direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D41, C164⋊2C2, C2.1D82, D82.2C2, Dic41⋊2C2, C82.2C22, C41⋊2(C2×C4), SmallGroup(328,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C4×D41 |
Generators and relations for C4×D41
G = < a,b,c | a4=b41=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D41
►On 164 points
Generators in S164
(1 130 57 86)(2 131 58 87)(3 132 59 88)(4 133 60 89)(5 134 61 90)(6 135 62 91)(7 136 63 92)(8 137 64 93)(9 138 65 94)(10 139 66 95)(11 140 67 96)(12 141 68 97)(13 142 69 98)(14 143 70 99)(15 144 71 100)(16 145 72 101)(17 146 73 102)(18 147 74 103)(19 148 75 104)(20 149 76 105)(21 150 77 106)(22 151 78 107)(23 152 79 108)(24 153 80 109)(25 154 81 110)(26 155 82 111)(27 156 42 112)(28 157 43 113)(29 158 44 114)(30 159 45 115)(31 160 46 116)(32 161 47 117)(33 162 48 118)(34 163 49 119)(35 164 50 120)(36 124 51 121)(37 125 52 122)(38 126 53 123)(39 127 54 83)(40 128 55 84)(41 129 56 85)
(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 41)(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)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(72 82)(73 81)(74 80)(75 79)(76 78)(83 88)(84 87)(85 86)(89 123)(90 122)(91 121)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(136 164)(137 163)(138 162)(139 161)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)
G:=sub<Sym(164)| (1,130,57,86)(2,131,58,87)(3,132,59,88)(4,133,60,89)(5,134,61,90)(6,135,62,91)(7,136,63,92)(8,137,64,93)(9,138,65,94)(10,139,66,95)(11,140,67,96)(12,141,68,97)(13,142,69,98)(14,143,70,99)(15,144,71,100)(16,145,72,101)(17,146,73,102)(18,147,74,103)(19,148,75,104)(20,149,76,105)(21,150,77,106)(22,151,78,107)(23,152,79,108)(24,153,80,109)(25,154,81,110)(26,155,82,111)(27,156,42,112)(28,157,43,113)(29,158,44,114)(30,159,45,115)(31,160,46,116)(32,161,47,117)(33,162,48,118)(34,163,49,119)(35,164,50,120)(36,124,51,121)(37,125,52,122)(38,126,53,123)(39,127,54,83)(40,128,55,84)(41,129,56,85), (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), (1,41)(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)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(72,82)(73,81)(74,80)(75,79)(76,78)(83,88)(84,87)(85,86)(89,123)(90,122)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)>;
G:=Group( (1,130,57,86)(2,131,58,87)(3,132,59,88)(4,133,60,89)(5,134,61,90)(6,135,62,91)(7,136,63,92)(8,137,64,93)(9,138,65,94)(10,139,66,95)(11,140,67,96)(12,141,68,97)(13,142,69,98)(14,143,70,99)(15,144,71,100)(16,145,72,101)(17,146,73,102)(18,147,74,103)(19,148,75,104)(20,149,76,105)(21,150,77,106)(22,151,78,107)(23,152,79,108)(24,153,80,109)(25,154,81,110)(26,155,82,111)(27,156,42,112)(28,157,43,113)(29,158,44,114)(30,159,45,115)(31,160,46,116)(32,161,47,117)(33,162,48,118)(34,163,49,119)(35,164,50,120)(36,124,51,121)(37,125,52,122)(38,126,53,123)(39,127,54,83)(40,128,55,84)(41,129,56,85), (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), (1,41)(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)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(72,82)(73,81)(74,80)(75,79)(76,78)(83,88)(84,87)(85,86)(89,123)(90,122)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151) );
G=PermutationGroup([[(1,130,57,86),(2,131,58,87),(3,132,59,88),(4,133,60,89),(5,134,61,90),(6,135,62,91),(7,136,63,92),(8,137,64,93),(9,138,65,94),(10,139,66,95),(11,140,67,96),(12,141,68,97),(13,142,69,98),(14,143,70,99),(15,144,71,100),(16,145,72,101),(17,146,73,102),(18,147,74,103),(19,148,75,104),(20,149,76,105),(21,150,77,106),(22,151,78,107),(23,152,79,108),(24,153,80,109),(25,154,81,110),(26,155,82,111),(27,156,42,112),(28,157,43,113),(29,158,44,114),(30,159,45,115),(31,160,46,116),(32,161,47,117),(33,162,48,118),(34,163,49,119),(35,164,50,120),(36,124,51,121),(37,125,52,122),(38,126,53,123),(39,127,54,83),(40,128,55,84),(41,129,56,85)], [(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)], [(1,41),(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),(42,71),(43,70),(44,69),(45,68),(46,67),(47,66),(48,65),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57),(72,82),(73,81),(74,80),(75,79),(76,78),(83,88),(84,87),(85,86),(89,123),(90,122),(91,121),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(124,135),(125,134),(126,133),(127,132),(128,131),(129,130),(136,164),(137,163),(138,162),(139,161),(140,160),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 41A | ··· | 41T | 82A | ··· | 82T | 164A | ··· | 164AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 41 | ··· | 41 | 82 | ··· | 82 | 164 | ··· | 164 |
| size | 1 | 1 | 41 | 41 | 1 | 1 | 41 | 41 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D41 | D82 | C4×D41 |
| kernel | C4×D41 | Dic41 | C164 | D82 | D41 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 40 |
Matrix representation of C4×D41 ►in GL3(𝔽821) generated by
| 295 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 820 | 532 |
| 820 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(821))| [295,0,0,0,1,0,0,0,1],[1,0,0,0,0,820,0,1,532],[820,0,0,0,0,1,0,1,0] >;
C4×D41 in GAP, Magma, Sage, TeX
C_4\times D_{41} % in TeX
G:=Group("C4xD41"); // GroupNames label
G:=SmallGroup(328,5);
// by ID
G=gap.SmallGroup(328,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-41,21,5123]);
// Polycyclic
G:=Group<a,b,c|a^4=b^41=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export