direct product, metabelian, supersoluble, monomial, A-group
Aliases: C13×C3⋊S3, C39⋊3S3, C32⋊2C26, C3⋊(S3×C13), (C3×C39)⋊5C2, SmallGroup(234,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C13×C3⋊S3 |
Generators and relations for C13×C3⋊S3
G = < a,b,c,d | a13=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Smallest permutation representation of C13×C3⋊S3
►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 18 106)(2 19 107)(3 20 108)(4 21 109)(5 22 110)(6 23 111)(7 24 112)(8 25 113)(9 26 114)(10 14 115)(11 15 116)(12 16 117)(13 17 105)(27 66 86)(28 67 87)(29 68 88)(30 69 89)(31 70 90)(32 71 91)(33 72 79)(34 73 80)(35 74 81)(36 75 82)(37 76 83)(38 77 84)(39 78 85)(40 53 95)(41 54 96)(42 55 97)(43 56 98)(44 57 99)(45 58 100)(46 59 101)(47 60 102)(48 61 103)(49 62 104)(50 63 92)(51 64 93)(52 65 94)
(1 72 94)(2 73 95)(3 74 96)(4 75 97)(5 76 98)(6 77 99)(7 78 100)(8 66 101)(9 67 102)(10 68 103)(11 69 104)(12 70 92)(13 71 93)(14 88 48)(15 89 49)(16 90 50)(17 91 51)(18 79 52)(19 80 40)(20 81 41)(21 82 42)(22 83 43)(23 84 44)(24 85 45)(25 86 46)(26 87 47)(27 59 113)(28 60 114)(29 61 115)(30 62 116)(31 63 117)(32 64 105)(33 65 106)(34 53 107)(35 54 108)(36 55 109)(37 56 110)(38 57 111)(39 58 112)
(14 115)(15 116)(16 117)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 40)(35 41)(36 42)(37 43)(38 44)(39 45)(53 80)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 79)(66 101)(67 102)(68 103)(69 104)(70 92)(71 93)(72 94)(73 95)(74 96)(75 97)(76 98)(77 99)(78 100)
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), (1,18,106)(2,19,107)(3,20,108)(4,21,109)(5,22,110)(6,23,111)(7,24,112)(8,25,113)(9,26,114)(10,14,115)(11,15,116)(12,16,117)(13,17,105)(27,66,86)(28,67,87)(29,68,88)(30,69,89)(31,70,90)(32,71,91)(33,72,79)(34,73,80)(35,74,81)(36,75,82)(37,76,83)(38,77,84)(39,78,85)(40,53,95)(41,54,96)(42,55,97)(43,56,98)(44,57,99)(45,58,100)(46,59,101)(47,60,102)(48,61,103)(49,62,104)(50,63,92)(51,64,93)(52,65,94), (1,72,94)(2,73,95)(3,74,96)(4,75,97)(5,76,98)(6,77,99)(7,78,100)(8,66,101)(9,67,102)(10,68,103)(11,69,104)(12,70,92)(13,71,93)(14,88,48)(15,89,49)(16,90,50)(17,91,51)(18,79,52)(19,80,40)(20,81,41)(21,82,42)(22,83,43)(23,84,44)(24,85,45)(25,86,46)(26,87,47)(27,59,113)(28,60,114)(29,61,115)(30,62,116)(31,63,117)(32,64,105)(33,65,106)(34,53,107)(35,54,108)(36,55,109)(37,56,110)(38,57,111)(39,58,112), (14,115)(15,116)(16,117)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(53,80)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,79)(66,101)(67,102)(68,103)(69,104)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100)>;
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), (1,18,106)(2,19,107)(3,20,108)(4,21,109)(5,22,110)(6,23,111)(7,24,112)(8,25,113)(9,26,114)(10,14,115)(11,15,116)(12,16,117)(13,17,105)(27,66,86)(28,67,87)(29,68,88)(30,69,89)(31,70,90)(32,71,91)(33,72,79)(34,73,80)(35,74,81)(36,75,82)(37,76,83)(38,77,84)(39,78,85)(40,53,95)(41,54,96)(42,55,97)(43,56,98)(44,57,99)(45,58,100)(46,59,101)(47,60,102)(48,61,103)(49,62,104)(50,63,92)(51,64,93)(52,65,94), (1,72,94)(2,73