direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C13×3- 1+2, C9⋊C39, C117⋊4C3, C32.C39, C39.8C32, (C3×C39).1C3, C3.2(C3×C39), SmallGroup(351,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×3- 1+2
G = < a,b,c | a13=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C13×3- 1+2
►On 117 points
Generators in S117
(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)
(1 36 20 68 79 41 63 92 110)(2 37 21 69 80 42 64 93 111)(3 38 22 70 81 43 65 94 112)(4 39 23 71 82 44 53 95 113)(5 27 24 72 83 45 54 96 114)(6 28 25 73 84 46 55 97 115)(7 29 26 74 85 47 56 98 116)(8 30 14 75 86 48 57 99 117)(9 31 15 76 87 49 58 100 105)(10 32 16 77 88 50 59 101 106)(11 33 17 78 89 51 60 102 107)(12 34 18 66 90 52 61 103 108)(13 35 19 67 91 40 62 104 109)
(14 48 117)(15 49 105)(16 50 106)(17 51 107)(18 52 108)(19 40 109)(20 41 110)(21 42 111)(22 43 112)(23 44 113)(24 45 114)(25 46 115)(26 47 116)(27 96 83)(28 97 84)(29 98 85)(30 99 86)(31 100 87)(32 101 88)(33 102 89)(34 103 90)(35 104 91)(36 92 79)(37 93 80)(38 94 81)(39 95 82)
G:=sub<Sym(117)| (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), (1,36,20,68,79,41,63,92,110)(2,37,21,69,80,42,64,93,111)(3,38,22,70,81,43,65,94,112)(4,39,23,71,82,44,53,95,113)(5,27,24,72,83,45,54,96,114)(6,28,25,73,84,46,55,97,115)(7,29,26,74,85,47,56,98,116)(8,30,14,75,86,48,57,99,117)(9,31,15,76,87,49,58,100,105)(10,32,16,77,88,50,59,101,106)(11,33,17,78,89,51,60,102,107)(12,34,18,66,90,52,61,103,108)(13,35,19,67,91,40,62,104,109), (14,48,117)(15,49,105)(16,50,106)(17,51,107)(18,52,108)(19,40,109)(20,41,110)(21,42,111)(22,43,112)(23,44,113)(24,45,114)(25,46,115)(26,47,116)(27,96,83)(28,97,84)(29,98,85)(30,99,86)(31,100,87)(32,101,88)(33,102,89)(34,103,90)(35,104,91)(36,92,79)(37,93,80)(38,94,81)(39,95,82)>;
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), (1,36,20,68,79,41,63,92,110)(2,37,21,69,80,42,64,93,111)(3,38,22,70,81,43,65,94,112)(4,39,23,71,82,44,53,95,113)(5,27,24,72,83,45,54,96,114)(6,28,25,73,84,46,55,97,115)(7,29,26,74,85,47,56,98,116)(8,30,14,75,86,48,57,99,117)(9,31,15,76,87,49,58,100,105)(10,32,16,77,88,50,59,101,106)(11,33,17,78,89,51,60,102,107)(12,34,18,66,90,52,61,103,108)(13,35,19,67,91,40,62,104,109), (14,48,117)(15,49,105)(16,50,106)(17,51,107)(18,52,108)(19,40,109)(20,41,110)(21,42,111)(22,43,112)(23,44,113)(24,45,114)(25,46,115)(26,47,116)(27,96,83)(28,97,84)(29,98,85)(30,99,86)(31,100,87)(32,101,88)(33,102,89)(34,103,90)(35,104,91)(36,92,79)(37,93,80)(38,94,81)(39,95,82) );
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)], [(1,36,20,68,79,41,63,92,110),(2,37,21,69,80,42,64,93,111),(3,38,22,70,81,43,65,94,112),(4,39,23,71,82,44,53,95,113),(5,27,24,72,83,45,54,96,114),(6,28,25,73,84,46,55,97,115),(7,29,26,74,85,47,56,98,116),(8,30,14,75,86,48,57,99,117),(9,31,15,76,87,49,58,100,105),(10,32,16,77,88,50,59,101,106),(11,33,17,78,89,51,60,102,107),(12,34,18,66,90,52,61,103,108),(13,35,19,67,91,40,62,104,109)], [(14,48,117),(15,49,105),(16,50,106),(17,51,107),(18,52,108),(19,40,109),(20,41,110),(21,42,111),(22,43,112),(23,44,113),(24,45,114),(25,46,115),(26,47,116),(27,96,83),(28,97,84),(29,98,85),(30,99,86),(31,100,87),(32,101,88),(33,102,89),(34,103,90),(35,104,91),(36,92,79),(37,93,80),(38,94,81),(39,95,82)]])
143 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 9A | ··· | 9F | 13A | ··· | 13L | 39A | ··· | 39X | 39Y | ··· | 39AV | 117A | ··· | 117BT |
| order | 1 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 13 | ··· | 13 | 39 | ··· | 39 | 39 | ··· | 39 | 117 | ··· | 117 |
| size | 1 | 1 | 1 | 3 | 3 | 3 | ··· | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C13 | C39 | C39 | 3- 1+2 | C13×3- 1+2 |
| kernel | C13×3- 1+2 | C117 | C3×C39 | 3- 1+2 | C9 | C32 | C13 | C1 |
| # reps | 1 | 6 | 2 | 12 | 72 | 24 | 2 | 24 |
Matrix representation of C13×3- 1+2 ►in GL3(𝔽937) generated by
| 931 | 0 | 0 |
| 0 | 931 | 0 |
| 0 | 0 | 931 |
| 0 | 1 | 0 |
| 889 | 401 | 292 |
| 420 | 287 | 536 |
| 1 | 0 | 0 |
| 0 | 322 | 0 |
| 464 | 401 | 614 |
G:=sub<GL(3,GF(937))| [931,0,0,0,931,0,0,0,931],[0,889,420,1,401,287,0,292,536],[1,0,464,0,322,401,0,0,614] >;
C13×3- 1+2 in GAP, Magma, Sage, TeX
C_{13}\times 3_-^{1+2} % in TeX
G:=Group("C13xES-(3,1)"); // GroupNames label
G:=SmallGroup(351,11);
// by ID
G=gap.SmallGroup(351,11);
# by ID
G:=PCGroup([4,-3,-3,-13,-3,468,961]);
// Polycyclic
G:=Group<a,b,c|a^13=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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