metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊29D4, C22⋊1D40, C23.25D20, (C2×C10)⋊5D8, (C2×D40)⋊9C2, C5⋊4(C8⋊7D4), (C22×C8)⋊6D5, C20⋊7D4⋊2C2, C8⋊13(C5⋊D4), C40⋊5C4⋊14C2, C10.17(C2×D8), (C2×C4).68D20, C2.17(C2×D40), D20⋊5C4⋊3C2, (C22×C40)⋊10C2, C20.413(C2×D4), (C2×C8).308D10, (C2×C20).356D4, C10.18(C4○D8), C4.112(C4○D20), C20.228(C4○D4), C10.71(C4⋊D4), C2.19(C20⋊7D4), (C2×C20).769C23, (C2×C40).380C22, (C2×D20).21C22, (C22×C10).141D4, C22.132(C2×D20), (C22×C4).431D10, C4⋊Dic5.24C22, C2.18(D40⋊7C2), (C22×C20).519C22, C4.106(C2×C5⋊D4), (C2×C10).159(C2×D4), (C2×C4).717(C22×D5), SmallGroup(320,742)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊29D4
G = < a,b,c | a40=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 694 in 134 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×2], C22×C4, C2×D4 [×4], Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4 [×2], C2.D8, C4⋊D4 [×2], C22×C8, 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⋊7D4, D40 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C2×C40 [×2], C2×C40 [×2], C2×D20 [×2], C2×C5⋊D4 [×2], C22×C20, C40⋊5C4, D20⋊5C4 [×2], C2×D40, C20⋊7D4 [×2], C22×C40, C40⋊29D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C4○D8, D20 [×2], C5⋊D4 [×2], C22×D5, C8⋊7D4, D40 [×2], C2×D20, C4○D20, C2×C5⋊D4, C2×D40, D40⋊7C2, C20⋊7D4, C40⋊29D4
►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 75 87 153)(2 74 88 152)(3 73 89 151)(4 72 90 150)(5 71 91 149)(6 70 92 148)(7 69 93 147)(8 68 94 146)(9 67 95 145)(10 66 96 144)(11 65 97 143)(12 64 98 142)(13 63 99 141)(14 62 100 140)(15 61 101 139)(16 60 102 138)(17 59 103 137)(18 58 104 136)(19 57 105 135)(20 56 106 134)(21 55 107 133)(22 54 108 132)(23 53 109 131)(24 52 110 130)(25 51 111 129)(26 50 112 128)(27 49 113 127)(28 48 114 126)(29 47 115 125)(30 46 116 124)(31 45 117 123)(32 44 118 122)(33 43 119 121)(34 42 120 160)(35 41 81 159)(36 80 82 158)(37 79 83 157)(38 78 84 156)(39 77 85 155)(40 76 86 154)
(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)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 140)(49 139)(50 138)(51 137)(52 136)(53 135)(54 134)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 126)(63 125)(64 124)(65 123)(66 122)(67 121)(68 160)(69 159)(70 158)(71 157)(72 156)(73 155)(74 154)(75 153)(76 152)(77 151)(78 150)(79 149)(80 148)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)
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), (1,75,87,153)(2,74,88,152)(3,73,89,151)(4,72,90,150)(5,71,91,149)(6,70,92,148)(7,69,93,147)(8,68,94,146)(9,67,95,145)(10,66,96,144)(11,65,97,143)(12,64,98,142)(13,63,99,141)(14,62,100,140)(15,61,101,139)(16,60,102,138)(17,59,103,137)(18,58,104,136)(19,57,105,135)(20,56,106,134)(21,55,107,133)(22,54,108,132)(23,53,109,131)(24,52,110,130)(25,51,111,129)(26,50,112,128)(27,49,113,127)(28,48,114,126)(29,47,115,125)(30,46,116,124)(31,45,117,123)(32,44,118,122)(33,43,119,121)(34,42,120,160)(35,41,81,159)(36,80,82,158)(37,79,83,157)(38,78,84,156)(39,77,85,155)(40,76,86,154), (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)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,140)(49,139)(50,138)(51,137)(52,136)(53,135)(54,134)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)>;
