(80,84)], [(1,51,145,119),(2,72,146,100),(3,53,147,81),(4,74,148,102),(5,55,149,83),(6,76,150,104),(7,57,151,85),(8,78,152,106),(9,59,153,87),(10,80,154,108),(11,61,155,89),(12,42,156,110),(13,63,157,91),(14,44,158,112),(15,65,159,93),(16,46,160,114),(17,67,121,95),(18,48,122,116),(19,69,123,97),(20,50,124,118),(21,71,125,99),(22,52,126,120),(23,73,127,101),(24,54,128,82),(25,75,129,103),(26,56,130,84),(27,77,131,105),(28,58,132,86),(29,79,133,107),(30,60,134,88),(31,41,135,109),(32,62,136,90),(33,43,137,111),(34,64,138,92),(35,45,139,113),(36,66,140,94),(37,47,141,115),(38,68,142,96),(39,49,143,117),(40,70,144,98)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D20 | C4○D20 | C8⋊C22 | C8⋊D10 |
| kernel | D40⋊9C4 | C40⋊6C4 | D20⋊5C4 | C5×C8⋊C4 | C4×D20 | C2×D40 | D40 | C2×C20 | C8⋊C4 | C20 | C42 | C2×C8 | C8 | C2×C4 | C4 | C10 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 2 | 8 |
Matrix representation of D40⋊9C4 ►in GL6(𝔽41)
| 39 | 1 | 0 | 0 | 0 | 0 |
| 36 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 17 | 15 | 1 |
| 0 | 0 | 15 | 9 | 38 | 14 |
| 0 | 0 | 2 | 14 | 2 | 4 |
| 0 | 0 | 3 | 21 | 39 | 23 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 14 | 0 | 0 |
| 0 | 0 | 30 | 14 | 0 | 0 |
| 0 | 0 | 6 | 15 | 16 | 30 |
| 0 | 0 | 1 | 15 | 12 | 25 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 20 | 3 | 3 |
| 0 | 0 | 2 | 7 | 38 | 0 |
| 0 | 0 | 25 | 2 | 36 | 2 |
| 0 | 0 | 38 | 35 | 15 | 29 |
G:=sub<GL(6,GF(41))| [39,36,0,0,0,0,1,2,0,0,0,0,0,0,7,15,2,3,0,0,17,9,14,21,0,0,15,38,2,39,0,0,1,14,4,23],[1,4,0,0,0,0,0,40,0,0,0,0,0,0,27,30,6,1,0,0,14,14,15,15,0,0,0,0,16,12,0,0,0,0,30,25],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,10,2,25,38,0,0,20,7,2,35,0,0,3,38,36,15,0,0,3,0,2,29] >;
D40⋊9C4 in GAP, Magma, Sage, TeX
D_{40}\rtimes_9C_4 % in TeX
G:=Group("D40:9C4"); // GroupNames label
G:=SmallGroup(320,338);
// by ID
G=gap.SmallGroup(320,338);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,387,58,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^20*b>;
// generators/relations