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, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊2C8, C40, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊9D4, C8×D5, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C5×M4(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, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C5⋊D4, 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
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)
(1 105 44 139)(2 94 45 128)(3 83 46 157)(4 112 47 146)(5 101 48 135)(6 90 49 124)(7 119 50 153)(8 108 51 142)(9 97 52 131)(10 86 53 160)(11 115 54 149)(12 104 55 138)(13 93 56 127)(14 82 57 156)(15 111 58 145)(16 100 59 134)(17 89 60 123)(18 118 61 152)(19 107 62 141)(20 96 63 130)(21 85 64 159)(22 114 65 148)(23 103 66 137)(24 92 67 126)(25 81 68 155)(26 110 69 144)(27 99 70 133)(28 88 71 122)(29 117 72 151)(30 106 73 140)(31 95 74 129)(32 84 75 158)(33 113 76 147)(34 102 77 136)(35 91 78 125)(36 120 79 154)(37 109 80 143)(38 98 41 132)(39 87 42 121)(40 116 43 150)
(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 57)(42 66)(43 75)(45 53)(46 62)(47 71)(48 80)(50 58)(51 67)(52 76)(55 63)(56 72)(60 68)(61 77)(65 73)(70 78)(81 123)(82 132)(83 141)(84 150)(85 159)(86 128)(87 137)(88 146)(89 155)(90 124)(91 133)(92 142)(93 151)(94 160)(95 129)(96 138)(97 147)(98 156)(99 125)(100 134)(101 143)(102 152)(103 121)(104 130)(105 139)(106 148)(107 157)(108 126)(109 135)(110 144)(111 153)(112 122)(113 131)(114 140)(115 149)(116 158)(117 127)(118 136)(119 145)(120 154)
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), (1,105,44,139)(2,94,45,128)(3,83,46,157)(4,112,47,146)(5,101,48,135)(6,90,49,124)(7,119,50,153)(8,108,51,142)(9,97,52,131)(10,86,53,160)(11,115,54,149)(12,104,55,138)(13,93,56,127)(14,82,57,156)(15,111,58,145)(16,100,59,134)(17,89,60,123)(18,118,61,152)(19,107,62,141)(20,96,63,130)(21,85,64,159)(22,114,65,148)(23,103,66,137)(24,92,67,126)(25,81,68,155)(26,110,69,144)(27,99,70,133)(28,88,71,122)(29,117,72,151)(30,106,73,140)(31,95,74,129)(32,84,75,158)(33,113,76,147)(34,102,77,136)(35,91,78,125)(36,120,79,154)(37,109,80,143)(38,98,41,132)(39,87,42,121)(40,116,43,150), (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)(41,57)(42,66)(43,75)(45,53)(46,62)(47,71)(48,80)(50,58)(51,67)(52,76)(55,63)(56,72)(60,68)(61,77)(65,73)(70,78)(81,123)(82,132)(83,141)(84,150)(85,159)(86,128)(87,137)(88,146)(89,155)(90,124)(91,133)(92,142)(93,151)(94,160)(95,129)(96,138)(97,147)(98,156)(99,125)(100,134)(101,143)(102,152)(103,121)(104,130)(105,139)(106,148)(107,157)(108,126)(109,135)(110,144)(111,153)(112,122)(113,131)(114,140)(115,149)(116,158)(117,127)(118,136)(119,145)(120,154)>;
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), (1,105,44,139)(2,94,45,128)(3,83,46,157)(4,112,47,146)(5,101,48,135)(6,90,49,124)(7,119,50,153)(8,108,51,142)(9,97,52,131)(10,86,53,160)(11,115,54,149)(12,104,55,138)(13,93,56,127)(14,82,57,156)(15,111,58,145)(16,100,59,134)(17,89,60,123)(18,118,61,152)(19,107,62,141)(20,96,63,130)(21,85,64,159)(22,114,65,148)(23,103,66,137)(24,92,67,126)(25,81,68,155)(26,110,69,144)(27,99,70,133)(28,88,71,122)(29,117,72,151)(30,106,73,140)(31,95,74,129)(32,84,75,158)(33,113,76,147)(34,102,77,136)(35,91,78,125)(36,120,79,154)(37,109,80,143)(38,98,41,132)(39,87,42,121)(40,116,43,150), (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)(41,57)(42,66)(43,75)(45,53)(46,62)(47,71)(48,80)(50,58)(51,67)(52,76)(55,63)(56,72)(60,68)(61,77)(65,73)(70,78)(81,123)(82,132)(83,141)(84,150)(85,159)(86,128)(87,137)(88,146)(89,155)(90,124)(91,133)(92,142)(93,151)(94,160)(95,129)(96,138)(97,147)(98,156)(99,125)(100,134)(101,143)(102,152)(103,121)(104,130)(105,139)(106,148)(107,157)(108,126)(109,135)(110,144)(111,153)(112,122)(113,131)(114,140)(115,149)(116,158)(117,127)(118,136)(119,145)(120,154) );
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)], [(1,105,44,139),(2,94,45,128),(3,83,46,157),(4,112,47,146),(5,101,48,135),(6,90,49,124),(7,119,50,153),(8,108,51,142),(9,97,52,131),(10,86,53,160),(11,115,54,149),(12,104,55,138),(13,93,56,127),(14,82,57,156),(15,111,58,145),(16,100,59,134),(17,89,60,123),(18,118,61,152),(19,107,62,141),(20,96,63,130),(21,85,64,159),(22,114,65,148),(23,103,66,137),(24,92,67,126),(25,81,68,155),(26,110,69,144),(27,99,70,133),(28,88,71,122),(29,117,72,151),(30,106,73,140),(31,95,74,129),(32,84,75,158),(33,113,76,147),(34,102,77,136),(35,91,78,125),(36,120,79,154),(37,109,80,143),(38,98,41,132),(39,87,42,121),(40,116,43,150)], [(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),(41,57),(42,66),(43,75),(45,53),(46,62),(47,71),(48,80),(50,58),(51,67),(52,76),(55,63),(56,72),(60,68),(61,77),(65,73),(70,78),(81,123),(82,132),(83,141),(84,150),(85,159),(86,128),(87,137),(88,146),(89,155),(90,124),(91,133),(92,142),(93,151),(94,160),(95,129),(96,138),(97,147),(98,156),(99,125),(100,134),(101,143),(102,152),(103,121),(104,130),(105,139),(106,148),(107,157),(108,126),(109,135),(110,144),(111,153),(112,122),(113,131),(114,140),(115,149),(116,158),(117,127),(118,136),(119,145),(120,154)]])
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