direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C13×He3, C32⋊C39, C39.7C32, (C3×C39)⋊1C3, C3.1(C3×C39), SmallGroup(351,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×He3
G = < a,b,c,d | a13=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >
Smallest permutation representation of C13×He3
►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)
(14 41 76)(15 42 77)(16 43 78)(17 44 66)(18 45 67)(19 46 68)(20 47 69)(21 48 70)(22 49 71)(23 50 72)(24 51 73)(25 52 74)(26 40 75)(27 84 63)(28 85 64)(29 86 65)(30 87 53)(31 88 54)(32 89 55)(33 90 56)(34 91 57)(35 79 58)(36 80 59)(37 81 60)(38 82 61)(39 83 62)
(1 117 97)(2 105 98)(3 106 99)(4 107 100)(5 108 101)(6 109 102)(7 110 103)(8 111 104)(9 112 92)(10 113 93)(11 114 94)(12 115 95)(13 116 96)(14 76 41)(15 77 42)(16 78 43)(17 66 44)(18 67 45)(19 68 46)(20 69 47)(21 70 48)(22 71 49)(23 72 50)(24 73 51)(25 74 52)(26 75 40)(27 84 63)(28 85 64)(29 86 65)(30 87 53)(31 88 54)(32 89 55)(33 90 56)(34 91 57)(35 79 58)(36 80 59)(37 81 60)(38 82 61)(39 83 62)
(1 25 38)(2 26 39)(3 14 27)(4 15 28)(5 16 29)(6 17 30)(7 18 31)(8 19 32)(9 20 33)(10 21 34)(11 22 35)(12 23 36)(13 24 37)(40 62 98)(41 63 99)(42 64 100)(43 65 101)(44 53 102)(45 54 103)(46 55 104)(47 56 92)(48 57 93)(49 58 94)(50 59 95)(51 60 96)(52 61 97)(66 87 109)(67 88 110)(68 89 111)(69 90 112)(70 91 113)(71 79 114)(72 80 115)(73 81 116)(74 82 117)(75 83 105)(76 84 106)(77 85 107)(78 86 108)
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), (14,41,76)(15,42,77)(16,43,78)(17,44,66)(18,45,67)(19,46,68)(20,47,69)(21,48,70)(22,49,71)(23,50,72)(24,51,73)(25,52,74)(26,40,75)(27,84,63)(28,85,64)(29,86,65)(30,87,53)(31,88,54)(32,89,55)(33,90,56)(34,91,57)(35,79,58)(36,80,59)(37,81,60)(38,82,61)(39,83,62), (1,117,97)(2,105,98)(3,106,99)(4,107,100)(5,108,101)(6,109,102)(7,110,103)(8,111,104)(9,112,92)(10,113,93)(11,114,94)(12,115,95)(13,116,96)(14,76,41)(15,77,42)(16,78,43)(17,66,44)(18,67,45)(19,68,46)(20,69,47)(21,70,48)(22,71,49)(23,72,50)(24,73,51)(25,74,52)(26,75,40)(27,84,63)(28,85,64)(29,86,65)(30,87,53)(31,88,54)(32,89,55)(33,90,56)(34,91,57)(35,79,58)(36,80,59)(37,81,60)(38,82,61)(39,83,62), (1,25,38)(2,26,39)(3,14,27)(4,15,28)(5,16,29)(6,17,30)(7,18,31)(8,19,32)(9,20,33)(10,21,34)(11,22,35)(12,23,36)(13,24,37)(40,62,98)(41,63,99)(42,64,100)(43,65,101)(44,53,102)(45,54,103)(46,55,104)(47,56,92)(48,57,93)(49,58,94)(50,59,95)(51,60,96)(52,61,97)(66,87,109)(67,88,110)(68,89,111)(69,90,112)(70,91,113)(71,79,114)(72,80,115)(73,81,116)(74,82,117)(75,83,105)(76,84,106)(77,85,107)(78,86,108)>;
