metabelian, supersoluble, monomial, 3-hyperelementary
Aliases: C39.5C32, C13⋊33- 1+2, C13⋊C9⋊3C3, (C3×C39).3C3, C32.(C13⋊C3), C3.5(C3×C13⋊C3), SmallGroup(351,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C39.C32
G = < a,b,c | a39=c3=1, b3=a13, bab-1=a22, ac=ca, cbc-1=a13b >
Smallest permutation representation of C39.C32
►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 91 64 14 104 77 27 117 51)(2 107 47 15 81 60 28 94 73)(3 84 69 16 97 43 29 110 56)(4 100 52 17 113 65 30 87 78)(5 116 74 18 90 48 31 103 61)(6 93 57 19 106 70 32 80 44)(7 109 40 20 83 53 33 96 66)(8 86 62 21 99 75 34 112 49)(9 102 45 22 115 58 35 89 71)(10 79 67 23 92 41 36 105 54)(11 95 50 24 108 63 37 82 76)(12 111 72 25 85 46 38 98 59)(13 88 55 26 101 68 39 114 42)
(40 53 66)(41 54 67)(42 55 68)(43 56 69)(44 57 70)(45 58 71)(46 59 72)(47 60 73)(48 61 74)(49 62 75)(50 63 76)(51 64 77)(52 65 78)(79 105 92)(80 106 93)(81 107 94)(82 108 95)(83 109 96)(84 110 97)(85 111 98)(86 112 99)(87 113 100)(88 114 101)(89 115 102)(90 116 103)(91 117 104)
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), (1,91,64,14,104,77,27,117,51)(2,107,47,15,81,60,28,94,73)(3,84,69,16,97,43,29,110,56)(4,100,52,17,113,65,30,87,78)(5,116,74,18,90,48,31,103,61)(6,93,57,19,106,70,32,80,44)(7,109,40,20,83,53,33,96,66)(8,86,62,21,99,75,34,112,49)(9,102,45,22,115,58,35,89,71)(10,79,67,23,92,41,36,105,54)(11,95,50,24,108,63,37,82,76)(12,111,72,25,85,46,38,98,59)(13,88,55,26,101,68,39,114,42), (40,53,66)(41,54,67)(42,55,68)(43,56,69)(44,57,70)(45,58,71)(46,59,72)(47,60,73)(48,61,74)(49,62,75)(50,63,76)(51,64,77)(52,65,78)(79,105,92)(80,106,93)(81,107,94)(82,108,95)(83,109,96)(84,110,97)(85,111,98)(86,112,99)(87,113,100)(88,114,101)(89,115,102)(90,116,103)(91,117,104)>;
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), (1,91,64,14,104,77,27,117,51)(2,107,47,15,81,60,28,94,73)(3,84,69,16,97,43,29,110,56)(4,100,52,17,113,65,30,87,78)(5,116,74,18,90,48,31,103,61)(6,93,57,19,106,70,32,80,44)(7,109,40,20,83,53,33,96,66)(8,86,62,21,99,75,34,112,49)(9,102,45,22,115,58,35,89,71)(10,79,67,23,92,41,36,105,54)(11,95,50,24,108,63,37,82,76)(12,111,72,25,85,46,38,98,59)(13,88,55,26,101,68,39,114,42), (40,53,66)(41,54,67)(42,55,68)(43,56,69)(44,57,70)(45,58,71)(46,59,72)(47,60,73)(48,61,74)(49,62,75)(50,63,76)(51,64,77)(52,65,78)(79,105,92)(80,106,93)(81,107,94)(82,108,95)(83,109,96)(84,110,97)(85,111,98)(86,112,99)(87,113,100)(88,114,101)(89,115,102)(90,116,103)(91,117,104) );
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)], [(1,91,64,14,104,77,27,117,51),(2,107,47,15,81,60,28,94,73),(3,84,69,16,97,43,29,110,56),(4,100,52,17,113,65,30,87,78),(5,116,74,18,90,48,31,103,61),(6,93,57,19,106,70,32,80,44),(7,109,40,20,83,53,33,96,66),(8,86,62,21,99,75,34,112,49),(9,102,45,22,115,58,35,89,71),(10,79,67,23,92,41,36,105,54),(11,95,50,24,108,63,37,82,76),(12,111,72,25,85,46,38,98,59),(13,88,55,26,101,68,39,114,42)], [(40,53,66),(41,54,67),(42,55,68),(43,56,69),(44,57,70),(45,58,71),(46,59,72),(47,60,73),(48,61,74),(49,62,75),(50,63,76),(51,64,77),(52,65,78),(79,105,92),(80,106,93),(81,107,94),(82,108,95),(83,109,96),(84,110,97),(85,111,98),(86,112,99),(87,113,100),(88,114,101),(89,115,102),(90,116,103),(91,117,104)]])
47 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 9A | ··· | 9F | 13A | 13B | 13C | 13D | 39A | ··· | 39AF |
| order | 1 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 13 | 13 | 13 | 13 | 39 | ··· | 39 |
| size | 1 | 1 | 1 | 3 | 3 | 39 | ··· | 39 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
47 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | 3- 1+2 | C13⋊C3 | C3×C13⋊C3 | C39.C32 |
| kernel | C39.C32 | C13⋊C9 | C3×C39 | C13 | C32 | C3 | C1 |
| # reps | 1 | 6 | 2 | 2 | 4 | 8 | 24 |
Matrix representation of C39.C32 ►in GL3(𝔽937) generated by
| 488 | 0 | 0 |
| 0 | 553 | 0 |
| 0 | 0 | 820 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 614 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 614 | 0 |
| 0 | 0 | 322 |
G:=sub<GL(3,GF(937))| [488,0,0,0,553,0,0,0,820],[0,0,614,1,0,0,0,1,0],[1,0,0,0,614,0,0,0,322] >;
C39.C32 in GAP, Magma, Sage, TeX
C_{39}.C_3^2 % in TeX
G:=Group("C39.C3^2"); // GroupNames label
G:=SmallGroup(351,7);
// by ID
G=gap.SmallGroup(351,7);
# by ID
G:=PCGroup([4,-3,-3,-3,-13,36,97,1299]);
// Polycyclic
G:=Group<a,b,c|a^39=c^3=1,b^3=a^13,b*a*b^-1=a^22,a*c=c*a,c*b*c^-1=a^13*b>;
// generators/relations
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