direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C11×D11, C11≀C2, AΣL1(𝔽121), C11⋊C22, C112⋊1C2, SmallGroup(242,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C11×D11 |
Generators and relations for C11×D11
G = < a,b,c | a11=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of C11×D11
►On 22 points - transitive group 22T7
Generators in S22
(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)
(1 11 10 9 8 7 6 5 4 3 2)(12 13 14 15 16 17 18 19 20 21 22)
(1 22)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)
G:=sub<Sym(22)| (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,11,10,9,8,7,6,5,4,3,2)(12,13,14,15,16,17,18,19,20,21,22), (1,22)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,11,10,9,8,7,6,5,4,3,2)(12,13,14,15,16,17,18,19,20,21,22), (1,22)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22)], [(1,11,10,9,8,7,6,5,4,3,2),(12,13,14,15,16,17,18,19,20,21,22)], [(1,22),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21)]])
G:=TransitiveGroup(22,7);
77 conjugacy classes
| class | 1 | 2 | 11A | ··· | 11J | 11K | ··· | 11BM | 22A | ··· | 22J |
| order | 1 | 2 | 11 | ··· | 11 | 11 | ··· | 11 | 22 | ··· | 22 |
| size | 1 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C11 | C22 | D11 | C11×D11 |
| kernel | C11×D11 | C112 | D11 | C11 | C11 | C1 |
| # reps | 1 | 1 | 10 | 10 | 5 | 50 |
Matrix representation of C11×D11 ►in GL2(𝔽23) generated by
| 9 | 0 |
| 0 | 9 |
| 4 | 0 |
| 0 | 6 |
| 0 | 6 |
| 4 | 0 |
G:=sub<GL(2,GF(23))| [9,0,0,9],[4,0,0,6],[0,4,6,0] >;
C11×D11 in GAP, Magma, Sage, TeX
C_{11}\times D_{11} % in TeX
G:=Group("C11xD11"); // GroupNames label
G:=SmallGroup(242,3);
// by ID
G=gap.SmallGroup(242,3);
# by ID
G:=PCGroup([3,-2,-11,-11,1982]);
// Polycyclic
G:=Group<a,b,c|a^11=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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