direct product, metacyclic, supersoluble, monomial
Aliases: D4×C11⋊C5, C44⋊3C10, (D4×C11)⋊C5, C11⋊3(C5×D4), (C2×C22)⋊3C10, C22.7(C2×C10), C4⋊(C2×C11⋊C5), C22⋊(C2×C11⋊C5), (C4×C11⋊C5)⋊3C2, (C22×C11⋊C5)⋊3C2, C2.2(C22×C11⋊C5), (C2×C11⋊C5).7C22, SmallGroup(440,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C11 — C22 — C2×C11⋊C5 — C22×C11⋊C5 — D4×C11⋊C5 |
Generators and relations for D4×C11⋊C5
G = < a,b,c,d | a4=b2=c11=d5=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Smallest permutation representation of D4×C11⋊C5
►On 44 points
Generators in S44
(1 23 12 34)(2 24 13 35)(3 25 14 36)(4 26 15 37)(5 27 16 38)(6 28 17 39)(7 29 18 40)(8 30 19 41)(9 31 20 42)(10 32 21 43)(11 33 22 44)
(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)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)
G:=sub<Sym(44)| (1,23,12,34)(2,24,13,35)(3,25,14,36)(4,26,15,37)(5,27,16,38)(6,28,17,39)(7,29,18,40)(8,30,19,41)(9,31,20,42)(10,32,21,43)(11,33,22,44), (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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)>;
G:=Group( (1,23,12,34)(2,24,13,35)(3,25,14,36)(4,26,15,37)(5,27,16,38)(6,28,17,39)(7,29,18,40)(8,30,19,41)(9,31,20,42)(10,32,21,43)(11,33,22,44), (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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40) );
G=PermutationGroup([[(1,23,12,34),(2,24,13,35),(3,25,14,36),(4,26,15,37),(5,27,16,38),(6,28,17,39),(7,29,18,40),(8,30,19,41),(9,31,20,42),(10,32,21,43),(11,33,22,44)], [(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)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18),(24,27,28,32,26),(25,31,33,30,29),(35,38,39,43,37),(36,42,44,41,40)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 11A | 11B | 20A | 20B | 20C | 20D | 22A | 22B | 22C | 22D | 22E | 22F | 44A | 44B |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 11 | 11 | 20 | 20 | 20 | 20 | 22 | 22 | 22 | 22 | 22 | 22 | 44 | 44 |
| size | 1 | 1 | 2 | 2 | 2 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 22 | ··· | 22 | 5 | 5 | 22 | 22 | 22 | 22 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 2 | 2 | 5 | 5 | 5 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D4×C11⋊C5 | D4 | C5×D4 | C11⋊C5 | C2×C11⋊C5 | C2×C11⋊C5 |
| kernel | D4×C11⋊C5 | C4×C11⋊C5 | C22×C11⋊C5 | D4×C11 | C44 | C2×C22 | C1 | C11⋊C5 | C11 | D4 | C4 | C22 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of D4×C11⋊C5 ►in GL7(𝔽661)
| 660 | 104 | 0 | 0 | 0 | 0 | 0 |
| 572 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 89 | 660 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 1 | 660 | 45 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 616 | 659 | 1 | 615 | 43 |
| 0 | 0 | 615 | 43 | 1 | 659 | 44 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(7,GF(661))| [660,572,0,0,0,0,0,104,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,89,0,0,0,0,0,0,660,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,44,1,0,0,0,0,0,1,0,1,0,0,0,0,660,0,0,1,0,0,0,45,0,0,0,1,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,616,615,0,0,0,0,0,659,43,1,0,0,0,0,1,1,0,0,0,0,1,615,659,0,0,0,0,0,43,44,0] >;
D4×C11⋊C5 in GAP, Magma, Sage, TeX
D_4\times C_{11}\rtimes C_5 % in TeX
G:=Group("D4xC11:C5"); // GroupNames label
G:=SmallGroup(440,13);
// by ID
G=gap.SmallGroup(440,13);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-11,221,1014]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^11=d^5=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^-1=c^3>;
// generators/relations
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