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), (1,75,87,153)(2,74,88,152)(3,73,89,151)(4,72,90,150)(5,71,91,149)(6,70,92,148)(7,69,93,147)(8,68,94,146)(9,67,95,145)(10,66,96,144)(11,65,97,143)(12,64,98,142)(13,63,99,141)(14,62,100,140)(15,61,101,139)(16,60,102,138)(17,59,103,137)(18,58,104,136)(19,57,105,135)(20,56,106,134)(21,55,107,133)(22,54,108,132)(23,53,109,131)(24,52,110,130)(25,51,111,129)(26,50,112,128)(27,49,113,127)(28,48,114,126)(29,47,115,125)(30,46,116,124)(31,45,117,123)(32,44,118,122)(33,43,119,121)(34,42,120,160)(35,41,81,159)(36,80,82,158)(37,79,83,157)(38,78,84,156)(39,77,85,155)(40,76,86,154), (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)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,140)(49,139)(50,138)(51,137)(52,136)(53,135)(54,134)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,126)(63,125)(64,124)(65,123)(66,122)(67,121)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108) );
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)], [(1,75,87,153),(2,74,88,152),(3,73,89,151),(4,72,90,150),(5,71,91,149),(6,70,92,148),(7,69,93,147),(8,68,94,146),(9,67,95,145),(10,66,96,144),(11,65,97,143),(12,64,98,142),(13,63,99,141),(14,62,100,140),(15,61,101,139),(16,60,102,138),(17,59,103,137),(18,58,104,136),(19,57,105,135),(20,56,106,134),(21,55,107,133),(22,54,108,132),(23,53,109,131),(24,52,110,130),(25,51,111,129),(26,50,112,128),(27,49,113,127),(28,48,114,126),(29,47,115,125),(30,46,116,124),(31,45,117,123),(32,44,118,122),(33,43,119,121),(34,42,120,160),(35,41,81,159),(36,80,82,158),(37,79,83,157),(38,78,84,156),(39,77,85,155),(40,76,86,154)], [(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),(41,147),(42,146),(43,145),(44,144),(45,143),(46,142),(47,141),(48,140),(49,139),(50,138),(51,137),(52,136),(53,135),(54,134),(55,133),(56,132),(57,131),(58,130),(59,129),(60,128),(61,127),(62,126),(63,125),(64,124),(65,123),(66,122),(67,121),(68,160),(69,159),(70,158),(71,157),(72,156),(73,155),(74,154),(75,153),(76,152),(77,151),(78,150),(79,149),(80,148),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(94,120),(95,119),(96,118),(97,117),(98,116),(99,115),(100,114),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108)])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D8 | D10 | D10 | C4○D8 | C5⋊D4 | D20 | D20 | C4○D20 | D40 | D40⋊7C2 |
| kernel | C40⋊29D4 | C40⋊5C4 | D20⋊5C4 | C2×D40 | C20⋊7D4 | C22×C40 | C40 | C2×C20 | C22×C10 | C22×C8 | C20 | C2×C10 | C2×C8 | C22×C4 | C10 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C40⋊29D4 ►in GL4(𝔽41) generated by
| 35 | 26 | 0 | 0 |
| 12 | 23 | 0 | 0 |
| 0 | 0 | 3 | 23 |
| 0 | 0 | 18 | 29 |
| 1 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 |
| 0 | 0 | 21 | 21 |
| 0 | 0 | 18 | 20 |
| 1 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 |
| 0 | 0 | 6 | 35 |
| 0 | 0 | 40 | 35 |
G:=sub<GL(4,GF(41))| [35,12,0,0,26,23,0,0,0,0,3,18,0,0,23,29],[1,9,0,0,0,40,0,0,0,0,21,18,0,0,21,20],[1,9,0,0,0,40,0,0,0,0,6,40,0,0,35,35] >;
C40⋊29D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{29}D_4 % in TeX
G:=Group("C40:29D4"); // GroupNames label
G:=SmallGroup(320,742);
// by ID
G=gap.SmallGroup(320,742);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations