metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊17D4, D10⋊6M4(2), C8⋊12(C5⋊D4), C5⋊11(C8⋊9D4), C40⋊8C4⋊28C2, (C2×C8).188D10, C10.109(C4×D4), C20.442(C2×D4), (C2×M4(2))⋊7D5, D10⋊1C8⋊38C2, C23.19(C4×D5), C10.56(C8○D4), (C10×M4(2))⋊6C2, C20.8Q8⋊40C2, C2.21(D5×M4(2)), C23.D5.19C4, D10⋊C4.26C4, C20.253(C4○D4), C4.137(C4○D20), C20.55D4⋊29C2, (C2×C40).234C22, (C2×C20).867C23, C10.D4.26C4, (C22×C4).136D10, C10.66(C2×M4(2)), C2.17(D20.2C4), (C22×C20).374C22, (C4×Dic5).209C22, (D5×C2×C8)⋊27C2, (C2×C4).50(C4×D5), C2.24(C4×C5⋊D4), (C4×C5⋊D4).16C2, (C2×C5⋊D4).19C4, C4.133(C2×C5⋊D4), C22.146(C2×C4×D5), (C2×C20).360(C2×C4), (C2×C4×D5).355C22, (C2×Dic5).36(C2×C4), (C22×D5).81(C2×C4), (C2×C4).809(C22×D5), (C22×C10).135(C2×C4), (C2×C10).238(C22×C4), (C2×C5⋊2C8).331C22, SmallGroup(320,754)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊D4
G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a9, cbc=b-1 >
Subgroups: 382 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5 [×2], C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊2C8 [×2], C40 [×2], C40, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C8⋊9D4, C8×D5 [×2], C2×C5⋊2C8 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40 [×2], C5×M4(2) [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, C20.8Q8, C40⋊8C4, D10⋊1C8, C20.55D4, D5×C2×C8, C4×C5⋊D4, C10×M4(2), C40⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C8⋊9D4, C2×C4×D5, C4○D20, C2×C5⋊D4, D5×M4(2), D20.2C4, C4×C5⋊D4, C40⋊D4
►On 160 points
(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)
(1 135 43 103)(2 124 44 92)(3 153 45 81)(4 142 46 110)(5 131 47 99)(6 160 48 88)(7 149 49 117)(8 138 50 106)(9 127 51 95)(10 156 52 84)(11 145 53 113)(12 134 54 102)(13 123 55 91)(14 152 56 120)(15 141 57 109)(16 130 58 98)(17 159 59 87)(18 148 60 116)(19 137 61 105)(20 126 62 94)(21 155 63 83)(22 144 64 112)(23 133 65 101)(24 122 66 90)(25 151 67 119)(26 140 68 108)(27 129 69 97)(28 158 70 86)(29 147 71 115)(30 136 72 104)(31 125 73 93)(32 154 74 82)(33 143 75 111)(34 132 76 100)(35 121 77 89)(36 150 78 118)(37 139 79 107)(38 128 80 96)(39 157 41 85)(40 146 42 114)
(2 10)(3 19)(4 28)(5 37)(7 15)(8 24)(9 33)(12 20)(13 29)(14 38)(17 25)(18 34)(22 30)(23 39)(27 35)(32 40)(41 65)(42 74)(44 52)(45 61)(46 70)(47 79)(49 57)(50 66)(51 75)(54 62)(55 71)(56 80)(59 67)(60 76)(64 72)(69 77)(81 137)(82 146)(83 155)(84 124)(85 133)(86 142)(87 151)(88 160)(89 129)(90 138)(91 147)(92 156)(93 125)(94 134)(95 143)(96 152)(97 121)(98 130)(99 139)(100 148)(101 157)(102 126)(103 135)(104 144)(105 153)(106 122)(107 131)(108 140)(109 149)(110 158)(111 127)(112 136)(113 145)(114 154)(115 123)(116 132)(117 141)(118 150)(119 159)(120 128)
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), (1,135,43,103)(2,124,44,92)(3,153,45,81)(4,142,46,110)(5,131,47,99)(6,160,48,88)(7,149,49,117)(8,138,50,106)(9,127,51,95)(10,156,52,84)(11,145,53,113)(12,134,54,102)(13,123,55,91)(14,152,56,120)(15,141,57,109)(16,130,58,98)(17,159,59,87)(18,148,60,116)(19,137,61,105)(20,126,62,94)(21,155,63,83)(22,144,64,112)(23,133,65,101)(24,122,66,90)(25,151,67,119)(26,140,68,108)(27,129,69,97)(28,158,70,86)(29,147,71,115)(30,136,72,104)(31,125,73,93)(32,154,74,82)(33,143,75,111)(34,132,76,100)(35,121,77,89)(36,150,78,118)(37,139,79,107)(38,128,80,96)(39,157,41,85)(40,146,42,114), (2,10)(3,19)(4,28)(5,37)(7,15)(8,24)(9,33)(12,20)(13,29)(14,38)(17,25)(18,34)(22,30)(23,39)(27,35)(32,40)(41,65)(42,74)(44,52)(45,61)(46,70)(47,79)(49,57)(50,66)(51,75)(54,62)(55,71)(56,80)(59,67)(60,76)(64,72)(69,77)(81,137)(82,146)(83,155)(84,124)(85,133)(86,142)(87,151)(88,160)(89,129)(90,138)(91,147)(92,156)(93,125)(94,134)(95,143)(96,152)(97,121)(98,130)(99,139)(100,148)(101,157)(102,126)(103,135)(104,144)(105,153)(106,122)(107,131)(108,140)(109,149)(110,158)(111,127)(112,136)(113,145)(114,154)(115,123)(116,132)(117,141)(118,150)(119,159)(120,128)>;
