metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C41⋊2D4, C22⋊D41, D82⋊2C2, Dic41⋊C2, C2.5D82, C82.5C22, (C2×C82)⋊2C2, SmallGroup(328,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C41⋊D4
G = < a,b,c | a41=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C41⋊D4
►On 164 points
Generators in S164
(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)(124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164)
(1 103 74 142)(2 102 75 141)(3 101 76 140)(4 100 77 139)(5 99 78 138)(6 98 79 137)(7 97 80 136)(8 96 81 135)(9 95 82 134)(10 94 42 133)(11 93 43 132)(12 92 44 131)(13 91 45 130)(14 90 46 129)(15 89 47 128)(16 88 48 127)(17 87 49 126)(18 86 50 125)(19 85 51 124)(20 84 52 164)(21 83 53 163)(22 123 54 162)(23 122 55 161)(24 121 56 160)(25 120 57 159)(26 119 58 158)(27 118 59 157)(28 117 60 156)(29 116 61 155)(30 115 62 154)(31 114 63 153)(32 113 64 152)(33 112 65 151)(34 111 66 150)(35 110 67 149)(36 109 68 148)(37 108 69 147)(38 107 70 146)(39 106 71 145)(40 105 72 144)(41 104 73 143)
(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(83 162)(84 161)(85 160)(86 159)(87 158)(88 157)(89 156)(90 155)(91 154)(92 153)(93 152)(94 151)(95 150)(96 149)(97 148)(98 147)(99 146)(100 145)(101 144)(102 143)(103 142)(104 141)(105 140)(106 139)(107 138)(108 137)(109 136)(110 135)(111 134)(112 133)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(121 124)(122 164)(123 163)
G:=sub<Sym(164)| (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)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,103,74,142)(2,102,75,141)(3,101,76,140)(4,100,77,139)(5,99,78,138)(6,98,79,137)(7,97,80,136)(8,96,81,135)(9,95,82,134)(10,94,42,133)(11,93,43,132)(12,92,44,131)(13,91,45,130)(14,90,46,129)(15,89,47,128)(16,88,48,127)(17,87,49,126)(18,86,50,125)(19,85,51,124)(20,84,52,164)(21,83,53,163)(22,123,54,162)(23,122,55,161)(24,121,56,160)(25,120,57,159)(26,119,58,158)(27,118,59,157)(28,117,60,156)(29,116,61,155)(30,115,62,154)(31,114,63,153)(32,113,64,152)(33,112,65,151)(34,111,66,150)(35,110,67,149)(36,109,68,148)(37,108,69,147)(38,107,70,146)(39,106,71,145)(40,105,72,144)(41,104,73,143), (2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(83,162)(84,161)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137)(109,136)(110,135)(111,134)(112,133)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,164)(123,163)>;
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)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,103,74,142)(2,102,75,141)(3,101,76,140)(4,100,77,139)(5,99,78,138)(6,98,79,137)(7,97,80,136)(8,96,81,135)(9,95,82,134)(10,94,42,133)(11,93,43,132)(12,92,44,131)(13,91,45,130)(14,90,46,129)(15,89,47,128)(16,88,48,127)(17,87,49,126)(18,86,50,125)(19,85,51,124)(20,84,52,164)(21,83,53,163)(22,123,54,162)(23,122,55,161)(24,121,56,160)(25,120,57,159)(26,119,58,158)(27,118,59,157)(28,117,60,156)(29,116,61,155)(30,115,62,154)(31,114,63,153)(32,113,64,152)(33,112,65,151)(34,111,66,150)(35,110,67,149)(36,109,68,148)(37,108,69,147)(38,107,70,146)(39,106,71,145)(40,105,72,144)(41,104,73,143), (2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(83,162)(84,161)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137)(109,136)(110,135)(111,134)(112,133)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,164)(123,163) );
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),(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164)], [(1,103,74,142),(2,102,75,141),(3,101,76,140),(4,100,77,139),(5,99,78,138),(6,98,79,137),(7,97,80,136),(8,96,81,135),(9,95,82,134),(10,94,42,133),(11,93,43,132),(12,92,44,131),(13,91,45,130),(14,90,46,129),(15,89,47,128),(16,88,48,127),(17,87,49,126),(18,86,50,125),(19,85,51,124),(20,84,52,164),(21,83,53,163),(22,123,54,162),(23,122,55,161),(24,121,56,160),(25,120,57,159),(26,119,58,158),(27,118,59,157),(28,117,60,156),(29,116,61,155),(30,115,62,154),(31,114,63,153),(32,113,64,152),(33,112,65,151),(34,111,66,150),(35,110,67,149),(36,109,68,148),(37,108,69,147),(38,107,70,146),(39,106,71,145),(40,105,72,144),(41,104,73,143)], [(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(83,162),(84,161),(85,160),(86,159),(87,158),(88,157),(89,156),(90,155),(91,154),(92,153),(93,152),(94,151),(95,150),(96,149),(97,148),(98,147),(99,146),(100,145),(101,144),(102,143),(103,142),(104,141),(105,140),(106,139),(107,138),(108,137),(109,136),(110,135),(111,134),(112,133),(113,132),(114,131),(115,130),(116,129),(117,128),(118,127),(119,126),(120,125),(121,124),(122,164),(123,163)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 41A | ··· | 41T | 82A | ··· | 82BH |
| order | 1 | 2 | 2 | 2 | 4 | 41 | ··· | 41 | 82 | ··· | 82 |
| size | 1 | 1 | 2 | 82 | 82 | 2 | ··· | 2 | 2 | ··· | 2 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D41 | D82 | C41⋊D4 |
| kernel | C41⋊D4 | Dic41 | D82 | C2×C82 | C41 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 20 | 20 | 40 |
Matrix representation of C41⋊D4 ►in GL2(𝔽821) generated by
| 635 | 1 |
| 476 | 302 |
| 71 | 62 |
| 210 | 750 |
| 461 | 534 |
| 340 | 360 |
G:=sub<GL(2,GF(821))| [635,476,1,302],[71,210,62,750],[461,340,534,360] >;
C41⋊D4 in GAP, Magma, Sage, TeX
C_{41}\rtimes D_4 % in TeX
G:=Group("C41:D4"); // GroupNames label
G:=SmallGroup(328,8);
// by ID
G=gap.SmallGroup(328,8);
# by ID
G:=PCGroup([4,-2,-2,-2,-41,49,5123]);
// Polycyclic
G:=Group<a,b,c|a^41=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export