direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C4×C13⋊C6, C52⋊2C6, D13⋊2C12, D26.2C6, Dic13⋊2C6, (C4×D13)⋊C3, C13⋊2(C2×C12), C26.C6⋊2C2, C26.2(C2×C6), C13⋊C3⋊2(C2×C4), (C4×C13⋊C3)⋊2C2, C2.1(C2×C13⋊C6), (C2×C13⋊C6).2C2, (C2×C13⋊C3).2C22, SmallGroup(312,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C4×C13⋊C6 |
Generators and relations for C4×C13⋊C6
G = < a,b,c | a4=b13=c6=1, ab=ba, ac=ca, cbc-1=b10 >
Smallest permutation representation of C4×C13⋊C6
►On 52 points
Generators in S52
(1 40 14 27)(2 41 15 28)(3 42 16 29)(4 43 17 30)(5 44 18 31)(6 45 19 32)(7 46 20 33)(8 47 21 34)(9 48 22 35)(10 49 23 36)(11 50 24 37)(12 51 25 38)(13 52 26 39)
(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 14)(2 18 4 26 10 24)(3 22 7 25 6 21)(5 17 13 23 11 15)(8 16 9 20 12 19)(27 40)(28 44 30 52 36 50)(29 48 33 51 32 47)(31 43 39 49 37 41)(34 42 35 46 38 45)
G:=sub<Sym(52)| (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (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), (1,14)(2,18,4,26,10,24)(3,22,7,25,6,21)(5,17,13,23,11,15)(8,16,9,20,12,19)(27,40)(28,44,30,52,36,50)(29,48,33,51,32,47)(31,43,39,49,37,41)(34,42,35,46,38,45)>;
G:=Group( (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (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), (1,14)(2,18,4,26,10,24)(3,22,7,25,6,21)(5,17,13,23,11,15)(8,16,9,20,12,19)(27,40)(28,44,30,52,36,50)(29,48,33,51,32,47)(31,43,39,49,37,41)(34,42,35,46,38,45) );
G=PermutationGroup([[(1,40,14,27),(2,41,15,28),(3,42,16,29),(4,43,17,30),(5,44,18,31),(6,45,19,32),(7,46,20,33),(8,47,21,34),(9,48,22,35),(10,49,23,36),(11,50,24,37),(12,51,25,38),(13,52,26,39)], [(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)], [(1,14),(2,18,4,26,10,24),(3,22,7,25,6,21),(5,17,13,23,11,15),(8,16,9,20,12,19),(27,40),(28,44,30,52,36,50),(29,48,33,51,32,47),(31,43,39,49,37,41),(34,42,35,46,38,45)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 13A | 13B | 26A | 26B | 52A | 52B | 52C | 52D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 13 | 13 | 26 | 26 | 52 | 52 | 52 | 52 |
| size | 1 | 1 | 13 | 13 | 13 | 13 | 1 | 1 | 13 | 13 | 13 | ··· | 13 | 13 | ··· | 13 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | C13⋊C6 | C2×C13⋊C6 | C4×C13⋊C6 |
| kernel | C4×C13⋊C6 | C26.C6 | C4×C13⋊C3 | C2×C13⋊C6 | C4×D13 | C13⋊C6 | Dic13 | C52 | D26 | D13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 4 |
Matrix representation of C4×C13⋊C6 ►in GL6(𝔽157)
| 129 | 0 | 0 | 0 | 0 | 0 |
| 0 | 129 | 0 | 0 | 0 | 0 |
| 0 | 0 | 129 | 0 | 0 | 0 |
| 0 | 0 | 0 | 129 | 0 | 0 |
| 0 | 0 | 0 | 0 | 129 | 0 |
| 0 | 0 | 0 | 0 | 0 | 129 |
| 91 | 65 | 92 | 66 | 90 | 156 |
| 92 | 65 | 92 | 66 | 90 | 156 |
| 91 | 66 | 92 | 66 | 90 | 156 |
| 91 | 65 | 93 | 66 | 90 | 156 |
| 91 | 65 | 92 | 67 | 90 | 156 |
| 91 | 65 | 92 | 66 | 91 | 156 |
| 1 | 155 | 91 | 67 | 90 | 155 |
| 90 | 66 | 92 | 65 | 91 | 67 |
| 0 | 0 | 0 | 156 | 0 | 0 |
| 156 | 0 | 0 | 0 | 0 | 0 |
| 91 | 133 | 25 | 64 | 92 | 67 |
| 92 | 64 | 25 | 133 | 91 | 155 |
G:=sub<GL(6,GF(157))| [129,0,0,0,0,0,0,129,0,0,0,0,0,0,129,0,0,0,0,0,0,129,0,0,0,0,0,0,129,0,0,0,0,0,0,129],[91,92,91,91,91,91,65,65,66,65,65,65,92,92,92,93,92,92,66,66,66,66,67,66,90,90,90,90,90,91,156,156,156,156,156,156],[1,90,0,156,91,92,155,66,0,0,133,64,91,92,0,0,25,25,67,65,156,0,64,133,90,91,0,0,92,91,155,67,0,0,67,155] >;
C4×C13⋊C6 in GAP, Magma, Sage, TeX
C_4\times C_{13}\rtimes C_6 % in TeX
G:=Group("C4xC13:C6"); // GroupNames label
G:=SmallGroup(312,9);
// by ID
G=gap.SmallGroup(312,9);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,66,7204,464]);
// Polycyclic
G:=Group<a,b,c|a^4=b^13=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations
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