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), (1,135,43,103)(2,124,44,92)(3,153,45,81)(4,142,46,110)(5,131,47,99)(6,160,48,88)(7,149,49,117)(8,138,50,106)(9,127,51,95)(10,156,52,84)(11,145,53,113)(12,134,54,102)(13,123,55,91)(14,152,56,120)(15,141,57,109)(16,130,58,98)(17,159,59,87)(18,148,60,116)(19,137,61,105)(20,126,62,94)(21,155,63,83)(22,144,64,112)(23,133,65,101)(24,122,66,90)(25,151,67,119)(26,140,68,108)(27,129,69,97)(28,158,70,86)(29,147,71,115)(30,136,72,104)(31,125,73,93)(32,154,74,82)(33,143,75,111)(34,132,76,100)(35,121,77,89)(36,150,78,118)(37,139,79,107)(38,128,80,96)(39,157,41,85)(40,146,42,114), (2,10)(3,19)(4,28)(5,37)(7,15)(8,24)(9,33)(12,20)(13,29)(14,38)(17,25)(18,34)(22,30)(23,39)(27,35)(32,40)(41,65)(42,74)(44,52)(45,61)(46,70)(47,79)(49,57)(50,66)(51,75)(54,62)(55,71)(56,80)(59,67)(60,76)(64,72)(69,77)(81,137)(82,146)(83,155)(84,124)(85,133)(86,142)(87,151)(88,160)(89,129)(90,138)(91,147)(92,156)(93,125)(94,134)(95,143)(96,152)(97,121)(98,130)(99,139)(100,148)(101,157)(102,126)(103,135)(104,144)(105,153)(106,122)(107,131)(108,140)(109,149)(110,158)(111,127)(112,136)(113,145)(114,154)(115,123)(116,132)(117,141)(118,150)(119,159)(120,128) );
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)], [(1,135,43,103),(2,124,44,92),(3,153,45,81),(4,142,46,110),(5,131,47,99),(6,160,48,88),(7,149,49,117),(8,138,50,106),(9,127,51,95),(10,156,52,84),(11,145,53,113),(12,134,54,102),(13,123,55,91),(14,152,56,120),(15,141,57,109),(16,130,58,98),(17,159,59,87),(18,148,60,116),(19,137,61,105),(20,126,62,94),(21,155,63,83),(22,144,64,112),(23,133,65,101),(24,122,66,90),(25,151,67,119),(26,140,68,108),(27,129,69,97),(28,158,70,86),(29,147,71,115),(30,136,72,104),(31,125,73,93),(32,154,74,82),(33,143,75,111),(34,132,76,100),(35,121,77,89),(36,150,78,118),(37,139,79,107),(38,128,80,96),(39,157,41,85),(40,146,42,114)], [(2,10),(3,19),(4,28),(5,37),(7,15),(8,24),(9,33),(12,20),(13,29),(14,38),(17,25),(18,34),(22,30),(23,39),(27,35),(32,40),(41,65),(42,74),(44,52),(45,61),(46,70),(47,79),(49,57),(50,66),(51,75),(54,62),(55,71),(56,80),(59,67),(60,76),(64,72),(69,77),(81,137),(82,146),(83,155),(84,124),(85,133),(86,142),(87,151),(88,160),(89,129),(90,138),(91,147),(92,156),(93,125),(94,134),(95,143),(96,152),(97,121),(98,130),(99,139),(100,148),(101,157),(102,126),(103,135),(104,144),(105,153),(106,122),(107,131),(108,140),(109,149),(110,158),(111,127),(112,136),(113,145),(114,154),(115,123),(116,132),(117,141),(118,150),(119,159),(120,128)])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D5 | C4○D4 | M4(2) | D10 | D10 | C8○D4 | C5⋊D4 | C4×D5 | C4×D5 | C4○D20 | D5×M4(2) | D20.2C4 |
| kernel | C40⋊D4 | C20.8Q8 | C40⋊8C4 | D10⋊1C8 | C20.55D4 | D5×C2×C8 | C4×C5⋊D4 | C10×M4(2) | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C40 | C2×M4(2) | C20 | D10 | C2×C8 | C22×C4 | C10 | C8 | C2×C4 | C23 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 8 | 4 | 4 |
Matrix representation of C40⋊D4 ►in GL4(𝔽41) generated by
| 0 | 34 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 11 | 8 |
| 0 | 0 | 35 | 30 |
| 20 | 3 | 0 | 0 |
| 3 | 21 | 0 | 0 |
| 0 | 0 | 11 | 21 |
| 0 | 0 | 6 | 30 |
| 6 | 7 | 0 | 0 |
| 36 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [0,6,0,0,34,6,0,0,0,0,11,35,0,0,8,30],[20,3,0,0,3,21,0,0,0,0,11,6,0,0,21,30],[6,36,0,0,7,35,0,0,0,0,1,0,0,0,0,1] >;
C40⋊D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes D_4 % in TeX
G:=Group("C40:D4"); // GroupNames label
G:=SmallGroup(320,754);
// by ID
G=gap.SmallGroup(320,754);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations