direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C11×Dic11, C11⋊C44, C22.C22, C112⋊2C4, C22.4D11, C2.(C11×D11), (C11×C22).1C2, SmallGroup(484,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C11×Dic11 |
Generators and relations for C11×Dic11
G = < a,b,c | a11=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C11×Dic11
►On 44 points
Generators in S44
(1 7 13 19 3 9 15 21 5 11 17)(2 8 14 20 4 10 16 22 6 12 18)(23 39 33 27 43 37 31 25 41 35 29)(24 40 34 28 44 38 32 26 42 36 30)
(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)
(1 23 12 34)(2 44 13 33)(3 43 14 32)(4 42 15 31)(5 41 16 30)(6 40 17 29)(7 39 18 28)(8 38 19 27)(9 37 20 26)(10 36 21 25)(11 35 22 24)
G:=sub<Sym(44)| (1,7,13,19,3,9,15,21,5,11,17)(2,8,14,20,4,10,16,22,6,12,18)(23,39,33,27,43,37,31,25,41,35,29)(24,40,34,28,44,38,32,26,42,36,30), (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), (1,23,12,34)(2,44,13,33)(3,43,14,32)(4,42,15,31)(5,41,16,30)(6,40,17,29)(7,39,18,28)(8,38,19,27)(9,37,20,26)(10,36,21,25)(11,35,22,24)>;
G:=Group( (1,7,13,19,3,9,15,21,5,11,17)(2,8,14,20,4,10,16,22,6,12,18)(23,39,33,27,43,37,31,25,41,35,29)(24,40,34,28,44,38,32,26,42,36,30), (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), (1,23,12,34)(2,44,13,33)(3,43,14,32)(4,42,15,31)(5,41,16,30)(6,40,17,29)(7,39,18,28)(8,38,19,27)(9,37,20,26)(10,36,21,25)(11,35,22,24) );
G=PermutationGroup([[(1,7,13,19,3,9,15,21,5,11,17),(2,8,14,20,4,10,16,22,6,12,18),(23,39,33,27,43,37,31,25,41,35,29),(24,40,34,28,44,38,32,26,42,36,30)], [(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)], [(1,23,12,34),(2,44,13,33),(3,43,14,32),(4,42,15,31),(5,41,16,30),(6,40,17,29),(7,39,18,28),(8,38,19,27),(9,37,20,26),(10,36,21,25),(11,35,22,24)]])
154 conjugacy classes
| class | 1 | 2 | 4A | 4B | 11A | ··· | 11J | 11K | ··· | 11BM | 22A | ··· | 22J | 22K | ··· | 22BM | 44A | ··· | 44T |
| order | 1 | 2 | 4 | 4 | 11 | ··· | 11 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 11 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C11 | C22 | C44 | D11 | Dic11 | C11×D11 | C11×Dic11 |
| kernel | C11×Dic11 | C11×C22 | C112 | Dic11 | C22 | C11 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 5 | 5 | 50 | 50 |
Matrix representation of C11×Dic11 ►in GL2(𝔽23) generated by
| 18 | 0 |
| 0 | 18 |
| 21 | 0 |
| 0 | 11 |
| 0 | 22 |
| 1 | 0 |
G:=sub<GL(2,GF(23))| [18,0,0,18],[21,0,0,11],[0,1,22,0] >;
C11×Dic11 in GAP, Magma, Sage, TeX
C_{11}\times {\rm Dic}_{11} % in TeX
G:=Group("C11xDic11"); // GroupNames label
G:=SmallGroup(484,5);
// by ID
G=gap.SmallGroup(484,5);
# by ID
G:=PCGroup([4,-2,-11,-2,-11,88,7043]);
// Polycyclic
G:=Group<a,b,c|a^11=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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