metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊18D4, Dic5⋊3M4(2), C8⋊9(C5⋊D4), C5⋊8(C8⋊6D4), (C8×Dic5)⋊31C2, C10.110(C4×D4), C20.443(C2×D4), (C2×C8).276D10, (C2×M4(2))⋊8D5, D10⋊1C8⋊39C2, C23.20(C4×D5), C10.57(C8○D4), (C10×M4(2))⋊7C2, C20.8Q8⋊41C2, C2.22(D5×M4(2)), C23.D5.20C4, D10⋊C4.27C4, C4.138(C4○D20), C20.254(C4○D4), C20.55D4⋊30C2, (C2×C20).868C23, (C2×C40).318C22, C10.D4.27C4, (C22×C4).137D10, C10.67(C2×M4(2)), C2.18(D20.2C4), (C22×C20).375C22, (C4×Dic5).317C22, (C2×C4).51(C4×D5), C2.25(C4×C5⋊D4), (C2×C8⋊D5)⋊25C2, (C2×C5⋊D4).20C4, (C4×C5⋊D4).17C2, C4.134(C2×C5⋊D4), C22.147(C2×C4×D5), (C2×C20).361(C2×C4), (C2×C4×D5).236C22, (C22×D5).30(C2×C4), (C2×C4).810(C22×D5), (C22×C10).136(C2×C4), (C2×C10).239(C22×C4), (C2×C5⋊2C8).332C22, (C2×Dic5).113(C2×C4), SmallGroup(320,755)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊18D4
G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a29, cbc=b-1 >
Subgroups: 382 in 122 conjugacy classes, 53 normal (47 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×6], C5, C8 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2), 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⋊6D4, 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, C8×Dic5, C20.8Q8, D10⋊1C8, C20.55D4, C2×C8⋊D5, C4×C5⋊D4, C10×M4(2), C40⋊18D4
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⋊6D4, C2×C4×D5, C4○D20, C2×C5⋊D4, D5×M4(2), D20.2C4, C4×C5⋊D4, C40⋊18D4
►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 84 48 122)(2 93 49 131)(3 102 50 140)(4 111 51 149)(5 120 52 158)(6 89 53 127)(7 98 54 136)(8 107 55 145)(9 116 56 154)(10 85 57 123)(11 94 58 132)(12 103 59 141)(13 112 60 150)(14 81 61 159)(15 90 62 128)(16 99 63 137)(17 108 64 146)(18 117 65 155)(19 86 66 124)(20 95 67 133)(21 104 68 142)(22 113 69 151)(23 82 70 160)(24 91 71 129)(25 100 72 138)(26 109 73 147)(27 118 74 156)(28 87 75 125)(29 96 76 134)(30 105 77 143)(31 114 78 152)(32 83 79 121)(33 92 80 130)(34 101 41 139)(35 110 42 148)(36 119 43 157)(37 88 44 126)(38 97 45 135)(39 106 46 144)(40 115 47 153)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 45)(42 74)(43 63)(44 52)(46 70)(47 59)(49 77)(50 66)(51 55)(53 73)(54 62)(56 80)(57 69)(60 76)(61 65)(64 72)(67 79)(71 75)(81 155)(82 144)(83 133)(84 122)(85 151)(86 140)(87 129)(88 158)(89 147)(90 136)(91 125)(92 154)(93 143)(94 132)(95 121)(96 150)(97 139)(98 128)(99 157)(100 146)(101 135)(102 124)(103 153)(104 142)(105 131)(106 160)(107 149)(108 138)(109 127)(110 156)(111 145)(112 134)(113 123)(114 152)(115 141)(116 130)(117 159)(118 148)(119 137)(120 126)
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), (1,84,48,122)(2,93,49,131)(3,102,50,140)(4,111,51,149)(5,120,52,158)(6,89,53,127)(7,98,54,136)(8,107,55,145)(9,116,56,154)(10,85,57,123)(11,94,58,132)(12,103,59,141)(13,112,60,150)(14,81,61,159)(15,90,62,128)(16,99,63,137)(17,108,64,146)(18,117,65,155)(19,86,66,124)(20,95,67,133)(21,104,68,142)(22,113,69,151)(23,82,70,160)(24,91,71,129)(25,100,72,138)(26,109,73,147)(27,118,74,156)(28,87,75,125)(29,96,76,134)(30,105,77,143)(31,114,78,152)(32,83,79,121)(33,92,80,130)(34,101,41,139)(35,110,42,148)(36,119,43,157)(37,88,44,126)(38,97,45,135)(39,106,46,144)(40,115,47,153), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,45)(42,74)(43,63)(44,52)(46,70)(47,59)(49,77)(50,66)(51,55)(53,73)(54,62)(56,80)(57,69)(60,76)(61,65)(64,72)(67,79)(71,75)(81,155)(82,144)(83,133)(84,122)(85,151)(86,140)(87,129)(88,158)(89,147)(90,136)(91,125)(92,154)(93,143)(94,132)(95,121)(96,150)(97,139)(98,128)(99,157)(100,146)(101,135)(102,124)(103,153)(104,142)(105,131)(106,160)(107,149)(108,138)(109,127)(110,156)(111,145)(112,134)(113,123)(114,152)(115,141)(116,130)(117,159)(118,148)(119,137)(120,126)>;
