direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D13, C52⋊2C2, C4○Dic13, C2.1D26, D26.2C2, Dic13⋊2C2, C26.2C22, C13⋊2(C2×C4), SmallGroup(104,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C4×D13 |
Generators and relations for C4×D13
G = < a,b,c | a4=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D13
►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)]])
C4×D13 is a maximal subgroup of
C8⋊D13 D13⋊C8 C52.C4 C52⋊C4 D52⋊5C2 D4⋊2D13 D52⋊C2 D78.C2
C4×D13 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 | C4×D13 |
| kernel | C4×D13 | Dic13 | C52 | D26 | D13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 |
Matrix representation of C4×D13 ►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] >;
C4×D13 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
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