,95)(3,74,96)(4,75,97)(5,76,98)(6,77,99)(7,78,100)(8,66,101)(9,67,102)(10,68,103)(11,69,104)(12,70,92)(13,71,93)(14,88,48)(15,89,49)(16,90,50)(17,91,51)(18,79,52)(19,80,40)(20,81,41)(21,82,42)(22,83,43)(23,84,44)(24,85,45)(25,86,46)(26,87,47)(27,59,113)(28,60,114)(29,61,115)(30,62,116)(31,63,117)(32,64,105)(33,65,106)(34,53,107)(35,54,108)(36,55,109)(37,56,110)(38,57,111)(39,58,112), (14,115)(15,116)(16,117)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(53,80)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,79)(66,101)(67,102)(68,103)(69,104)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100) );
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)], [(1,18,106),(2,19,107),(3,20,108),(4,21,109),(5,22,110),(6,23,111),(7,24,112),(8,25,113),(9,26,114),(10,14,115),(11,15,116),(12,16,117),(13,17,105),(27,66,86),(28,67,87),(29,68,88),(30,69,89),(31,70,90),(32,71,91),(33,72,79),(34,73,80),(35,74,81),(36,75,82),(37,76,83),(38,77,84),(39,78,85),(40,53,95),(41,54,96),(42,55,97),(43,56,98),(44,57,99),(45,58,100),(46,59,101),(47,60,102),(48,61,103),(49,62,104),(50,63,92),(51,64,93),(52,65,94)], [(1,72,94),(2,73,95),(3,74,96),(4,75,97),(5,76,98),(6,77,99),(7,78,100),(8,66,101),(9,67,102),(10,68,103),(11,69,104),(12,70,92),(13,71,93),(14,88,48),(15,89,49),(16,90,50),(17,91,51),(18,79,52),(19,80,40),(20,81,41),(21,82,42),(22,83,43),(23,84,44),(24,85,45),(25,86,46),(26,87,47),(27,59,113),(28,60,114),(29,61,115),(30,62,116),(31,63,117),(32,64,105),(33,65,106),(34,53,107),(35,54,108),(36,55,109),(37,56,110),(38,57,111),(39,58,112)], [(14,115),(15,116),(16,117),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,46),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,40),(35,41),(36,42),(37,43),(38,44),(39,45),(53,80),(54,81),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87),(61,88),(62,89),(63,90),(64,91),(65,79),(66,101),(67,102),(68,103),(69,104),(70,92),(71,93),(72,94),(73,95),(74,96),(75,97),(76,98),(77,99),(78,100)]])
C13×C3⋊S3 is a maximal subgroup of
C32⋊Dic13 S32×C13 D39⋊S3
78 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39AV |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C13 | C26 | S3 | S3×C13 |
| kernel | C13×C3⋊S3 | C3×C39 | C3⋊S3 | C32 | C39 | C3 |
| # reps | 1 | 1 | 12 | 12 | 4 | 48 |
Matrix representation of C13×C3⋊S3 ►in GL4(𝔽79) generated by
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 65 |
| 0 | 1 | 0 | 0 |
| 78 | 78 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 78 | 78 |
| 78 | 78 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 78 | 78 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 78 |
G:=sub<GL(4,GF(79))| [46,0,0,0,0,46,0,0,0,0,65,0,0,0,0,65],[0,78,0,0,1,78,0,0,0,0,0,78,0,0,1,78],[78,1,0,0,78,0,0,0,0,0,1,0,0,0,0,1],[1,78,0,0,0,78,0,0,0,0,1,0,0,0,1,78] >;
C13×C3⋊S3 in GAP, Magma, Sage, TeX
C_{13}\times C_3\rtimes S_3 % in TeX
G:=Group("C13xC3:S3"); // GroupNames label
G:=SmallGroup(234,14);
// by ID
G=gap.SmallGroup(234,14);
# by ID
G:=PCGroup([4,-2,-13,-3,-3,626,2499]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export