direct product, metabelian, supersoluble, monomial, A-group
Aliases: C5×C9⋊S3, C45⋊2S3, C15⋊3D9, C9⋊(C5×S3), C3⋊(C5×D9), (C3×C45)⋊5C2, (C3×C9)⋊3C10, (C3×C15).6S3, C15.3(C3⋊S3), C32.3(C5×S3), C3.(C5×C3⋊S3), SmallGroup(270,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C5×C9⋊S3 |
Generators and relations for C5×C9⋊S3
G = < a,b,c,d | a5=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Smallest permutation representation of C5×C9⋊S3
►On 135 points
Generators in S135
(1 17 114 87 60)(2 18 115 88 61)(3 10 116 89 62)(4 11 117 90 63)(5 12 109 82 55)(6 13 110 83 56)(7 14 111 84 57)(8 15 112 85 58)(9 16 113 86 59)(19 122 95 68 41)(20 123 96 69 42)(21 124 97 70 43)(22 125 98 71 44)(23 126 99 72 45)(24 118 91 64 37)(25 119 92 65 38)(26 120 93 66 39)(27 121 94 67 40)(28 133 106 79 52)(29 134 107 80 53)(30 135 108 81 54)(31 127 100 73 46)(32 128 101 74 47)(33 129 102 75 48)(34 130 103 76 49)(35 131 104 77 50)(36 132 105 78 51)
(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)
(1 28 42)(2 29 43)(3 30 44)(4 31 45)(5 32 37)(6 33 38)(7 34 39)(8 35 40)(9 36 41)(10 135 22)(11 127 23)(12 128 24)(13 129 25)(14 130 26)(15 131 27)(16 132 19)(17 133 20)(18 134 21)(46 72 63)(47 64 55)(48 65 56)(49 66 57)(50 67 58)(51 68 59)(52 69 60)(53 70 61)(54 71 62)(73 99 90)(74 91 82)(75 92 83)(76 93 84)(77 94 85)(78 95 86)(79 96 87)(80 97 88)(81 98 89)(100 126 117)(101 118 109)(102 119 110)(103 120 111)(104 121 112)(105 122 113)(106 123 114)(107 124 115)(108 125 116)
(2 9)(3 8)(4 7)(5 6)(10 15)(11 14)(12 13)(16 18)(19 134)(20 133)(21 132)(22 131)(23 130)(24 129)(25 128)(26 127)(27 135)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 45)(35 44)(36 43)(46 66)(47 65)(48 64)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 56)(57 63)(58 62)(59 61)(73 93)(74 92)(75 91)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 83)(84 90)(85 89)(86 88)(100 120)(101 119)(102 118)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 110)(111 117)(112 116)(113 115)
G:=sub<Sym(135)| (1,17,114,87,60)(2,18,115,88,61)(3,10,116,89,62)(4,11,117,90,63)(5,12,109,82,55)(6,13,110,83,56)(7,14,111,84,57)(8,15,112,85,58)(9,16,113,86,59)(19,122,95,68,41)(20,123,96,69,42)(21,124,97,70,43)(22,125,98,71,44)(23,126,99,72,45)(24,118,91,64,37)(25,119,92,65,38)(26,120,93,66,39)(27,121,94,67,40)(28,133,106,79,52)(29,134,107,80,53)(30,135,108,81,54)(31,127,100,73,46)(32,128,101,74,47)(33,129,102,75,48)(34,130,103,76,49)(35,131,104,77,50)(36,132,105,78,51), (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), (1,28,42)(2,29,43)(3,30,44)(4,31,45)(5,32,37)(6,33,38)(7,34,39)(8,35,40)(9,36,41)(10,135,22)(11,127,23)(12,128,24)(13,129,25)(14,130,26)(15,131,27)(16,132,19)(17,133,20)(18,134,21)(46,72,63)(47,64,55)(48,65,56)(49,66,57)(50,67,58)(51,68,59)(52,69,60)(53,70,61)(54,71,62)(73,99,90)(74,91,82)(75,92,83)(76,93,84)(77,94,85)(78,95,86)(79,96,87)(80,97,88)(81,98,89)(100,126,117)(101,118,109)(102,119,110)(103,120,111)(104,121,112)(105,122,113)(106,123,114)(107,124,115)(108,125,116), (2,9)(3,8)(4,7)(5,6)(10,15)(11,14)(12,13)(16,18)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,135)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,45)(35,44)(36,43)(46,66)(47,65)(48,64)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,56)(57,63)(58,62)(59,61)(73,93)(74,92)(75,91)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,83)(84,90)(85,89)(86,88)(100,120)(101,119)(102,118)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,110)(111,117)(112,116)(113,115)>;
