direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D44, C33⋊5D4, C44⋊1C6, C132⋊3C2, D22⋊1C6, C12⋊3D11, C6.15D22, C66.15C22, C4⋊(C3×D11), C11⋊1(C3×D4), (C6×D11)⋊4C2, C22.3(C2×C6), C2.4(C6×D11), SmallGroup(264,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D44
G = < a,b,c | a3=b44=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D44
►On 132 points
Generators in S132
(1 89 50)(2 90 51)(3 91 52)(4 92 53)(5 93 54)(6 94 55)(7 95 56)(8 96 57)(9 97 58)(10 98 59)(11 99 60)(12 100 61)(13 101 62)(14 102 63)(15 103 64)(16 104 65)(17 105 66)(18 106 67)(19 107 68)(20 108 69)(21 109 70)(22 110 71)(23 111 72)(24 112 73)(25 113 74)(26 114 75)(27 115 76)(28 116 77)(29 117 78)(30 118 79)(31 119 80)(32 120 81)(33 121 82)(34 122 83)(35 123 84)(36 124 85)(37 125 86)(38 126 87)(39 127 88)(40 128 45)(41 129 46)(42 130 47)(43 131 48)(44 132 49)
(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)
(1 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(45 54)(46 53)(47 52)(48 51)(49 50)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(89 132)(90 131)(91 130)(92 129)(93 128)(94 127)(95 126)(96 125)(97 124)(98 123)(99 122)(100 121)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)
G:=sub<Sym(132)| (1,89,50)(2,90,51)(3,91,52)(4,92,53)(5,93,54)(6,94,55)(7,95,56)(8,96,57)(9,97,58)(10,98,59)(11,99,60)(12,100,61)(13,101,62)(14,102,63)(15,103,64)(16,104,65)(17,105,66)(18,106,67)(19,107,68)(20,108,69)(21,109,70)(22,110,71)(23,111,72)(24,112,73)(25,113,74)(26,114,75)(27,115,76)(28,116,77)(29,117,78)(30,118,79)(31,119,80)(32,120,81)(33,121,82)(34,122,83)(35,123,84)(36,124,85)(37,125,86)(38,126,87)(39,127,88)(40,128,45)(41,129,46)(42,130,47)(43,131,48)(44,132,49), (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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(45,54)(46,53)(47,52)(48,51)(49,50)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(89,132)(90,131)(91,130)(92,129)(93,128)(94,127)(95,126)(96,125)(97,124)(98,123)(99,122)(100,121)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)>;
G:=Group( (1,89,50)(2,90,51)(3,91,52)(4,92,53)(5,93,54)(6,94,55)(7,95,56)(8,96,57)(9,97,58)(10,98,59)(11,99,60)(12,100,61)(13,101,62)(14,102,63)(15,103,64)(16,104,65)(17,105,66)(18,106,67)(19,107,68)(20,108,69)(21,109,70)(22,110,71)(23,111,72)(24,112,73)(25,113,74)(26,114,75)(27,115,76)(28,116,77)(29,117,78)(30,118,79)(31,119,80)(32,120,81)(33,121,82)(34,122,83)(35,123,84)(36,124,85)(37,125,86)(38,126,87)(39,127,88)(40,128,45)(41,129,46)(42,130,47)(43,131,48)(44,132,49), (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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(45,54)(46,53)(47,52)(48,51)(49,50)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(89,132)(90,131)(91,130)(92,129)(93,128)(94,127)(95,126)(96,125)(97,124)(98,123)(99,122)(100,121)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111) );
G=PermutationGroup([[(1,89,50),(2,90,51),(3,91,52),(4,92,53),(5,93,54),(6,94,55),(7,95,56),(8,96,57),(9,97,58),(10,98,59),(11,99,60),(12,100,61),(13,101,62),(14,102,63),(15,103,64),(16,104,65),(17,105,66),(18,106,67),(19,107,68),(20,108,69),(21,109,70),(22,110,71),(23,111,72),(24,112,73),(25,113,74),(26,114,75),(27,115,76),(28,116,77),(29,117,78),(30,118,79),(31,119,80),(32,120,81),(33,121,82),(34,122,83),(35,123,84),(36,124,85),(37,125,86),(38,126,87),(39,127,88),(40,128,45),(41,129,46),(42,130,47),(43,131,48),(44,132,49)], [(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)], [(1,44),(2,43),(3,42),(4,41),(5,40),(6,39),(7,38),(8,37),(9,36),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,24),(22,23),(45,54),(46,53),(47,52),(48,51),(49,50),(55,88),(56,87),(57,86),(58,85),(59,84),(60,83),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(89,132),(90,131),(91,130),(92,129),(93,128),(94,127),(95,126),(96,125),(97,124),(98,123),(99,122),(100,121),(101,120),(102,119),(103,118),(104,117),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22E | 33A | ··· | 33J | 44A | ··· | 44J | 66A | ··· | 66J | 132A | ··· | 132T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 22 | 22 | 1 | 1 | 2 | 1 | 1 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D11 | C3×D4 | D22 | C3×D11 | D44 | C6×D11 | C3×D44 |
| kernel | C3×D44 | C132 | C6×D11 | D44 | C44 | D22 | C33 | C12 | C11 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 5 | 2 | 5 | 10 | 10 | 10 | 20 |
Matrix representation of C3×D44 ►in GL2(𝔽43) generated by
| 36 | 0 |
| 0 | 36 |
| 5 | 18 |
| 40 | 41 |
| 41 | 42 |
| 3 | 2 |
G:=sub<GL(2,GF(43))| [36,0,0,36],[5,40,18,41],[41,3,42,2] >;
C3×D44 in GAP, Magma, Sage, TeX
C_3\times D_{44} % in TeX
G:=Group("C3xD44"); // GroupNames label
G:=SmallGroup(264,15);
// by ID
G=gap.SmallGroup(264,15);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-11,141,66,6004]);
// Polycyclic
G:=Group<a,b,c|a^3=b^44=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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