metabelian, soluble, monomial, A-group
Aliases: C37⋊A4, (C2×C74)⋊2C3, C22⋊(C37⋊C3), SmallGroup(444,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C74 — C37⋊A4 |
Generators and relations for C37⋊A4
G = < a,b,c,d | a37=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a10, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C37⋊A4
►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 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 112)(72 113)(73 114)(74 115)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(81 139)(82 140)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)
(2 27 11)(3 16 21)(4 5 31)(6 20 14)(7 9 24)(8 35 34)(10 13 17)(12 28 37)(15 32 30)(18 36 23)(19 25 33)(22 29 26)(38 143 84)(39 132 94)(40 121 104)(41 147 77)(42 136 87)(43 125 97)(44 114 107)(45 140 80)(46 129 90)(47 118 100)(48 144 110)(49 133 83)(50 122 93)(51 148 103)(52 137 76)(53 126 86)(54 115 96)(55 141 106)(56 130 79)(57 119 89)(58 145 99)(59 134 109)(60 123 82)(61 112 92)(62 138 102)(63 127 75)(64 116 85)(65 142 95)(66 131 105)(67 120 78)(68 146 88)(69 135 98)(70 124 108)(71 113 81)(72 139 91)(73 128 101)(74 117 111)
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), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,112)(72,113)(73,114)(74,115), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)(38,143,84)(39,132,94)(40,121,104)(41,147,77)(42,136,87)(43,125,97)(44,114,107)(45,140,80)(46,129,90)(47,118,100)(48,144,110)(49,133,83)(50,122,93)(51,148,103)(52,137,76)(53,126,86)(54,115,96)(55,141,106)(56,130,79)(57,119,89)(58,145,99)(59,134,109)(60,123,82)(61,112,92)(62,138,102)(63,127,75)(64,116,85)(65,142,95)(66,131,105)(67,120,78)(68,146,88)(69,135,98)(70,124,108)(71,113,81)(72,139,91)(73,128,101)(74,117,111)>;
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), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,112)(72,113)(73,114)(74,115), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)(38,143,84)(39,132,94)(40,121,104)(41,147,77)(42,136,87)(43,125,97)(44,114,107)(45,140,80)(46,129,90)(47,118,100)(48,144,110)(49,133,83)(50,122,93)(51,148,103)(52,137,76)(53,126,86)(54,115,96)(55,141,106)(56,130,79)(57,119,89)(58,145,99)(59,134,109)(60,123,82)(61,112,92)(62,138,102)(63,127,75)(64,116,85)(65,142,95)(66,131,105)(67,120,78)(68,146,88)(69,135,98)(70,124,108)(71,113,81)(72,139,91)(73,128,101)(74,117,111) );
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)], [(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,81),(32,82),(33,83),(34,84),(35,85),(36,86),(37,87),(38,116),(39,117),(40,118),(41,119),(42,120),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,127),(50,128),(51,129),(52,130),(53,131),(54,132),(55,133),(56,134),(57,135),(58,136),(59,137),(60,138),(61,139),(62,140),(63,141),(64,142),(65,143),(66,144),(67,145),(68,146),(69,147),(70,148),(71,112),(72,113),(73,114),(74,115)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(75,133),(76,134),(77,135),(78,136),(79,137),(80,138),(81,139),(82,140),(83,141),(84,142),(85,143),(86,144),(87,145),(88,146),(89,147),(90,148),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117),(97,118),(98,119),(99,120),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132)], [(2,27,11),(3,16,21),(4,5,31),(6,20,14),(7,9,24),(8,35,34),(10,13,17),(12,28,37),(15,32,30),(18,36,23),(19,25,33),(22,29,26),(38,143,84),(39,132,94),(40,121,104),(41,147,77),(42,136,87),(43,125,97),(44,114,107),(45,140,80),(46,129,90),(47,118,100),(48,144,110),(49,133,83),(50,122,93),(51,148,103),(52,137,76),(53,126,86),(54,115,96),(55,141,106),(56,130,79),(57,119,89),(58,145,99),(59,134,109),(60,123,82),(61,112,92),(62,138,102),(63,127,75),(64,116,85),(65,142,95),(66,131,105),(67,120,78),(68,146,88),(69,135,98),(70,124,108),(71,113,81),(72,139,91),(73,128,101),(74,117,111)]])
52 conjugacy classes
| class | 1 | 2 | 3A | 3B | 37A | ··· | 37L | 74A | ··· | 74AJ |
| order | 1 | 2 | 3 | 3 | 37 | ··· | 37 | 74 | ··· | 74 |
| size | 1 | 3 | 148 | 148 | 3 | ··· | 3 | 3 | ··· | 3 |
52 irreducible representations
| dim | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||
| image | C1 | C3 | A4 | C37⋊C3 | C37⋊A4 |
| kernel | C37⋊A4 | C2×C74 | C37 | C22 | C1 |
| # reps | 1 | 2 | 1 | 12 | 36 |
Matrix representation of C37⋊A4 ►in GL3(𝔽223) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 63 | 211 |
| 221 | 153 | 191 |
| 191 | 212 | 91 |
| 91 | 126 | 12 |
| 183 | 8 | 71 |
| 71 | 196 | 48 |
| 48 | 196 | 66 |
| 1 | 0 | 0 |
| 186 | 218 | 72 |
| 212 | 9 | 4 |
G:=sub<GL(3,GF(223))| [0,0,1,1,0,63,0,1,211],[221,191,91,153,212,126,191,91,12],[183,71,48,8,196,196,71,48,66],[1,186,212,0,218,9,0,72,4] >;
C37⋊A4 in GAP, Magma, Sage, TeX
C_{37}\rtimes A_4 % in TeX
G:=Group("C37:A4"); // GroupNames label
G:=SmallGroup(444,14);
// by ID
G=gap.SmallGroup(444,14);
# by ID
G:=PCGroup([4,-3,-2,2,-37,49,110,4995]);
// Polycyclic
G:=Group<a,b,c,d|a^37=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^10,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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