metabelian, supersoluble, monomial, A-group
Aliases: C3⋊D45, C9⋊D15, C45⋊1S3, C15⋊1D9, C32.3D15, C5⋊(C9⋊S3), (C3×C9)⋊3D5, (C3×C45)⋊3C2, C3.(C3⋊D15), (C3×C15).3S3, C15.1(C3⋊S3), SmallGroup(270,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C45 — C3⋊D45 |
Generators and relations for C3⋊D45
G = < a,b,c | a3=b45=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D45
►On 135 points
Generators in S135
(1 126 83)(2 127 84)(3 128 85)(4 129 86)(5 130 87)(6 131 88)(7 132 89)(8 133 90)(9 134 46)(10 135 47)(11 91 48)(12 92 49)(13 93 50)(14 94 51)(15 95 52)(16 96 53)(17 97 54)(18 98 55)(19 99 56)(20 100 57)(21 101 58)(22 102 59)(23 103 60)(24 104 61)(25 105 62)(26 106 63)(27 107 64)(28 108 65)(29 109 66)(30 110 67)(31 111 68)(32 112 69)(33 113 70)(34 114 71)(35 115 72)(36 116 73)(37 117 74)(38 118 75)(39 119 76)(40 120 77)(41 121 78)(42 122 79)(43 123 80)(44 124 81)(45 125 82)
(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)
(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(46 118)(47 117)(48 116)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 108)(57 107)(58 106)(59 105)(60 104)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 135)(75 134)(76 133)(77 132)(78 131)(79 130)(80 129)(81 128)(82 127)(83 126)(84 125)(85 124)(86 123)(87 122)(88 121)(89 120)(90 119)
G:=sub<Sym(135)| (1,126,83)(2,127,84)(3,128,85)(4,129,86)(5,130,87)(6,131,88)(7,132,89)(8,133,90)(9,134,46)(10,135,47)(11,91,48)(12,92,49)(13,93,50)(14,94,51)(15,95,52)(16,96,53)(17,97,54)(18,98,55)(19,99,56)(20,100,57)(21,101,58)(22,102,59)(23,103,60)(24,104,61)(25,105,62)(26,106,63)(27,107,64)(28,108,65)(29,109,66)(30,110,67)(31,111,68)(32,112,69)(33,113,70)(34,114,71)(35,115,72)(36,116,73)(37,117,74)(38,118,75)(39,119,76)(40,120,77)(41,121,78)(42,122,79)(43,123,80)(44,124,81)(45,125,82), (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), (2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(46,118)(47,117)(48,116)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,108)(57,107)(58,106)(59,105)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)>;
G:=Group( (1,126,83)(2,127,84)(3,128,85)(4,129,86)(5,130,87)(6,131,88)(7,132,89)(8,133,90)(9,134,46)(10,135,47)(11,91,48)(12,92,49)(13,93,50)(14,94,51)(15,95,52)(16,96,53)(17,97,54)(18,98,55)(19,99,56)(20,100,57)(21,101,58)(22,102,59)(23,103,60)(24,104,61)(25,105,62)(26,106,63)(27,107,64)(28,108,65)(29,109,66)(30,110,67)(31,111,68)(32,112,69)(33,113,70)(34,114,71)(35,115,72)(36,116,73)(37,117,74)(38,118,75)(39,119,76)(40,120,77)(41,121,78)(42,122,79)(43,123,80)(44,124,81)(45,125,82), (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), (2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(46,118)(47,117)(48,116)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,108)(57,107)(58,106)(59,105)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119) );
G=PermutationGroup([[(1,126,83),(2,127,84),(3,128,85),(4,129,86),(5,130,87),(6,131,88),(7,132,89),(8,133,90),(9,134,46),(10,135,47),(11,91,48),(12,92,49),(13,93,50),(14,94,51),(15,95,52),(16,96,53),(17,97,54),(18,98,55),(19,99,56),(20,100,57),(21,101,58),(22,102,59),(23,103,60),(24,104,61),(25,105,62),(26,106,63),(27,107,64),(28,108,65),(29,109,66),(30,110,67),(31,111,68),(32,112,69),(33,113,70),(34,114,71),(35,115,72),(36,116,73),(37,117,74),(38,118,75),(39,119,76),(40,120,77),(41,121,78),(42,122,79),(43,123,80),(44,124,81),(45,125,82)], [(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)], [(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(46,118),(47,117),(48,116),(49,115),(50,114),(51,113),(52,112),(53,111),(54,110),(55,109),(56,108),(57,107),(58,106),(59,105),(60,104),(61,103),(62,102),(63,101),(64,100),(65,99),(66,98),(67,97),(68,96),(69,95),(70,94),(71,93),(72,92),(73,91),(74,135),(75,134),(76,133),(77,132),(78,131),(79,130),(80,129),(81,128),(82,127),(83,126),(84,125),(85,124),(86,123),(87,122),(88,121),(89,120),(90,119)]])
69 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 5A | 5B | 9A | ··· | 9I | 15A | ··· | 15P | 45A | ··· | 45AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 5 | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 135 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
69 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | S3 | D5 | D9 | D15 | D15 | D45 |
| kernel | C3⋊D45 | C3×C45 | C45 | C3×C15 | C3×C9 | C15 | C9 | C32 | C3 |
| # reps | 1 | 1 | 3 | 1 | 2 | 9 | 12 | 4 | 36 |
Matrix representation of C3⋊D45 ►in GL4(𝔽181) generated by
| 180 | 180 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 180 | 180 |
| 118 | 28 | 0 | 0 |
| 153 | 90 | 0 | 0 |
| 0 | 0 | 4 | 54 |
| 0 | 0 | 127 | 131 |
| 1 | 0 | 0 | 0 |
| 180 | 180 | 0 | 0 |
| 0 | 0 | 112 | 29 |
| 0 | 0 | 98 | 69 |
G:=sub<GL(4,GF(181))| [180,1,0,0,180,0,0,0,0,0,0,180,0,0,1,180],[118,153,0,0,28,90,0,0,0,0,4,127,0,0,54,131],[1,180,0,0,0,180,0,0,0,0,112,98,0,0,29,69] >;
C3⋊D45 in GAP, Magma, Sage, TeX
C_3\rtimes D_{45} % in TeX
G:=Group("C3:D45"); // GroupNames label
G:=SmallGroup(270,18);
// by ID
G=gap.SmallGroup(270,18);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-3,941,906,182,1443,4504]);
// Polycyclic
G:=Group<a,b,c|a^3=b^45=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export