direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D13, C78⋊2C2, C26⋊3C6, C39⋊3C22, C13⋊3(C2×C6), SmallGroup(156,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C6×D13 |
Generators and relations for C6×D13
G = < a,b,c | a6=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D13
►On 78 points
Generators in S78
(1 55 36 50 16 67)(2 56 37 51 17 68)(3 57 38 52 18 69)(4 58 39 40 19 70)(5 59 27 41 20 71)(6 60 28 42 21 72)(7 61 29 43 22 73)(8 62 30 44 23 74)(9 63 31 45 24 75)(10 64 32 46 25 76)(11 65 33 47 26 77)(12 53 34 48 14 78)(13 54 35 49 15 66)
(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)
(1 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 52)(12 51)(13 50)(14 56)(15 55)(16 54)(17 53)(18 65)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 58)(26 57)(27 75)(28 74)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 78)(38 77)(39 76)
G:=sub<Sym(78)| (1,55,36,50,16,67)(2,56,37,51,17,68)(3,57,38,52,18,69)(4,58,39,40,19,70)(5,59,27,41,20,71)(6,60,28,42,21,72)(7,61,29,43,22,73)(8,62,30,44,23,74)(9,63,31,45,24,75)(10,64,32,46,25,76)(11,65,33,47,26,77)(12,53,34,48,14,78)(13,54,35,49,15,66), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,52)(12,51)(13,50)(14,56)(15,55)(16,54)(17,53)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,78)(38,77)(39,76)>;
G:=Group( (1,55,36,50,16,67)(2,56,37,51,17,68)(3,57,38,52,18,69)(4,58,39,40,19,70)(5,59,27,41,20,71)(6,60,28,42,21,72)(7,61,29,43,22,73)(8,62,30,44,23,74)(9,63,31,45,24,75)(10,64,32,46,25,76)(11,65,33,47,26,77)(12,53,34,48,14,78)(13,54,35,49,15,66), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,52)(12,51)(13,50)(14,56)(15,55)(16,54)(17,53)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,78)(38,77)(39,76) );
G=PermutationGroup([[(1,55,36,50,16,67),(2,56,37,51,17,68),(3,57,38,52,18,69),(4,58,39,40,19,70),(5,59,27,41,20,71),(6,60,28,42,21,72),(7,61,29,43,22,73),(8,62,30,44,23,74),(9,63,31,45,24,75),(10,64,32,46,25,76),(11,65,33,47,26,77),(12,53,34,48,14,78),(13,54,35,49,15,66)], [(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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,52),(12,51),(13,50),(14,56),(15,55),(16,54),(17,53),(18,65),(19,64),(20,63),(21,62),(22,61),(23,60),(24,59),(25,58),(26,57),(27,75),(28,74),(29,73),(30,72),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,78),(38,77),(39,76)]])
C6×D13 is a maximal subgroup of
C39⋊D4 C3⋊D52
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 78A | ··· | 78L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D13 | D26 | C3×D13 | C6×D13 |
| kernel | C6×D13 | C3×D13 | C78 | D26 | D13 | C26 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 6 | 12 | 12 |
Matrix representation of C6×D13 ►in GL3(𝔽79) generated by
| 78 | 0 | 0 |
| 0 | 55 | 0 |
| 0 | 0 | 55 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 78 | 6 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(79))| [78,0,0,0,55,0,0,0,55],[1,0,0,0,0,78,0,1,6],[1,0,0,0,0,1,0,1,0] >;
C6×D13 in GAP, Magma, Sage, TeX
C_6\times D_{13} % in TeX
G:=Group("C6xD13"); // GroupNames label
G:=SmallGroup(156,15);
// by ID
G=gap.SmallGroup(156,15);
# by ID
G:=PCGroup([4,-2,-2,-3,-13,2307]);
// Polycyclic
G:=Group<a,b,c|a^6=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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