direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C26, C6⋊C26, C78⋊3C2, C39⋊4C22, C3⋊(C2×C26), SmallGroup(156,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C26 |
Generators and relations for S3×C26
G = < a,b,c | a26=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C26
►On 78 points
Generators in S78
(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 55)(2 50 56)(3 51 57)(4 52 58)(5 27 59)(6 28 60)(7 29 61)(8 30 62)(9 31 63)(10 32 64)(11 33 65)(12 34 66)(13 35 67)(14 36 68)(15 37 69)(16 38 70)(17 39 71)(18 40 72)(19 41 73)(20 42 74)(21 43 75)(22 44 76)(23 45 77)(24 46 78)(25 47 53)(26 48 54)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)
G:=sub<Sym(78)| (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), (1,49,55)(2,50,56)(3,51,57)(4,52,58)(5,27,59)(6,28,60)(7,29,61)(8,30,62)(9,31,63)(10,32,64)(11,33,65)(12,34,66)(13,35,67)(14,36,68)(15,37,69)(16,38,70)(17,39,71)(18,40,72)(19,41,73)(20,42,74)(21,43,75)(22,44,76)(23,45,77)(24,46,78)(25,47,53)(26,48,54), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)>;
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), (1,49,55)(2,50,56)(3,51,57)(4,52,58)(5,27,59)(6,28,60)(7,29,61)(8,30,62)(9,31,63)(10,32,64)(11,33,65)(12,34,66)(13,35,67)(14,36,68)(15,37,69)(16,38,70)(17,39,71)(18,40,72)(19,41,73)(20,42,74)(21,43,75)(22,44,76)(23,45,77)(24,46,78)(25,47,53)(26,48,54), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71) );
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)], [(1,49,55),(2,50,56),(3,51,57),(4,52,58),(5,27,59),(6,28,60),(7,29,61),(8,30,62),(9,31,63),(10,32,64),(11,33,65),(12,34,66),(13,35,67),(14,36,68),(15,37,69),(16,38,70),(17,39,71),(18,40,72),(19,41,73),(20,42,74),(21,43,75),(22,44,76),(23,45,77),(24,46,78),(25,47,53),(26,48,54)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71)]])
S3×C26 is a maximal subgroup of
C39⋊D4 C13⋊D12
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AJ | 39A | ··· | 39L | 78A | ··· | 78L |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | S3 | D6 | S3×C13 | S3×C26 |
| kernel | S3×C26 | S3×C13 | C78 | D6 | S3 | C6 | C26 | C13 | C2 | C1 |
| # reps | 1 | 2 | 1 | 12 | 24 | 12 | 1 | 1 | 12 | 12 |
Matrix representation of S3×C26 ►in GL4(𝔽79) generated by
| 38 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 78 | 78 |
| 0 | 0 | 1 | 0 |
| 78 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 78 | 78 |
G:=sub<GL(4,GF(79))| [38,0,0,0,0,78,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,78,1,0,0,78,0],[78,0,0,0,0,1,0,0,0,0,1,78,0,0,0,78] >;
S3×C26 in GAP, Magma, Sage, TeX
S_3\times C_{26} % in TeX
G:=Group("S3xC26"); // GroupNames label
G:=SmallGroup(156,16);
// by ID
G=gap.SmallGroup(156,16);
# by ID
G:=PCGroup([4,-2,-2,-13,-3,1667]);
// Polycyclic
G:=Group<a,b,c|a^26=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export