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), (1,84,48,122)(2,93,49,131)(3,102,50,140)(4,111,51,149)(5,120,52,158)(6,89,53,127)(7,98,54,136)(8,107,55,145)(9,116,56,154)(10,85,57,123)(11,94,58,132)(12,103,59,141)(13,112,60,150)(14,81,61,159)(15,90,62,128)(16,99,63,137)(17,108,64,146)(18,117,65,155)(19,86,66,124)(20,95,67,133)(21,104,68,142)(22,113,69,151)(23,82,70,160)(24,91,71,129)(25,100,72,138)(26,109,73,147)(27,118,74,156)(28,87,75,125)(29,96,76,134)(30,105,77,143)(31,114,78,152)(32,83,79,121)(33,92,80,130)(34,101,41,139)(35,110,42,148)(36,119,43,157)(37,88,44,126)(38,97,45,135)(39,106,46,144)(40,115,47,153), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,45)(42,74)(43,63)(44,52)(46,70)(47,59)(49,77)(50,66)(51,55)(53,73)(54,62)(56,80)(57,69)(60,76)(61,65)(64,72)(67,79)(71,75)(81,155)(82,144)(83,133)(84,122)(85,151)(86,140)(87,129)(88,158)(89,147)(90,136)(91,125)(92,154)(93,143)(94,132)(95,121)(96,150)(97,139)(98,128)(99,157)(100,146)(101,135)(102,124)(103,153)(104,142)(105,131)(106,160)(107,149)(108,138)(109,127)(110,156)(111,145)(112,134)(113,123)(114,152)(115,141)(116,130)(117,159)(118,148)(119,137)(120,126) );
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)], [(1,84,48,122),(2,93,49,131),(3,102,50,140),(4,111,51,149),(5,120,52,158),(6,89,53,127),(7,98,54,136),(8,107,55,145),(9,116,56,154),(10,85,57,123),(11,94,58,132),(12,103,59,141),(13,112,60,150),(14,81,61,159),(15,90,62,128),(16,99,63,137),(17,108,64,146),(18,117,65,155),(19,86,66,124),(20,95,67,133),(21,104,68,142),(22,113,69,151),(23,82,70,160),(24,91,71,129),(25,100,72,138),(26,109,73,147),(27,118,74,156),(28,87,75,125),(29,96,76,134),(30,105,77,143),(31,114,78,152),(32,83,79,121),(33,92,80,130),(34,101,41,139),(35,110,42,148),(36,119,43,157),(37,88,44,126),(38,97,45,135),(39,106,46,144),(40,115,47,153)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,45),(42,74),(43,63),(44,52),(46,70),(47,59),(49,77),(50,66),(51,55),(53,73),(54,62),(56,80),(57,69),(60,76),(61,65),(64,72),(67,79),(71,75),(81,155),(82,144),(83,133),(84,122),(85,151),(86,140),(87,129),(88,158),(89,147),(90,136),(91,125),(92,154),(93,143),(94,132),(95,121),(96,150),(97,139),(98,128),(99,157),(100,146),(101,135),(102,124),(103,153),(104,142),(105,131),(106,160),(107,149),(108,138),(109,127),(110,156),(111,145),(112,134),(113,123),(114,152),(115,141),(116,130),(117,159),(118,148),(119,137),(120,126)])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 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 | 4 | 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 | 20 | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 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 | M4(2) | C4○D4 | D10 | D10 | C8○D4 | C5⋊D4 | C4×D5 | C4×D5 | C4○D20 | D5×M4(2) | D20.2C4 |
| kernel | C40⋊18D4 | C8×Dic5 | C20.8Q8 | D10⋊1C8 | C20.55D4 | C2×C8⋊D5 | C4×C5⋊D4 | C10×M4(2) | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C40 | C2×M4(2) | Dic5 | C20 | 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 | 4 | 2 | 4 | 2 | 4 | 8 | 4 | 4 | 8 | 4 | 4 |
Matrix representation of C40⋊18D4 ►in GL4(𝔽41) generated by
| 40 | 34 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 32 | 0 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,7,0,0,34,7,0,0,0,0,0,32,0,0,1,0],[17,3,0,0,40,24,0,0,0,0,40,0,0,0,0,40],[1,34,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C40⋊18D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{18}D_4 % in TeX
G:=Group("C40:18D4"); // GroupNames label
G:=SmallGroup(320,755);
// by ID
G=gap.SmallGroup(320,755);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations