direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C41, C22⋊C123, (C2×C82)⋊C3, SmallGroup(492,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C41 |
Generators and relations for A4×C41
G = < a,b,c,d | a41=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C41
►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 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 158)(110 159)(111 160)(112 161)(113 162)(114 163)(115 164)(116 124)(117 125)(118 126)(119 127)(120 128)(121 129)(122 130)(123 131)
(1 160)(2 161)(3 162)(4 163)(5 164)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 139)(22 140)(23 141)(24 142)(25 143)(26 144)(27 145)(28 146)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 159)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)
(42 125 117)(43 126 118)(44 127 119)(45 128 120)(46 129 121)(47 130 122)(48 131 123)(49 132 83)(50 133 84)(51 134 85)(52 135 86)(53 136 87)(54 137 88)(55 138 89)(56 139 90)(57 140 91)(58 141 92)(59 142 93)(60 143 94)(61 144 95)(62 145 96)(63 146 97)(64 147 98)(65 148 99)(66 149 100)(67 150 101)(68 151 102)(69 152 103)(70 153 104)(71 154 105)(72 155 106)(73 156 107)(74 157 108)(75 158 109)(76 159 110)(77 160 111)(78 161 112)(79 162 113)(80 163 114)(81 164 115)(82 124 116)
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), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,124)(117,125)(118,126)(119,127)(120,128)(121,129)(122,130)(123,131), (1,160)(2,161)(3,162)(4,163)(5,164)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116), (42,125,117)(43,126,118)(44,127,119)(45,128,120)(46,129,121)(47,130,122)(48,131,123)(49,132,83)(50,133,84)(51,134,85)(52,135,86)(53,136,87)(54,137,88)(55,138,89)(56,139,90)(57,140,91)(58,141,92)(59,142,93)(60,143,94)(61,144,95)(62,145,96)(63,146,97)(64,147,98)(65,148,99)(66,149,100)(67,150,101)(68,151,102)(69,152,103)(70,153,104)(71,154,105)(72,155,106)(73,156,107)(74,157,108)(75,158,109)(76,159,110)(77,160,111)(78,161,112)(79,162,113)(80,163,114)(81,164,115)(82,124,116)>;
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)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,124)(117,125)(118,126)(119,127)(120,128)(121,129)(122,130)(123,131), (1,160)(2,161)(3,162)(4,163)(5,164)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116), (42,125,117)(43,126,118)(44,127,119)(45,128,120)(46,129,121)(47,130,122)(48,131,123)(49,132,83)(50,133,84)(51,134,85)(52,135,86)(53,136,87)(54,137,88)(55,138,89)(56,139,90)(57,140,91)(58,141,92)(59,142,93)(60,143,94)(61,144,95)(62,145,96)(63,146,97)(64,147,98)(65,148,99)(66,149,100)(67,150,101)(68,151,102)(69,152,103)(70,153,104)(71,154,105)(72,155,106)(73,156,107)(74,157,108)(75,158,109)(76,159,110)(77,160,111)(78,161,112)(79,162,113)(80,163,114)(81,164,115)(82,124,116) );
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),(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(83,132),(84,133),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149),(101,150),(102,151),(103,152),(104,153),(105,154),(106,155),(107,156),(108,157),(109,158),(110,159),(111,160),(112,161),(113,162),(114,163),(115,164),(116,124),(117,125),(118,126),(119,127),(120,128),(121,129),(122,130),(123,131)], [(1,160),(2,161),(3,162),(4,163),(5,164),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(21,139),(22,140),(23,141),(24,142),(25,143),(26,144),(27,145),(28,146),(29,147),(30,148),(31,149),(32,150),(33,151),(34,152),(35,153),(36,154),(37,155),(38,156),(39,157),(40,158),(41,159),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94),(61,95),(62,96),(63,97),(64,98),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,105),(72,106),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,113),(80,114),(81,115),(82,116)], [(42,125,117),(43,126,118),(44,127,119),(45,128,120),(46,129,121),(47,130,122),(48,131,123),(49,132,83),(50,133,84),(51,134,85),(52,135,86),(53,136,87),(54,137,88),(55,138,89),(56,139,90),(57,140,91),(58,141,92),(59,142,93),(60,143,94),(61,144,95),(62,145,96),(63,146,97),(64,147,98),(65,148,99),(66,149,100),(67,150,101),(68,151,102),(69,152,103),(70,153,104),(71,154,105),(72,155,106),(73,156,107),(74,157,108),(75,158,109),(76,159,110),(77,160,111),(78,161,112),(79,162,113),(80,163,114),(81,164,115),(82,124,116)]])
164 conjugacy classes
| class | 1 | 2 | 3A | 3B | 41A | ··· | 41AN | 82A | ··· | 82AN | 123A | ··· | 123CB |
| order | 1 | 2 | 3 | 3 | 41 | ··· | 41 | 82 | ··· | 82 | 123 | ··· | 123 |
| size | 1 | 3 | 4 | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
164 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C3 | C41 | C123 | A4 | A4×C41 |
| kernel | A4×C41 | C2×C82 | A4 | C22 | C41 | C1 |
| # reps | 1 | 2 | 40 | 80 | 1 | 40 |
Matrix representation of A4×C41 ►in GL3(𝔽739) generated by
| 400 | 0 | 0 |
| 0 | 400 | 0 |
| 0 | 0 | 400 |
| 738 | 0 | 0 |
| 738 | 0 | 1 |
| 738 | 1 | 0 |
| 0 | 1 | 738 |
| 1 | 0 | 738 |
| 0 | 0 | 738 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(739))| [400,0,0,0,400,0,0,0,400],[738,738,738,0,0,1,0,1,0],[0,1,0,1,0,0,738,738,738],[0,0,1,1,0,0,0,1,0] >;
A4×C41 in GAP, Magma, Sage, TeX
A_4\times C_{41} % in TeX
G:=Group("A4xC41"); // GroupNames label
G:=SmallGroup(492,8);
// by ID
G=gap.SmallGroup(492,8);
# by ID
G:=PCGroup([4,-3,-41,-2,2,2954,5907]);
// Polycyclic
G:=Group<a,b,c,d|a^41=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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