direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D11, C55⋊2C2, C11⋊3C10, SmallGroup(110,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C5×D11 |
Generators and relations for C5×D11
G = < a,b,c | a5=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D11
►On 55 points
Generators in S55
(1 54 43 32 21)(2 55 44 33 22)(3 45 34 23 12)(4 46 35 24 13)(5 47 36 25 14)(6 48 37 26 15)(7 49 38 27 16)(8 50 39 28 17)(9 51 40 29 18)(10 52 41 30 19)(11 53 42 31 20)
(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 53 54 55)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)
G:=sub<Sym(55)| (1,54,43,32,21)(2,55,44,33,22)(3,45,34,23,12)(4,46,35,24,13)(5,47,36,25,14)(6,48,37,26,15)(7,49,38,27,16)(8,50,39,28,17)(9,51,40,29,18)(10,52,41,30,19)(11,53,42,31,20), (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)(45,46,47,48,49,50,51,52,53,54,55), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)>;
G:=Group( (1,54,43,32,21)(2,55,44,33,22)(3,45,34,23,12)(4,46,35,24,13)(5,47,36,25,14)(6,48,37,26,15)(7,49,38,27,16)(8,50,39,28,17)(9,51,40,29,18)(10,52,41,30,19)(11,53,42,31,20), (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)(45,46,47,48,49,50,51,52,53,54,55), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54) );
G=PermutationGroup([[(1,54,43,32,21),(2,55,44,33,22),(3,45,34,23,12),(4,46,35,24,13),(5,47,36,25,14),(6,48,37,26,15),(7,49,38,27,16),(8,50,39,28,17),(9,51,40,29,18),(10,52,41,30,19),(11,53,42,31,20)], [(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),(45,46,47,48,49,50,51,52,53,54,55)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54)]])
35 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 11A | ··· | 11E | 55A | ··· | 55T |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 55 | ··· | 55 |
| size | 1 | 11 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 2 | ··· | 2 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D11 | C5×D11 |
| kernel | C5×D11 | C55 | D11 | C11 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 5 | 20 |
Matrix representation of C5×D11 ►in GL2(𝔽331) generated by
| 323 | 0 |
| 0 | 323 |
| 330 | 1 |
| 206 | 124 |
| 330 | 0 |
| 206 | 1 |
G:=sub<GL(2,GF(331))| [323,0,0,323],[330,206,1,124],[330,206,0,1] >;
C5×D11 in GAP, Magma, Sage, TeX
C_5\times D_{11} % in TeX
G:=Group("C5xD11"); // GroupNames label
G:=SmallGroup(110,4);
// by ID
G=gap.SmallGroup(110,4);
# by ID
G:=PCGroup([3,-2,-5,-11,902]);
// Polycyclic
G:=Group<a,b,c|a^5=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export