direct product, non-abelian, soluble, monomial
Aliases: C13×S4, A4⋊C26, (C2×C26)⋊1S3, C22⋊(S3×C13), (A4×C13)⋊3C2, SmallGroup(312,47)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C13×S4 |
Generators and relations for C13×S4
G = < a,b,c,d,e | a13=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Smallest permutation representation of C13×S4
►On 52 points
Generators in S52
(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)
(1 25)(2 26)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(27 50)(28 51)(29 52)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)
(1 39)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 51)(15 52)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)
(14 28 51)(15 29 52)(16 30 40)(17 31 41)(18 32 42)(19 33 43)(20 34 44)(21 35 45)(22 36 46)(23 37 47)(24 38 48)(25 39 49)(26 27 50)
(14 51)(15 52)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)
G:=sub<Sym(52)| (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), (1,25)(2,26)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(27,50)(28,51)(29,52)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49), (1,39)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50), (14,28,51)(15,29,52)(16,30,40)(17,31,41)(18,32,42)(19,33,43)(20,34,44)(21,35,45)(22,36,46)(23,37,47)(24,38,48)(25,39,49)(26,27,50), (14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)>;
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), (1,25)(2,26)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(27,50)(28,51)(29,52)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49), (1,39)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50), (14,28,51)(15,29,52)(16,30,40)(17,31,41)(18,32,42)(19,33,43)(20,34,44)(21,35,45)(22,36,46)(23,37,47)(24,38,48)(25,39,49)(26,27,50), (14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50) );
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)], [(1,25),(2,26),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(27,50),(28,51),(29,52),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49)], [(1,39),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,51),(15,52),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50)], [(14,28,51),(15,29,52),(16,30,40),(17,31,41),(18,32,42),(19,33,43),(20,34,44),(21,35,45),(22,36,46),(23,37,47),(24,38,48),(25,39,49),(26,27,50)], [(14,51),(15,52),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4 | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26X | 39A | ··· | 39L | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 3 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 |
| size | 1 | 3 | 6 | 8 | 6 | 1 | ··· | 1 | 3 | ··· | 3 | 6 | ··· | 6 | 8 | ··· | 8 | 6 | ··· | 6 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C13 | C26 | S3 | S3×C13 | S4 | C13×S4 |
| kernel | C13×S4 | A4×C13 | S4 | A4 | C2×C26 | C22 | C13 | C1 |
| # reps | 1 | 1 | 12 | 12 | 1 | 12 | 2 | 24 |
Matrix representation of C13×S4 ►in GL3(𝔽157) generated by
| 99 | 0 | 0 |
| 0 | 99 | 0 |
| 0 | 0 | 99 |
| 0 | 0 | 1 |
| 156 | 156 | 156 |
| 1 | 0 | 0 |
| 156 | 156 | 156 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 156 | 156 | 156 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(157))| [99,0,0,0,99,0,0,0,99],[0,156,1,0,156,0,1,156,0],[156,0,0,156,0,1,156,1,0],[1,156,0,0,156,1,0,156,0],[1,0,0,0,0,1,0,1,0] >;
C13×S4 in GAP, Magma, Sage, TeX
C_{13}\times S_4 % in TeX
G:=Group("C13xS4"); // GroupNames label
G:=SmallGroup(312,47);
// by ID
G=gap.SmallGroup(312,47);
# by ID
G:=PCGroup([5,-2,-13,-3,-2,2,782,3123,133,1954,239]);
// Polycyclic
G:=Group<a,b,c,d,e|a^13=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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