direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C9×C13⋊C3, C117⋊1C3, C39.1C32, C13⋊C9⋊4C3, C13⋊1(C3×C9), C3.1(C3×C13⋊C3), (C3×C13⋊C3).3C3, SmallGroup(351,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C9×C13⋊C3 |
Generators and relations for C9×C13⋊C3
G = < a,b,c | a9=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C9×C13⋊C3
►On 117 points
Generators in S117
(1 105 66 27 92 53 14 79 40)(2 106 67 28 93 54 15 80 41)(3 107 68 29 94 55 16 81 42)(4 108 69 30 95 56 17 82 43)(5 109 70 31 96 57 18 83 44)(6 110 71 32 97 58 19 84 45)(7 111 72 33 98 59 20 85 46)(8 112 73 34 99 60 21 86 47)(9 113 74 35 100 61 22 87 48)(10 114 75 36 101 62 23 88 49)(11 115 76 37 102 63 24 89 50)(12 116 77 38 103 64 25 90 51)(13 117 78 39 104 65 26 91 52)
(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 14 27)(2 17 36)(3 20 32)(4 23 28)(5 26 37)(6 16 33)(7 19 29)(8 22 38)(9 25 34)(10 15 30)(11 18 39)(12 21 35)(13 24 31)(40 53 66)(41 56 75)(42 59 71)(43 62 67)(44 65 76)(45 55 72)(46 58 68)(47 61 77)(48 64 73)(49 54 69)(50 57 78)(51 60 74)(52 63 70)(79 92 105)(80 95 114)(81 98 110)(82 101 106)(83 104 115)(84 94 111)(85 97 107)(86 100 116)(87 103 112)(88 93 108)(89 96 117)(90 99 113)(91 102 109)
G:=sub<Sym(117)| (1,105,66,27,92,53,14,79,40)(2,106,67,28,93,54,15,80,41)(3,107,68,29,94,55,16,81,42)(4,108,69,30,95,56,17,82,43)(5,109,70,31,96,57,18,83,44)(6,110,71,32,97,58,19,84,45)(7,111,72,33,98,59,20,85,46)(8,112,73,34,99,60,21,86,47)(9,113,74,35,100,61,22,87,48)(10,114,75,36,101,62,23,88,49)(11,115,76,37,102,63,24,89,50)(12,116,77,38,103,64,25,90,51)(13,117,78,39,104,65,26,91,52), (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), (1,14,27)(2,17,36)(3,20,32)(4,23,28)(5,26,37)(6,16,33)(7,19,29)(8,22,38)(9,25,34)(10,15,30)(11,18,39)(12,21,35)(13,24,31)(40,53,66)(41,56,75)(42,59,71)(43,62,67)(44,65,76)(45,55,72)(46,58,68)(47,61,77)(48,64,73)(49,54,69)(50,57,78)(51,60,74)(52,63,70)(79,92,105)(80,95,114)(81,98,110)(82,101,106)(83,104,115)(84,94,111)(85,97,107)(86,100,116)(87,103,112)(88,93,108)(89,96,117)(90,99,113)(91,102,109)>;
G:=Group( (1,105,66,27,92,53,14,79,40)(2,106,67,28,93,54,15,80,41)(3,107,68,29,94,55,16,81,42)(4,108,69,30,95,56,17,82,43)(5,109,70,31,96,57,18,83,44)(6,110,71,32,97,58,19,84,45)(7,111,72,33,98,59,20,85,46)(8,112,73,34,99,60,21,86,47)(9,113,74,35,100,61,22,87,48)(10,114,75,36,101,62,23,88,49)(11,115,76,37,102,63,24,89,50)(12,116,77,38,103,64,25,90,51)(13,117,78,39,104,65,26,91,52), (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), (1,14,27)(2,17,36)(3,20,32)(4,23,28)(5,26,37)(6,16,33)(7,19,29)(8,22,38)(9,25,34)(10,15,30)(11,18,39)(12,21,35)(13,24,31)(40,53,66)(41,56,75)(42,59,71)(43,62,67)(44,65,76)(45,55,72)(46,58,68)(47,61,77)(48,64,73)(49,54,69)(50,57,78)(51,60,74)(52,63,70)(79,92,105)(80,95,114)(81,98,110)(82,101,106)(83,104,115)(84,94,111)(85,97,107)(86,100,116)(87,103,112)(88,93,108)(89,96,117)(90,99,113)(91,102,109) );