G:=Group( (1,17,114,87,60)(2,18,115,88,61)(3,10,116,89,62)(4,11,117,90,63)(5,12,109,82,55)(6,13,110,83,56)(7,14,111,84,57)(8,15,112,85,58)(9,16,113,86,59)(19,122,95,68,41)(20,123,96,69,42)(21,124,97,70,43)(22,125,98,71,44)(23,126,99,72,45)(24,118,91,64,37)(25,119,92,65,38)(26,120,93,66,39)(27,121,94,67,40)(28,133,106,79,52)(29,134,107,80,53)(30,135,108,81,54)(31,127,100,73,46)(32,128,101,74,47)(33,129,102,75,48)(34,130,103,76,49)(35,131,104,77,50)(36,132,105,78,51), (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), (1,28,42)(2,29,43)(3,30,44)(4,31,45)(5,32,37)(6,33,38)(7,34,39)(8,35,40)(9,36,41)(10,135,22)(11,127,23)(12,128,24)(13,129,25)(14,130,26)(15,131,27)(16,132,19)(17,133,20)(18,134,21)(46,72,63)(47,64,55)(48,65,56)(49,66,57)(50,67,58)(51,68,59)(52,69,60)(53,70,61)(54,71,62)(73,99,90)(74,91,82)(75,92,83)(76,93,84)(77,94,85)(78,95,86)(79,96,87)(80,97,88)(81,98,89)(100,126,117)(101,118,109)(102,119,110)(103,120,111)(104,121,112)(105,122,113)(106,123,114)(107,124,115)(108,125,116), (2,9)(3,8)(4,7)(5,6)(10,15)(11,14)(12,13)(16,18)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,135)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,45)(35,44)(36,43)(46,66)(47,65)(48,64)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,56)(57,63)(58,62)(59,61)(73,93)(74,92)(75,91)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,83)(84,90)(85,89)(86,88)(100,120)(101,119)(102,118)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,110)(111,117)(112,116)(113,115) );
G=PermutationGroup([[(1,17,114,87,60),(2,18,115,88,61),(3,10,116,89,62),(4,11,117,90,63),(5,12,109,82,55),(6,13,110,83,56),(7,14,111,84,57),(8,15,112,85,58),(9,16,113,86,59),(19,122,95,68,41),(20,123,96,69,42),(21,124,97,70,43),(22,125,98,71,44),(23,126,99,72,45),(24,118,91,64,37),(25,119,92,65,38),(26,120,93,66,39),(27,121,94,67,40),(28,133,106,79,52),(29,134,107,80,53),(30,135,108,81,54),(31,127,100,73,46),(32,128,101,74,47),(33,129,102,75,48),(34,130,103,76,49),(35,131,104,77,50),(36,132,105,78,51)], [(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)], [(1,28,42),(2,29,43),(3,30,44),(4,31,45),(5,32,37),(6,33,38),(7,34,39),(8,35,40),(9,36,41),(10,135,22),(11,127,23),(12,128,24),(13,129,25),(14,130,26),(15,131,27),(16,132,19),(17,133,20),(18,134,21),(46,72,63),(47,64,55),(48,65,56),(49,66,57),(50,67,58),(51,68,59),(52,69,60),(53,70,61),(54,71,62),(73,99,90),(74,91,82),(75,92,83),(76,93,84),(77,94,85),(78,95,86),(79,96,87),(80,97,88),(81,98,89),(100,126,117),(101,118,109),(102,119,110),(103,120,111),(104,121,112),(105,122,113),(106,123,114),(107,124,115),(108,125,116)], [(2,9),(3,8),(4,7),(5,6),(10,15),(11,14),(12,13),(16,18),(19,134),(20,133),(21,132),(22,131),(23,130),(24,129),(25,128),(26,127),(27,135),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,45),(35,44),(36,43),(46,66),(47,65),(48,64),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,56),(57,63),(58,62),(59,61),(73,93),(74,92),(75,91),(76,99),(77,98),(78,97),(79,96),(80,95),(81,94),(82,83),(84,90),(85,89),(86,88),(100,120),(101,119),(102,118),(103,126),(104,125),(105,124),(106,123),(107,122),(108,121),(109,110),(111,117),(112,116),(113,115)]])
75 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 5A | 5B | 5C | 5D | 9A | ··· | 9I | 10A | 10B | 10C | 10D | 15A | ··· | 15P | 45A | ··· | 45AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 27 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 27 | 27 | 27 | 27 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C5 | C10 | S3 | S3 | D9 | C5×S3 | C5×S3 | C5×D9 |
| kernel | C5×C9⋊S3 | C3×C45 | C9⋊S3 | C3×C9 | C45 | C3×C15 | C15 | C9 | C32 | C3 |
| # reps | 1 | 1 | 4 | 4 | 3 | 1 | 9 | 12 | 4 | 36 |
Matrix representation of C5×C9⋊S3 ►in GL4(𝔽181) generated by
| 125 | 0 | 0 | 0 |
| 0 | 125 | 0 | 0 |
| 0 | 0 | 125 | 0 |
| 0 | 0 | 0 | 125 |
| 127 | 131 | 0 | 0 |
| 50 | 177 | 0 | 0 |
| 0 | 0 | 127 | 131 |
| 0 | 0 | 50 | 177 |
| 0 | 1 | 0 | 0 |
| 180 | 180 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 180 | 180 | 0 | 0 |
| 0 | 0 | 54 | 50 |
| 0 | 0 | 177 | 127 |
G:=sub<GL(4,GF(181))| [125,0,0,0,0,125,0,0,0,0,125,0,0,0,0,125],[127,50,0,0,131,177,0,0,0,0,127,50,0,0,131,177],[0,180,0,0,1,180,0,0,0,0,1,0,0,0,0,1],[1,180,0,0,0,180,0,0,0,0,54,177,0,0,50,127] >;
C5×C9⋊S3 in GAP, Magma, Sage, TeX
C_5\times C_9\rtimes S_3
% in TeX
G:=Group("C5xC9:S3"); // GroupNames label
G:=SmallGroup(270,16);
// by ID
G=gap.SmallGroup(270,16);
# by ID
G:=PCGroup([5,-2,-5,-3,-3,-3,1652,282,1203,4504]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^9=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export