metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊3D4, C23.18D20, C8⋊2(C5⋊D4), C5⋊4(C8⋊2D4), C40⋊6C4⋊4C2, (C2×D40)⋊12C2, (C2×C4).52D20, (C2×C8).78D10, C20⋊7D4⋊41C2, C20.421(C2×D4), (C2×C20).298D4, (C2×M4(2))⋊2D5, D20⋊5C4⋊42C2, (C10×M4(2))⋊2C2, (C2×C40).64C22, C4.115(C4○D20), C20.231(C4○D4), C2.23(C8⋊D10), C10.74(C4⋊D4), C2.22(C20⋊7D4), C10.23(C8⋊C22), (C2×C20).776C23, (C2×D20).23C22, C22.135(C2×D20), (C22×C4).143D10, (C22×C10).104D4, C4⋊Dic5.27C22, (C22×C20).305C22, C4.114(C2×C5⋊D4), (C2×C10).166(C2×D4), (C2×C4).725(C22×D5), SmallGroup(320,762)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊3D4
G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >
Subgroups: 694 in 130 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], D8 [×2], C22×C4, C2×D4 [×4], Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×3], D4⋊C4 [×2], C4.Q8, C4⋊D4 [×2], C2×M4(2), C2×D8, C40 [×2], C40, D20 [×4], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, C8⋊2D4, D40 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C2×C40 [×2], C5×M4(2) [×2], C2×D20 [×2], C2×C5⋊D4 [×2], C22×C20, C40⋊6C4, D20⋊5C4 [×2], C2×D40, C20⋊7D4 [×2], C10×M4(2), C40⋊3D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C8⋊2D4, C2×D20, C4○D20, C2×C5⋊D4, C8⋊D10 [×2], C20⋊7D4, C40⋊3D4
►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)
(1 128 102 46)(2 147 103 65)(3 126 104 44)(4 145 105 63)(5 124 106 42)(6 143 107 61)(7 122 108 80)(8 141 109 59)(9 160 110 78)(10 139 111 57)(11 158 112 76)(12 137 113 55)(13 156 114 74)(14 135 115 53)(15 154 116 72)(16 133 117 51)(17 152 118 70)(18 131 119 49)(19 150 120 68)(20 129 81 47)(21 148 82 66)(22 127 83 45)(23 146 84 64)(24 125 85 43)(25 144 86 62)(26 123 87 41)(27 142 88 60)(28 121 89 79)(29 140 90 58)(30 159 91 77)(31 138 92 56)(32 157 93 75)(33 136 94 54)(34 155 95 73)(35 134 96 52)(36 153 97 71)(37 132 98 50)(38 151 99 69)(39 130 100 48)(40 149 101 67)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 133)(42 132)(43 131)(44 130)(45 129)(46 128)(47 127)(48 126)(49 125)(50 124)(51 123)(52 122)(53 121)(54 160)(55 159)(56 158)(57 157)(58 156)(59 155)(60 154)(61 153)(62 152)(63 151)(64 150)(65 149)(66 148)(67 147)(68 146)(69 145)(70 144)(71 143)(72 142)(73 141)(74 140)(75 139)(76 138)(77 137)(78 136)(79 135)(80 134)(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)
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), (1,128,102,46)(2,147,103,65)(3,126,104,44)(4,145,105,63)(5,124,106,42)(6,143,107,61)(7,122,108,80)(8,141,109,59)(9,160,110,78)(10,139,111,57)(11,158,112,76)(12,137,113,55)(13,156,114,74)(14,135,115,53)(15,154,116,72)(16,133,117,51)(17,152,118,70)(18,131,119,49)(19,150,120,68)(20,129,81,47)(21,148,82,66)(22,127,83,45)(23,146,84,64)(24,125,85,43)(25,144,86,62)(26,123,87,41)(27,142,88,60)(28,121,89,79)(29,140,90,58)(30,159,91,77)(31,138,92,56)(32,157,93,75)(33,136,94,54)(34,155,95,73)(35,134,96,52)(36,153,97,71)(37,132,98,50)(38,151,99,69)(39,130,100,48)(40,149,101,67), (2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,133)(42,132)(43,131)(44,130)(45,129)(46,128)(47,127)(48,126)(49,125)(50,124)(51,123)(52,122)(53,121)(54,160)(55,159)(56,158)(57,157)(58,156)(59,155)(60,154)(61,153)(62,152)(63,151)(64,150)(65,149)(66,148)(67,147)(68,146)(69,145)(70,144)(71,143)(72,142)(73,141)(74,140)(75,139)(76,138)(77,137)(78,136)(79,135)(80,134)(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)>;
