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