metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊12C4, Dic5⋊5D8, C5⋊4(C4×D8), C8⋊7(C4×D5), C2.3(D5×D8), C40⋊14(C2×C4), D20⋊17(C2×C4), C2.D8⋊13D5, (C8×Dic5)⋊3C2, (C2×D40).9C2, C10.82(C4×D4), C10.26(C2×D8), D20⋊8C4⋊7C2, C4⋊C4.166D10, (C2×C8).225D10, D20⋊6C4⋊19C2, C22.87(D4×D5), C20.35(C4○D4), C10.73(C4○D8), (C2×C40).77C22, C4.7(Q8⋊2D5), C2.3(Q8.D10), C20.105(C22×C4), (C2×C20).288C23, (C2×Dic5).275D4, (C2×D20).84C22, C2.12(D20⋊8C4), (C4×Dic5).262C22, C4.43(C2×C4×D5), (C5×C2.D8)⋊2C2, (C2×C10).293(C2×D4), (C5×C4⋊C4).81C22, (C2×C4).391(C22×D5), (C2×C5⋊2C8).237C22, SmallGroup(320,499)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊12C4
G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a31, cbc-1=a30b >
Subgroups: 646 in 134 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×D8, D40, C2×C5⋊2C8, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, D20⋊6C4, C8×Dic5, C5×C2.D8, D20⋊8C4, C2×D40, D40⋊12C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, D8, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×D8, C4○D8, C4×D5, C22×D5, C4×D8, C2×C4×D5, D4×D5, Q8⋊2D5, D20⋊8C4, D5×D8, Q8.D10, D40⋊12C4
►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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(121 155)(122 154)(123 153)(124 152)(125 151)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)(156 160)(157 159)
(1 113 57 151)(2 104 58 142)(3 95 59 133)(4 86 60 124)(5 117 61 155)(6 108 62 146)(7 99 63 137)(8 90 64 128)(9 81 65 159)(10 112 66 150)(11 103 67 141)(12 94 68 132)(13 85 69 123)(14 116 70 154)(15 107 71 145)(16 98 72 136)(17 89 73 127)(18 120 74 158)(19 111 75 149)(20 102 76 140)(21 93 77 131)(22 84 78 122)(23 115 79 153)(24 106 80 144)(25 97 41 135)(26 88 42 126)(27 119 43 157)(28 110 44 148)(29 101 45 139)(30 92 46 130)(31 83 47 121)(32 114 48 152)(33 105 49 143)(34 96 50 134)(35 87 51 125)(36 118 52 156)(37 109 53 147)(38 100 54 138)(39 91 55 129)(40 82 56 160)
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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(156,160)(157,159), (1,113,57,151)(2,104,58,142)(3,95,59,133)(4,86,60,124)(5,117,61,155)(6,108,62,146)(7,99,63,137)(8,90,64,128)(9,81,65,159)(10,112,66,150)(11,103,67,141)(12,94,68,132)(13,85,69,123)(14,116,70,154)(15,107,71,145)(16,98,72,136)(17,89,73,127)(18,120,74,158)(19,111,75,149)(20,102,76,140)(21,93,77,131)(22,84,78,122)(23,115,79,153)(24,106,80,144)(25,97,41,135)(26,88,42,126)(27,119,43,157)(28,110,44,148)(29,101,45,139)(30,92,46,130)(31,83,47,121)(32,114,48,152)(33,105,49,143)(34,96,50,134)(35,87,51,125)(36,118,52,156)(37,109,53,147)(38,100,54,138)(39,91,55,129)(40,82,56,160)>;
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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(156,160)(157,159), (1,113,57,151)(2,104,58,142)(3,95,59,133)(4,86,60,124)(5,117,61,155)(6,108,62,146)(7,99,63,137)(8,90,64,128)(9,81,65,159)(10,112,66,150)(11,103,67,141)(12,94,68,132)(13,85,69,123)(14,116,70,154)(15,107,71,145)(16,98,72,136)(17,89,73,127)(18,120,74,158)(19,111,75,149)(20,102,76,140)(21,93,77,131)(22,84,78,122)(23,115,79,153)(24,106,80,144)(25,97,41,135)(26,88,42,126)(27,119,43,157)(28,110,44,148)(29,101,45,139)(30,92,46,130)(31,83,47,121)(32,114,48,152)(33,105,49,143)(34,96,50,134)(35,87,51,125)(36,118,52,156)(37,109,53,147)(38,100,54,138)(39,91,55,129)(40,82,56,160) );
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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(81,119),(82,118),(83,117),(84,116),(85,115),(86,114),(87,113),(88,112),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101),(121,155),(122,154),(123,153),(124,152),(125,151),(126,150),(127,149),(128,148),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(136,140),(137,139),(156,160),(157,159)], [(1,113,57,151),(2,104,58,142),(3,95,59,133),(4,86,60,124),(5,117,61,155),(6,108,62,146),(7,99,63,137),(8,90,64,128),(9,81,65,159),(10,112,66,150),(11,103,67,141),(12,94,68,132),(13,85,69,123),(14,116,70,154),(15,107,71,145),(16,98,72,136),(17,89,73,127),(18,120,74,158),(19,111,75,149),(20,102,76,140),(21,93,77,131),(22,84,78,122),(23,115,79,153),(24,106,80,144),(25,97,41,135),(26,88,42,126),(27,119,43,157),(28,110,44,148),(29,101,45,139),(30,92,46,130),(31,83,47,121),(32,114,48,152),(33,105,49,143),(34,96,50,134),(35,87,51,125),(36,118,52,156),(37,109,53,147),(38,100,54,138),(39,91,55,129),(40,82,56,160)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | D8 | C4○D4 | D10 | D10 | C4○D8 | C4×D5 | Q8⋊2D5 | D4×D5 | D5×D8 | Q8.D10 |
| kernel | D40⋊12C4 | D20⋊6C4 | C8×Dic5 | C5×C2.D8 | D20⋊8C4 | C2×D40 | D40 | C2×Dic5 | C2.D8 | Dic5 | C20 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of D40⋊12C4 ►in GL4(𝔽41) generated by
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 12 | 17 |
| 40 | 34 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 30 | 11 |
| 0 | 0 | 15 | 11 |
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,0,12,0,0,17,17],[40,0,0,0,34,1,0,0,0,0,1,1,0,0,0,40],[9,0,0,0,0,9,0,0,0,0,30,15,0,0,11,11] >;
D40⋊12C4 in GAP, Magma, Sage, TeX
D_{40}\rtimes_{12}C_4 % in TeX
G:=Group("D40:12C4"); // GroupNames label
G:=SmallGroup(320,499);
// by ID
G=gap.SmallGroup(320,499);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,120,135,100,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^30*b>;
// generators/relations