direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4xD13, C52:2C2, C4oDic13, C2.1D26, D26.2C2, Dic13:2C2, C26.2C22, C13:2(C2xC4), SmallGroup(104,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C4xD13 |
Generators and relations for C4xD13
G = < a,b,c | a4=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C4, C22, C2xC4, D13, D26, C4xD13
Smallest permutation representation of C4xD13
►On 52 points
Generators in S52
(1 44 16 33)(2 45 17 34)(3 46 18 35)(4 47 19 36)(5 48 20 37)(6 49 21 38)(7 50 22 39)(8 51 23 27)(9 52 24 28)(10 40 25 29)(11 41 26 30)(12 42 14 31)(13 43 15 32)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 17)(15 16)(18 26)(19 25)(20 24)(21 23)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(40 47)(41 46)(42 45)(43 44)(48 52)(49 51)
G:=sub<Sym(52)| (1,44,16,33)(2,45,17,34)(3,46,18,35)(4,47,19,36)(5,48,20,37)(6,49,21,38)(7,50,22,39)(8,51,23,27)(9,52,24,28)(10,40,25,29)(11,41,26,30)(12,42,14,31)(13,43,15,32), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,17)(15,16)(18,26)(19,25)(20,24)(21,23)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51)>;
G:=Group( (1,44,16,33)(2,45,17,34)(3,46,18,35)(4,47,19,36)(5,48,20,37)(6,49,21,38)(7,50,22,39)(8,51,23,27)(9,52,24,28)(10,40,25,29)(11,41,26,30)(12,42,14,31)(13,43,15,32), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,17)(15,16)(18,26)(19,25)(20,24)(21,23)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51) );
G=PermutationGroup([[(1,44,16,33),(2,45,17,34),(3,46,18,35),(4,47,19,36),(5,48,20,37),(6,49,21,38),(7,50,22,39),(8,51,23,27),(9,52,24,28),(10,40,25,29),(11,41,26,30),(12,42,14,31),(13,43,15,32)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,17),(15,16),(18,26),(19,25),(20,24),(21,23),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(40,47),(41,46),(42,45),(43,44),(48,52),(49,51)]])
C4xD13 is a maximal subgroup of
C8:D13 D13:C8 C52.C4 C52:C4 D52:5C2 D4:2D13 D52:C2 D78.C2
C4xD13 is a maximal quotient of
C8:D13 C26.D4 D26:C4 D78.C2
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 13 | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D13 | D26 | C4xD13 |
| kernel | C4xD13 | Dic13 | C52 | D26 | D13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 |
Matrix representation of C4xD13 ►in GL3(F53) generated by
| 30 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 52 | 8 |
| 52 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(53))| [30,0,0,0,1,0,0,0,1],[1,0,0,0,0,52,0,1,8],[52,0,0,0,0,1,0,1,0] >;
C4xD13 in GAP, Magma, Sage, TeX
C_4\times D_{13} % in TeX
G:=Group("C4xD13"); // GroupNames label
G:=SmallGroup(104,5);
// by ID
G=gap.SmallGroup(104,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-13,21,1539]);
// Polycyclic
G:=Group<a,b,c|a^4=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export