direct product, abelian, monomial, 3-elementary
Aliases: C3xC39, SmallGroup(117,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3xC39 |
| C1 — C3xC39 |
| C1 — C3xC39 |
Generators and relations for C3xC39
G = < a,b | a3=b39=1, ab=ba >
Quotients: C1, C3, C32, C13, C39, C3xC39
Smallest permutation representation of C3xC39
►Regular action on 117 points
Generators in S117
(1 92 52)(2 93 53)(3 94 54)(4 95 55)(5 96 56)(6 97 57)(7 98 58)(8 99 59)(9 100 60)(10 101 61)(11 102 62)(12 103 63)(13 104 64)(14 105 65)(15 106 66)(16 107 67)(17 108 68)(18 109 69)(19 110 70)(20 111 71)(21 112 72)(22 113 73)(23 114 74)(24 115 75)(25 116 76)(26 117 77)(27 79 78)(28 80 40)(29 81 41)(30 82 42)(31 83 43)(32 84 44)(33 85 45)(34 86 46)(35 87 47)(36 88 48)(37 89 49)(38 90 50)(39 91 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)
G:=sub<Sym(117)| (1,92,52)(2,93,53)(3,94,54)(4,95,55)(5,96,56)(6,97,57)(7,98,58)(8,99,59)(9,100,60)(10,101,61)(11,102,62)(12,103,63)(13,104,64)(14,105,65)(15,106,66)(16,107,67)(17,108,68)(18,109,69)(19,110,70)(20,111,71)(21,112,72)(22,113,73)(23,114,74)(24,115,75)(25,116,76)(26,117,77)(27,79,78)(28,80,40)(29,81,41)(30,82,42)(31,83,43)(32,84,44)(33,85,45)(34,86,46)(35,87,47)(36,88,48)(37,89,49)(38,90,50)(39,91,51), (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)>;
G:=Group( (1,92,52)(2,93,53)(3,94,54)(4,95,55)(5,96,56)(6,97,57)(7,98,58)(8,99,59)(9,100,60)(10,101,61)(11,102,62)(12,103,63)(13,104,64)(14,105,65)(15,106,66)(16,107,67)(17,108,68)(18,109,69)(19,110,70)(20,111,71)(21,112,72)(22,113,73)(23,114,74)(24,115,75)(25,116,76)(26,117,77)(27,79,78)(28,80,40)(29,81,41)(30,82,42)(31,83,43)(32,84,44)(33,85,45)(34,86,46)(35,87,47)(36,88,48)(37,89,49)(38,90,50)(39,91,51), (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) );
G=PermutationGroup([[(1,92,52),(2,93,53),(3,94,54),(4,95,55),(5,96,56),(6,97,57),(7,98,58),(8,99,59),(9,100,60),(10,101,61),(11,102,62),(12,103,63),(13,104,64),(14,105,65),(15,106,66),(16,107,67),(17,108,68),(18,109,69),(19,110,70),(20,111,71),(21,112,72),(22,113,73),(23,114,74),(24,115,75),(25,116,76),(26,117,77),(27,79,78),(28,80,40),(29,81,41),(30,82,42),(31,83,43),(32,84,44),(33,85,45),(34,86,46),(35,87,47),(36,88,48),(37,89,49),(38,90,50),(39,91,51)], [(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)]])
C3xC39 is a maximal subgroup of
C3:D39 C39.C32 C13:He3
117 conjugacy classes
| class | 1 | 3A | ··· | 3H | 13A | ··· | 13L | 39A | ··· | 39CR |
| order | 1 | 3 | ··· | 3 | 13 | ··· | 13 | 39 | ··· | 39 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C13 | C39 |
| kernel | C3xC39 | C39 | C32 | C3 |
| # reps | 1 | 8 | 12 | 96 |
Matrix representation of C3xC39 ►in GL2(F79) generated by
| 1 | 0 |
| 0 | 55 |
| 2 | 0 |
| 0 | 64 |
G:=sub<GL(2,GF(79))| [1,0,0,55],[2,0,0,64] >;
C3xC39 in GAP, Magma, Sage, TeX
C_3\times C_{39} % in TeX
G:=Group("C3xC39"); // GroupNames label
G:=SmallGroup(117,4);
// by ID
G=gap.SmallGroup(117,4);
# by ID
G:=PCGroup([3,-3,-3,-13]);
// Polycyclic
G:=Group<a,b|a^3=b^39=1,a*b=b*a>;
// generators/relations
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