direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C37, C22⋊C111, (C2×C74)⋊1C3, SmallGroup(444,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C37 |
Generators and relations for A4×C37
G = < a,b,c,d | a37=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C37
►On 148 points
Generators in S148
(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)
(1 144)(2 145)(3 146)(4 147)(5 148)(6 112)(7 113)(8 114)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 141)(36 142)(37 143)(38 107)(39 108)(40 109)(41 110)(42 111)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)
(38 107 125)(39 108 126)(40 109 127)(41 110 128)(42 111 129)(43 75 130)(44 76 131)(45 77 132)(46 78 133)(47 79 134)(48 80 135)(49 81 136)(50 82 137)(51 83 138)(52 84 139)(53 85 140)(54 86 141)(55 87 142)(56 88 143)(57 89 144)(58 90 145)(59 91 146)(60 92 147)(61 93 148)(62 94 112)(63 95 113)(64 96 114)(65 97 115)(66 98 116)(67 99 117)(68 100 118)(69 101 119)(70 102 120)(71 103 121)(72 104 122)(73 105 123)(74 106 124)
G:=sub<Sym(148)| (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), (1,144)(2,145)(3,146)(4,147)(5,148)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,141)(36,142)(37,143)(38,107)(39,108)(40,109)(41,110)(42,111)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129), (38,107,125)(39,108,126)(40,109,127)(41,110,128)(42,111,129)(43,75,130)(44,76,131)(45,77,132)(46,78,133)(47,79,134)(48,80,135)(49,81,136)(50,82,137)(51,83,138)(52,84,139)(53,85,140)(54,86,141)(55,87,142)(56,88,143)(57,89,144)(58,90,145)(59,91,146)(60,92,147)(61,93,148)(62,94,112)(63,95,113)(64,96,114)(65,97,115)(66,98,116)(67,99,117)(68,100,118)(69,101,119)(70,102,120)(71,103,121)(72,104,122)(73,105,123)(74,106,124)>;
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), (1,144)(2,145)(3,146)(4,147)(5,148)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,137)(32,138)(33,139)(34,140)(35,141)(36,142)(37,143)(38,107)(39,108)(40,109)(41,110)(42,111)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129), (38,107,125)(39,108,126)(40,109,127)(41,110,128)(42,111,129)(43,75,130)(44,76,131)(45,77,132)(46,78,133)(47,79,134)(48,80,135)(49,81,136)(50,82,137)(51,83,138)(52,84,139)(53,85,140)(54,86,141)(55,87,142)(56,88,143)(57,89,144)(58,90,145)(59,91,146)(60,92,147)(61,93,148)(62,94,112)(63,95,113)(64,96,114)(65,97,115)(66,98,116)(67,99,117)(68,100,118)(69,101,119)(70,102,120)(71,103,121)(72,104,122)(73,105,123)(74,106,124) );
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)], [(1,144),(2,145),(3,146),(4,147),(5,148),(6,112),(7,113),(8,114),(9,115),(10,116),(11,117),(12,118),(13,119),(14,120),(15,121),(16,122),(17,123),(18,124),(19,125),(20,126),(21,127),(22,128),(23,129),(24,130),(25,131),(26,132),(27,133),(28,134),(29,135),(30,136),(31,137),(32,138),(33,139),(34,140),(35,141),(36,142),(37,143),(38,107),(39,108),(40,109),(41,110),(42,111),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(73,105),(74,106)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135),(81,136),(82,137),(83,138),(84,139),(85,140),(86,141),(87,142),(88,143),(89,144),(90,145),(91,146),(92,147),(93,148),(94,112),(95,113),(96,114),(97,115),(98,116),(99,117),(100,118),(101,119),(102,120),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,127),(110,128),(111,129)], [(38,107,125),(39,108,126),(40,109,127),(41,110,128),(42,111,129),(43,75,130),(44,76,131),(45,77,132),(46,78,133),(47,79,134),(48,80,135),(49,81,136),(50,82,137),(51,83,138),(52,84,139),(53,85,140),(54,86,141),(55,87,142),(56,88,143),(57,89,144),(58,90,145),(59,91,146),(60,92,147),(61,93,148),(62,94,112),(63,95,113),(64,96,114),(65,97,115),(66,98,116),(67,99,117),(68,100,118),(69,101,119),(70,102,120),(71,103,121),(72,104,122),(73,105,123),(74,106,124)]])
148 conjugacy classes
| class | 1 | 2 | 3A | 3B | 37A | ··· | 37AJ | 74A | ··· | 74AJ | 111A | ··· | 111BT |
| order | 1 | 2 | 3 | 3 | 37 | ··· | 37 | 74 | ··· | 74 | 111 | ··· | 111 |
| size | 1 | 3 | 4 | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
148 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C3 | C37 | C111 | A4 | A4×C37 |
| kernel | A4×C37 | C2×C74 | A4 | C22 | C37 | C1 |
| # reps | 1 | 2 | 36 | 72 | 1 | 36 |
Matrix representation of A4×C37 ►in GL3(𝔽223) generated by
| 136 | 0 | 0 |
| 0 | 136 | 0 |
| 0 | 0 | 136 |
| 222 | 0 | 0 |
| 222 | 0 | 1 |
| 222 | 1 | 0 |
| 0 | 1 | 222 |
| 1 | 0 | 222 |
| 0 | 0 | 222 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(223))| [136,0,0,0,136,0,0,0,136],[222,222,222,0,0,1,0,1,0],[0,1,0,1,0,0,222,222,222],[0,0,1,1,0,0,0,1,0] >;
A4×C37 in GAP, Magma, Sage, TeX
A_4\times C_{37} % in TeX
G:=Group("A4xC37"); // GroupNames label
G:=SmallGroup(444,13);
// by ID
G=gap.SmallGroup(444,13);
# by ID
G:=PCGroup([4,-3,-37,-2,2,2666,5331]);
// Polycyclic
G:=Group<a,b,c,d|a^37=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
Export