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), (14,41,76)(15,42,77)(16,43,78)(17,44,66)(18,45,67)(19,46,68)(20,47,69)(21,48,70)(22,49,71)(23,50,72)(24,51,73)(25,52,74)(26,40,75)(27,84,63)(28,85,64)(29,86,65)(30,87,53)(31,88,54)(32,89,55)(33,90,56)(34,91,57)(35,79,58)(36,80,59)(37,81,60)(38,82,61)(39,83,62), (1,117,97)(2,105,98)(3,106,99)(4,107,100)(5,108,101)(6,109,102)(7,110,103)(8,111,104)(9,112,92)(10,113,93)(11,114,94)(12,115,95)(13,116,96)(14,76,41)(15,77,42)(16,78,43)(17,66,44)(18,67,45)(19,68,46)(20,69,47)(21,70,48)(22,71,49)(23,72,50)(24,73,51)(25,74,52)(26,75,40)(27,84,63)(28,85,64)(29,86,65)(30,87,53)(31,88,54)(32,89,55)(33,90,56)(34,91,57)(35,79,58)(36,80,59)(37,81,60)(38,82,61)(39,83,62), (1,25,38)(2,26,39)(3,14,27)(4,15,28)(5,16,29)(6,17,30)(7,18,31)(8,19,32)(9,20,33)(10,21,34)(11,22,35)(12,23,36)(13,24,37)(40,62,98)(41,63,99)(42,64,100)(43,65,101)(44,53,102)(45,54,103)(46,55,104)(47,56,92)(48,57,93)(49,58,94)(50,59,95)(51,60,96)(52,61,97)(66,87,109)(67,88,110)(68,89,111)(69,90,112)(70,91,113)(71,79,114)(72,80,115)(73,81,116)(74,82,117)(75,83,105)(76,84,106)(77,85,107)(78,86,108) );
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)], [(14,41,76),(15,42,77),(16,43,78),(17,44,66),(18,45,67),(19,46,68),(20,47,69),(21,48,70),(22,49,71),(23,50,72),(24,51,73),(25,52,74),(26,40,75),(27,84,63),(28,85,64),(29,86,65),(30,87,53),(31,88,54),(32,89,55),(33,90,56),(34,91,57),(35,79,58),(36,80,59),(37,81,60),(38,82,61),(39,83,62)], [(1,117,97),(2,105,98),(3,106,99),(4,107,100),(5,108,101),(6,109,102),(7,110,103),(8,111,104),(9,112,92),(10,113,93),(11,114,94),(12,115,95),(13,116,96),(14,76,41),(15,77,42),(16,78,43),(17,66,44),(18,67,45),(19,68,46),(20,69,47),(21,70,48),(22,71,49),(23,72,50),(24,73,51),(25,74,52),(26,75,40),(27,84,63),(28,85,64),(29,86,65),(30,87,53),(31,88,54),(32,89,55),(33,90,56),(34,91,57),(35,79,58),(36,80,59),(37,81,60),(38,82,61),(39,83,62)], [(1,25,38),(2,26,39),(3,14,27),(4,15,28),(5,16,29),(6,17,30),(7,18,31),(8,19,32),(9,20,33),(10,21,34),(11,22,35),(12,23,36),(13,24,37),(40,62,98),(41,63,99),(42,64,100),(43,65,101),(44,53,102),(45,54,103),(46,55,104),(47,56,92),(48,57,93),(49,58,94),(50,59,95),(51,60,96),(52,61,97),(66,87,109),(67,88,110),(68,89,111),(69,90,112),(70,91,113),(71,79,114),(72,80,115),(73,81,116),(74,82,117),(75,83,105),(76,84,106),(77,85,107),(78,86,108)]])
143 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3J | 13A | ··· | 13L | 39A | ··· | 39X | 39Y | ··· | 39DP |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 13 | ··· | 13 | 39 | ··· | 39 | 39 | ··· | 39 |
| size | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C13 | C39 | He3 | C13×He3 |
| kernel | C13×He3 | C3×C39 | He3 | C32 | C13 | C1 |
| # reps | 1 | 8 | 12 | 96 | 2 | 24 |
Matrix representation of C13×He3 ►in GL3(𝔽79) generated by
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 23 | 0 |
| 47 | 16 | 55 |
| 23 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 23 |
| 0 | 1 | 0 |
| 25 | 11 | 22 |
| 45 | 58 | 68 |
G:=sub<GL(3,GF(79))| [10,0,0,0,10,0,0,0,10],[1,0,47,0,23,16,0,0,55],[23,0,0,0,23,0,0,0,23],[0,25,45,1,11,58,0,22,68] >;
C13×He3 in GAP, Magma, Sage, TeX
C_{13}\times {\rm He}_3 % in TeX
G:=Group("C13xHe3"); // GroupNames label
G:=SmallGroup(351,10);
// by ID
G=gap.SmallGroup(351,10);
# by ID
G:=PCGroup([4,-3,-3,-13,-3,961]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations
Export