G=PermutationGroup([[(1,105,66,27,92,53,14,79,40),(2,106,67,28,93,54,15,80,41),(3,107,68,29,94,55,16,81,42),(4,108,69,30,95,56,17,82,43),(5,109,70,31,96,57,18,83,44),(6,110,71,32,97,58,19,84,45),(7,111,72,33,98,59,20,85,46),(8,112,73,34,99,60,21,86,47),(9,113,74,35,100,61,22,87,48),(10,114,75,36,101,62,23,88,49),(11,115,76,37,102,63,24,89,50),(12,116,77,38,103,64,25,90,51),(13,117,78,39,104,65,26,91,52)], [(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)], [(1,14,27),(2,17,36),(3,20,32),(4,23,28),(5,26,37),(6,16,33),(7,19,29),(8,22,38),(9,25,34),(10,15,30),(11,18,39),(12,21,35),(13,24,31),(40,53,66),(41,56,75),(42,59,71),(43,62,67),(44,65,76),(45,55,72),(46,58,68),(47,61,77),(48,64,73),(49,54,69),(50,57,78),(51,60,74),(52,63,70),(79,92,105),(80,95,114),(81,98,110),(82,101,106),(83,104,115),(84,94,111),(85,97,107),(86,100,116),(87,103,112),(88,93,108),(89,96,117),(90,99,113),(91,102,109)]])
63 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3H | 9A | ··· | 9F | 9G | ··· | 9R | 13A | 13B | 13C | 13D | 39A | ··· | 39H | 117A | ··· | 117X |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 | 13 | 13 | 13 | 13 | 39 | ··· | 39 | 117 | ··· | 117 |
| size | 1 | 1 | 1 | 13 | ··· | 13 | 1 | ··· | 1 | 13 | ··· | 13 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C3 | C9 | C13⋊C3 | C3×C13⋊C3 | C9×C13⋊C3 |
| kernel | C9×C13⋊C3 | C13⋊C9 | C117 | C3×C13⋊C3 | C13⋊C3 | C9 | C3 | C1 |
| # reps | 1 | 4 | 2 | 2 | 18 | 4 | 8 | 24 |
Matrix representation of C9×C13⋊C3 ►in GL4(𝔽937) generated by
| 169 | 0 | 0 | 0 |
| 0 | 614 | 0 | 0 |
| 0 | 0 | 614 | 0 |
| 0 | 0 | 0 | 614 |
| 1 | 0 | 0 | 0 |
| 0 | 230 | 731 | 775 |
| 0 | 1 | 0 | 921 |
| 0 | 0 | 1 | 224 |
| 1 | 0 | 0 | 0 |
| 0 | 420 | 407 | 752 |
| 0 | 680 | 248 | 716 |
| 0 | 274 | 277 | 269 |
G:=sub<GL(4,GF(937))| [169,0,0,0,0,614,0,0,0,0,614,0,0,0,0,614],[1,0,0,0,0,230,1,0,0,731,0,1,0,775,921,224],[1,0,0,0,0,420,680,274,0,407,248,277,0,752,716,269] >;
C9×C13⋊C3 in GAP, Magma, Sage, TeX
C_9\times C_{13}\rtimes C_3 % in TeX
G:=Group("C9xC13:C3"); // GroupNames label
G:=SmallGroup(351,3);
// by ID
G=gap.SmallGroup(351,3);
# by ID
G:=PCGroup([4,-3,-3,-3,-13,29,1299]);
// Polycyclic
G:=Group<a,b,c|a^9=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
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