direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C13, C22⋊C39, (C2×C26)⋊1C3, SmallGroup(156,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C13 |
Generators and relations for A4×C13
G = < a,b,c,d | a13=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C13
►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 35)(2 36)(3 37)(4 38)(5 39)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 49)(15 50)(16 51)(17 52)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 40)(11 41)(12 42)(13 43)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)
(14 27 49)(15 28 50)(16 29 51)(17 30 52)(18 31 40)(19 32 41)(20 33 42)(21 34 43)(22 35 44)(23 36 45)(24 37 46)(25 38 47)(26 39 48)
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), (1,35)(2,36)(3,37)(4,38)(5,39)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,40)(11,41)(12,42)(13,43)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39), (14,27,49)(15,28,50)(16,29,51)(17,30,52)(18,31,40)(19,32,41)(20,33,42)(21,34,43)(22,35,44)(23,36,45)(24,37,46)(25,38,47)(26,39,48)>;
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), (1,35)(2,36)(3,37)(4,38)(5,39)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,40)(11,41)(12,42)(13,43)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39), (14,27,49)(15,28,50)(16,29,51)(17,30,52)(18,31,40)(19,32,41)(20,33,42)(21,34,43)(22,35,44)(23,36,45)(24,37,46)(25,38,47)(26,39,48) );
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)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,49),(15,50),(16,51),(17,52),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,40),(11,41),(12,42),(13,43),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39)], [(14,27,49),(15,28,50),(16,29,51),(17,30,52),(18,31,40),(19,32,41),(20,33,42),(21,34,43),(22,35,44),(23,36,45),(24,37,46),(25,38,47),(26,39,48)]])
A4×C13 is a maximal subgroup of
C13⋊S4
52 conjugacy classes
| class | 1 | 2 | 3A | 3B | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39X |
| order | 1 | 2 | 3 | 3 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 |
| size | 1 | 3 | 4 | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C3 | C13 | C39 | A4 | A4×C13 |
| kernel | A4×C13 | C2×C26 | A4 | C22 | C13 | C1 |
| # reps | 1 | 2 | 12 | 24 | 1 | 12 |
Matrix representation of A4×C13 ►in GL3(𝔽79) generated by
| 46 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 78 | 0 | 0 |
| 78 | 0 | 1 |
| 78 | 1 | 0 |
| 0 | 1 | 78 |
| 1 | 0 | 78 |
| 0 | 0 | 78 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(79))| [46,0,0,0,46,0,0,0,46],[78,78,78,0,0,1,0,1,0],[0,1,0,1,0,0,78,78,78],[0,0,1,1,0,0,0,1,0] >;
A4×C13 in GAP, Magma, Sage, TeX
A_4\times C_{13} % in TeX
G:=Group("A4xC13"); // GroupNames label
G:=SmallGroup(156,13);
// by ID
G=gap.SmallGroup(156,13);
# by ID
G:=PCGroup([4,-3,-13,-2,2,938,1875]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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