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), (1,128,102,46)(2,147,103,65)(3,126,104,44)(4,145,105,63)(5,124,106,42)(6,143,107,61)(7,122,108,80)(8,141,109,59)(9,160,110,78)(10,139,111,57)(11,158,112,76)(12,137,113,55)(13,156,114,74)(14,135,115,53)(15,154,116,72)(16,133,117,51)(17,152,118,70)(18,131,119,49)(19,150,120,68)(20,129,81,47)(21,148,82,66)(22,127,83,45)(23,146,84,64)(24,125,85,43)(25,144,86,62)(26,123,87,41)(27,142,88,60)(28,121,89,79)(29,140,90,58)(30,159,91,77)(31,138,92,56)(32,157,93,75)(33,136,94,54)(34,155,95,73)(35,134,96,52)(36,153,97,71)(37,132,98,50)(38,151,99,69)(39,130,100,48)(40,149,101,67), (2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,133)(42,132)(43,131)(44,130)(45,129)(46,128)(47,127)(48,126)(49,125)(50,124)(51,123)(52,122)(53,121)(54,160)(55,159)(56,158)(57,157)(58,156)(59,155)(60,154)(61,153)(62,152)(63,151)(64,150)(65,149)(66,148)(67,147)(68,146)(69,145)(70,144)(71,143)(72,142)(73,141)(74,140)(75,139)(76,138)(77,137)(78,136)(79,135)(80,134)(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) );
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)], [(1,128,102,46),(2,147,103,65),(3,126,104,44),(4,145,105,63),(5,124,106,42),(6,143,107,61),(7,122,108,80),(8,141,109,59),(9,160,110,78),(10,139,111,57),(11,158,112,76),(12,137,113,55),(13,156,114,74),(14,135,115,53),(15,154,116,72),(16,133,117,51),(17,152,118,70),(18,131,119,49),(19,150,120,68),(20,129,81,47),(21,148,82,66),(22,127,83,45),(23,146,84,64),(24,125,85,43),(25,144,86,62),(26,123,87,41),(27,142,88,60),(28,121,89,79),(29,140,90,58),(30,159,91,77),(31,138,92,56),(32,157,93,75),(33,136,94,54),(34,155,95,73),(35,134,96,52),(36,153,97,71),(37,132,98,50),(38,151,99,69),(39,130,100,48),(40,149,101,67)], [(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(41,133),(42,132),(43,131),(44,130),(45,129),(46,128),(47,127),(48,126),(49,125),(50,124),(51,123),(52,122),(53,121),(54,160),(55,159),(56,158),(57,157),(58,156),(59,155),(60,154),(61,153),(62,152),(63,151),(64,150),(65,149),(66,148),(67,147),(68,146),(69,145),(70,144),(71,143),(72,142),(73,141),(74,140),(75,139),(76,138),(77,137),(78,136),(79,135),(80,134),(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)])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 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 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 40 | 40 | 2 | 2 | 4 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 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 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | D20 | C4○D20 | C8⋊C22 | C8⋊D10 |
| kernel | C40⋊3D4 | C40⋊6C4 | D20⋊5C4 | C2×D40 | C20⋊7D4 | C10×M4(2) | C40 | C2×C20 | C22×C10 | C2×M4(2) | C20 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C10 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 8 | 2 | 8 |
Matrix representation of C40⋊3D4 ►in GL6(𝔽41)
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 29 | 8 | 13 |
| 0 | 0 | 12 | 2 | 28 | 4 |
| 0 | 0 | 33 | 28 | 8 | 12 |
| 0 | 0 | 13 | 37 | 29 | 39 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 35 | 1 |
G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,9,0,0,0,0,0,0,0,33,12,33,13,0,0,29,2,28,37,0,0,8,28,8,29,0,0,13,4,12,39],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,40,35,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1] >;
C40⋊3D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_3D_4 % in TeX
G:=Group("C40:3D4"); // GroupNames label
G:=SmallGroup(320,762);
// by ID
G=gap.SmallGroup(320,762);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations