direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D11, C4⋊1D22, C44⋊C22, D44⋊3C2, C22⋊1D22, D22⋊2C22, C22.5C23, Dic11⋊1C22, C11⋊2(C2×D4), (C2×C22)⋊C22, (C4×D11)⋊1C2, (D4×C11)⋊2C2, C11⋊D4⋊1C2, (C22×D11)⋊2C2, C2.6(C22×D11), SmallGroup(176,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D11
G = < a,b,c,d | a4=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 320 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C11, C2×D4, D11, D11, C22, C22, Dic11, C44, D22, D22, D22, C2×C22, C4×D11, D44, C11⋊D4, D4×C11, C22×D11, D4×D11
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, D22, C22×D11, D4×D11
►On 44 points
(1 32 21 43)(2 33 22 44)(3 23 12 34)(4 24 13 35)(5 25 14 36)(6 26 15 37)(7 27 16 38)(8 28 17 39)(9 29 18 40)(10 30 19 41)(11 31 20 42)
(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 22)(11 21)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)
G:=sub<Sym(44)| (1,32,21,43)(2,33,22,44)(3,23,12,34)(4,24,13,35)(5,25,14,36)(6,26,15,37)(7,27,16,38)(8,28,17,39)(9,29,18,40)(10,30,19,41)(11,31,20,42), (23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)>;
G:=Group( (1,32,21,43)(2,33,22,44)(3,23,12,34)(4,24,13,35)(5,25,14,36)(6,26,15,37)(7,27,16,38)(8,28,17,39)(9,29,18,40)(10,30,19,41)(11,31,20,42), (23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41) );
G=PermutationGroup([[(1,32,21,43),(2,33,22,44),(3,23,12,34),(4,24,13,35),(5,25,14,36),(6,26,15,37),(7,27,16,38),(8,28,17,39),(9,29,18,40),(10,30,19,41),(11,31,20,42)], [(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,22),(11,21),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,44),(31,43),(32,42),(33,41)]])
D4×D11 is a maximal subgroup of
D4⋊D22 D88⋊C2 D4⋊6D22 D4⋊8D22
D4×D11 is a maximal quotient of
C22⋊Dic22 Dic11⋊4D4 C22⋊D44 D22.D4 D22⋊D4 Dic11.D4 C44⋊Q8 D44⋊C4 D22.5D4 C4⋊2D44 D22⋊Q8 D4⋊D22 D8⋊3D11 D88⋊C2 D4.D22 Q8.D22 Q16⋊D11 D88⋊5C2 C23⋊D22 C44⋊2D4 Dic11⋊D4 C44⋊D4
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22O | 44A | ··· | 44E |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 2 | 11 | 11 | 22 | 22 | 2 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D11 | D22 | D22 | D4×D11 |
| kernel | D4×D11 | C4×D11 | D44 | C11⋊D4 | D4×C11 | C22×D11 | D11 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 5 | 5 | 10 | 5 |
Matrix representation of D4×D11 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 53 | 4 |
| 0 | 0 | 54 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 18 | 88 |
| 7 | 1 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 41 | 0 | 0 |
| 30 | 62 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,53,54,0,0,4,36],[1,0,0,0,0,1,0,0,0,0,1,18,0,0,0,88],[7,88,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[27,30,0,0,41,62,0,0,0,0,88,0,0,0,0,88] >;
D4×D11 in GAP, Magma, Sage, TeX
D_4\times D_{11} % in TeX
G:=Group("D4xD11"); // GroupNames label
G:=SmallGroup(176,31);
// by ID
G=gap.SmallGroup(176,31);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,97,4004]);
// Polycyclic
G:=Group<a,b,c,d|a^4=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