direct product, metabelian, supersoluble, monomial, A-group
Aliases: D112, C11⋊1D22, C112⋊C22, C11⋊D11⋊C2, (C11×D11)⋊C2, SmallGroup(484,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C112 — D112 |
Generators and relations for D112
G = < a,b,c,d | a11=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
11C2
11C2
121C2
2C11
2C11
2C11
2C11
2C11
121C22
11D11
11C22
11C22
11D11
22D11
22D11
22D11
22D11
22D11
11D22
11D22
Permutation representations of D112
►On 22 points - transitive group 22T9
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 12)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)
(1 3 5 7 9 11 2 4 6 8 10)(12 21 19 17 15 13 22 20 18 16 14)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 12)(10 13)(11 14)
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), (1,12)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13), (1,3,5,7,9,11,2,4,6,8,10)(12,21,19,17,15,13,22,20,18,16,14), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,12)(10,13)(11,14)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,12)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13), (1,3,5,7,9,11,2,4,6,8,10)(12,21,19,17,15,13,22,20,18,16,14), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,12)(10,13)(11,14) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22)], [(1,12),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13)], [(1,3,5,7,9,11,2,4,6,8,10),(12,21,19,17,15,13,22,20,18,16,14)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,12),(10,13),(11,14)]])
G:=TransitiveGroup(22,9);
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 11A | ··· | 11J | 11K | ··· | 11AI | 22A | ··· | 22J |
| order | 1 | 2 | 2 | 2 | 11 | ··· | 11 | 11 | ··· | 11 | 22 | ··· | 22 |
| size | 1 | 11 | 11 | 121 | 2 | ··· | 2 | 4 | ··· | 4 | 22 | ··· | 22 |
49 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D11 | D22 | D112 |
| kernel | D112 | C11×D11 | C11⋊D11 | D11 | C11 | C1 |
| # reps | 1 | 2 | 1 | 10 | 10 | 25 |
Matrix representation of D112 ►in GL4(𝔽23) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 20 |
| 0 | 0 | 1 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 22 |
| 14 | 10 | 0 | 0 |
| 2 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 3 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 22 |
G:=sub<GL(4,GF(23))| [1,0,0,0,0,1,0,0,0,0,14,1,0,0,20,13],[1,0,0,0,0,1,0,0,0,0,1,8,0,0,0,22],[14,2,0,0,10,13,0,0,0,0,1,0,0,0,0,1],[16,7,0,0,3,7,0,0,0,0,22,0,0,0,0,22] >;
D112 in GAP, Magma, Sage, TeX
D_{11}^2 % in TeX
G:=Group("D11^2"); // GroupNames label
G:=SmallGroup(484,9);
// by ID
G=gap.SmallGroup(484,9);
# by ID
G:=PCGroup([4,-2,-2,-11,-11,246,7043]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^2=c^11=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export