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 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C8, C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×8], C2×C10, C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4 [×2], C2×D8, C40 [×2], C40, C4×D5 [×4], D20 [×4], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], D8⋊C4, D40 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20 [×2], C40⋊6C4, D20⋊5C4 [×2], C5×C8⋊C4, C4×D20 [×2], C2×D40, D40⋊9C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22 [×2], C4×D5 [×2], D20 [×2], C22×D5, D8⋊C4, C2×C4×D5, C2×D20, C4○D20, C4×D20, C8⋊D10 [×2], 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)
(1 138)(2 137)(3 136)(4 135)(5 134)(6 133)(7 132)(8 131)(9 130)(10 129)(11 128)(12 127)(13 126)(14 125)(15 124)(16 123)(17 122)(18 121)(19 160)(20 159)(21 158)(22 157)(23 156)(24 155)(25 154)(26 153)(27 152)(28 151)(29 150)(30 149)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 141)(39 140)(40 139)(41 84)(42 83)(43 82)(44 81)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 105)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)
(1 119 124 52)(2 100 125 73)(3 81 126 54)(4 102 127 75)(5 83 128 56)(6 104 129 77)(7 85 130 58)(8 106 131 79)(9 87 132 60)(10 108 133 41)(11 89 134 62)(12 110 135 43)(13 91 136 64)(14 112 137 45)(15 93 138 66)(16 114 139 47)(17 95 140 68)(18 116 141 49)(19 97 142 70)(20 118 143 51)(21 99 144 72)(22 120 145 53)(23 101 146 74)(24 82 147 55)(25 103 148 76)(26 84 149 57)(27 105 150 78)(28 86 151 59)(29 107 152 80)(30 88 153 61)(31 109 154 42)(32 90 155 63)(33 111 156 44)(34 92 157 65)(35 113 158 46)(36 94 159 67)(37 115 160 48)(38 96 121 69)(39 117 122 50)(40 98 123 71)
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), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,130)(10,129)(11,128)(12,127)(13,126)(14,125)(15,124)(16,123)(17,122)(18,121)(19,160)(20,159)(21,158)(22,157)(23,156)(24,155)(25,154)(26,153)(27,152)(28,151)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,140)(40,139)(41,84)(42,83)(43,82)(44,81)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85), (1,119,124,52)(2,100,125,73)(3,81,126,54)(4,102,127,75)(5,83,128,56)(6,104,129,77)(7,85,130,58)(8,106,131,79)(9,87,132,60)(10,108,133,41)(11,89,134,62)(12,110,135,43)(13,91,136,64)(14,112,137,45)(15,93,138,66)(16,114,139,47)(17,95,140,68)(18,116,141,49)(19,97,142,70)(20,118,143,51)(21,99,144,72)(22,120,145,53)(23,101,146,74)(24,82,147,55)(25,103,148,76)(26,84,149,57)(27,105,150,78)(28,86,151,59)(29,107,152,80)(30,88,153,61)(31,109,154,42)(32,90,155,63)(33,111,156,44)(34,92,157,65)(35,113,158,46)(36,94,159,67)(37,115,160,48)(38,96,121,69)(39,117,122,50)(40,98,123,71)>;
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), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,130)(10,129)(11,128)(12,127)(13,126)(14,125)(15,124)(16,123)(17,122)(18,121)(19,160)(20,159)(21,158)(22,157)(23,156)(24,155)(25,154)(26,153)(27,152)(28,151)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,140)(40,139)(41,84)(42,83)(43,82)(44,81)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85), (1,119,124,52)(2,100,125,73)(3,81,126,54)(4,102,127,75)(5,83,128,56)(6,104,129,77)(7,85,130,58)(8,106,131,79)(9,87,132,60)(10,108,133,41)(11,89,134,62)(12,110,135,43)(13,91,136,64)(14,112,137,45)(15,93,138,66)(16,114,139,47)(17,95,140,68)(18,116,141,49)(19,97,142,70)(20,118,143,51)(21,99,144,72)(22,120,145,53)(23,101,146,74)(24,82,147,55)(25,103,148,76)(26,84,149,57)(27,105,150,78)(28,86,151,59)(29,107,152,80)(30,88,153,61)(31,109,154,42)(32,90,155,63)(33,111,156,44)(34,92,157,65)(35,113,158,46)(36,94,159,67)(37,115,160,48)(38,96,121,69)(39,117,122,50)(40,98,123,71) );
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)], [(1,138),(2,137),(3,136),(4,135),(5,134),(6,133),(7,132),(8,131),(9,130),(10,129),(11,128),(12,127),(13,126),(14,125),(15,124),(16,123),(17,122),(18,121),(19,160),(20,159),(21,158),(22,157),(23,156),(24,155),(25,154),(26,153),(27,152),(28,151),(29,150),(30,149),(31,148),(32,147),(33,146),(34,145),(35,144),(36,143),(37,142),(38,141),(39,140),(40,139),(41,84),(42,83),(43,82),(44,81),(45,120),(46,119),(47,118),(48,117),(49,116),(50,115),(51,114),(52,113),(53,112),(54,111),(55,110),(56,109),(57,108),(58,107),(59,106),(60,105),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,98),(68,97),(69,96),(70,95),(71,94),(72,93),(73,92),(74,91),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85)], [(1,119,124,52),(2,100,125,73),(3,81,126,54),(4,102,127,75),(5,83,128,56),(6,104,129,77),(7,85,130,58),(8,106,131,79),(9,87,132,60),(10,108,133,41),(11,89,134,62),(12,110,135,43),(13,91,136,64),(14,112,137,45),(15,93,138,66),(16,114,139,47),(17,95,140,68),(18,116,141,49),(19,97,142,70),(20,118,143,51),(21,99,144,72),(22,120,145,53),(23,101,146,74),(24,82,147,55),(25,103,148,76),(26,84,149,57),(27,105,150,78),(28,86,151,59),(29,107,152,80),(30,88,153,61),(31,109,154,42),(32,90,155,63),(33,111,156,44),(34,92,157,65),(35,113,158,46),(36,94,159,67),(37,115,160,48),(38,96,121,69),(39,117,122,50),(40,98,123,71)])
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