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, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C40, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊2D4, D40, C4⋊Dic5, D10⋊C4, C2×C40, C5×M4(2), C2×D20, C2×C5⋊D4, C22×C20, C40⋊6C4, D20⋊5C4, C2×D40, C20⋊7D4, C10×M4(2), C40⋊3D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, D20, C5⋊D4, C22×D5, C8⋊2D4, C2×D20, C4○D20, C2×C5⋊D4, C8⋊D10, 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 50 115 137)(2 69 116 156)(3 48 117 135)(4 67 118 154)(5 46 119 133)(6 65 120 152)(7 44 81 131)(8 63 82 150)(9 42 83 129)(10 61 84 148)(11 80 85 127)(12 59 86 146)(13 78 87 125)(14 57 88 144)(15 76 89 123)(16 55 90 142)(17 74 91 121)(18 53 92 140)(19 72 93 159)(20 51 94 138)(21 70 95 157)(22 49 96 136)(23 68 97 155)(24 47 98 134)(25 66 99 153)(26 45 100 132)(27 64 101 151)(28 43 102 130)(29 62 103 149)(30 41 104 128)(31 60 105 147)(32 79 106 126)(33 58 107 145)(34 77 108 124)(35 56 109 143)(36 75 110 122)(37 54 111 141)(38 73 112 160)(39 52 113 139)(40 71 114 158)
(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)(41 146)(42 145)(43 144)(44 143)(45 142)(46 141)(47 140)(48 139)(49 138)(50 137)(51 136)(52 135)(53 134)(54 133)(55 132)(56 131)(57 130)(58 129)(59 128)(60 127)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 160)(68 159)(69 158)(70 157)(71 156)(72 155)(73 154)(74 153)(75 152)(76 151)(77 150)(78 149)(79 148)(80 147)(81 109)(82 108)(83 107)(84 106)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(110 120)(111 119)(112 118)(113 117)(114 116)
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), (1,50,115,137)(2,69,116,156)(3,48,117,135)(4,67,118,154)(5,46,119,133)(6,65,120,152)(7,44,81,131)(8,63,82,150)(9,42,83,129)(10,61,84,148)(11,80,85,127)(12,59,86,146)(13,78,87,125)(14,57,88,144)(15,76,89,123)(16,55,90,142)(17,74,91,121)(18,53,92,140)(19,72,93,159)(20,51,94,138)(21,70,95,157)(22,49,96,136)(23,68,97,155)(24,47,98,134)(25,66,99,153)(26,45,100,132)(27,64,101,151)(28,43,102,130)(29,62,103,149)(30,41,104,128)(31,60,105,147)(32,79,106,126)(33,58,107,145)(34,77,108,124)(35,56,109,143)(36,75,110,122)(37,54,111,141)(38,73,112,160)(39,52,113,139)(40,71,114,158), (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)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,160)(68,159)(69,158)(70,157)(71,156)(72,155)(73,154)(74,153)(75,152)(76,151)(77,150)(78,149)(79,148)(80,147)(81,109)(82,108)(83,107)(84,106)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(110,120)(111,119)(112,118)(113,117)(114,116)>;
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), (1,50,115,137)(2,69,116,156)(3,48,117,135)(4,67,118,154)(5,46,119,133)(6,65,120,152)(7,44,81,131)(8,63,82,150)(9,42,83,129)(10,61,84,148)(11,80,85,127)(12,59,86,146)(13,78,87,125)(14,57,88,144)(15,76,89,123)(16,55,90,142)(17,74,91,121)(18,53,92,140)(19,72,93,159)(20,51,94,138)(21,70,95,157)(22,49,96,136)(23,68,97,155)(24,47,98,134)(25,66,99,153)(26,45,100,132)(27,64,101,151)(28,43,102,130)(29,62,103,149)(30,41,104,128)(31,60,105,147)(32,79,106,126)(33,58,107,145)(34,77,108,124)(35,56,109,143)(36,75,110,122)(37,54,111,141)(38,73,112,160)(39,52,113,139)(40,71,114,158), (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)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,160)(68,159)(69,158)(70,157)(71,156)(72,155)(73,154)(74,153)(75,152)(76,151)(77,150)(78,149)(79,148)(80,147)(81,109)(82,108)(83,107)(84,106)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(110,120)(111,119)(112,118)(113,117)(114,116) );
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)], [(1,50,115,137),(2,69,116,156),(3,48,117,135),(4,67,118,154),(5,46,119,133),(6,65,120,152),(7,44,81,131),(8,63,82,150),(9,42,83,129),(10,61,84,148),(11,80,85,127),(12,59,86,146),(13,78,87,125),(14,57,88,144),(15,76,89,123),(16,55,90,142),(17,74,91,121),(18,53,92,140),(19,72,93,159),(20,51,94,138),(21,70,95,157),(22,49,96,136),(23,68,97,155),(24,47,98,134),(25,66,99,153),(26,45,100,132),(27,64,101,151),(28,43,102,130),(29,62,103,149),(30,41,104,128),(31,60,105,147),(32,79,106,126),(33,58,107,145),(34,77,108,124),(35,56,109,143),(36,75,110,122),(37,54,111,141),(38,73,112,160),(39,52,113,139),(40,71,114,158)], [(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),(41,146),(42,145),(43,144),(44,143),(45,142),(46,141),(47,140),(48,139),(49,138),(50,137),(51,136),(52,135),(53,134),(54,133),(55,132),(56,131),(57,130),(58,129),(59,128),(60,127),(61,126),(62,125),(63,124),(64,123),(65,122),(66,121),(67,160),(68,159),(69,158),(70,157),(71,156),(72,155),(73,154),(74,153),(75,152),(76,151),(77,150),(78,149),(79,148),(80,147),(81,109),(82,108),(83,107),(84,106),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(110,120),(111,119),(112,118),(113,117),(114,116)]])
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