metacyclic, supersoluble, monomial
Aliases: D44⋊C5, C4⋊F11, C44⋊1C10, D22⋊1C10, C11⋊C5⋊1D4, C11⋊1(C5×D4), (C2×F11)⋊1C2, C2.4(C2×F11), C22.3(C2×C10), (C4×C11⋊C5)⋊1C2, (C2×C11⋊C5).3C22, SmallGroup(440,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D44⋊C5
G = < a,b,c | a44=b2=c5=1, bab=a-1, cac-1=a5, cbc-1=a4b >
22C2
22C2
11C5
11C22
11C22
11C10
22C10
22C10
2D11
2D11
11D4
11C2×C10
11C2×C10
11C20
2F11
2F11
11C5×D4
Character table of D44⋊C5
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 11 | 20A | 20B | 20C | 20D | 22 | 44A | 44B | |
| size | 1 | 1 | 22 | 22 | 2 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 22 | 22 | 22 | 22 | 22 | 22 | 22 | 22 | 10 | 22 | 22 | 22 | 22 | 10 | 10 | 10 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | ζ52 | ζ54 | ζ53 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | ζ54 | -ζ53 | -ζ54 | -ζ5 | ζ5 | ζ52 | ζ53 | -ζ52 | 1 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | 1 | -1 | -1 | linear of order 10 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ54 | ζ53 | ζ5 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | ζ53 | ζ5 | ζ53 | ζ52 | ζ52 | ζ54 | ζ5 | ζ54 | 1 | ζ5 | ζ54 | ζ53 | ζ52 | 1 | 1 | 1 | linear of order 5 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | ζ53 | ζ5 | ζ52 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | ζ5 | ζ52 | ζ5 | ζ54 | ζ54 | ζ53 | ζ52 | ζ53 | 1 | ζ52 | ζ53 | ζ5 | ζ54 | 1 | 1 | 1 | linear of order 5 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | ζ54 | ζ53 | ζ5 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | -ζ53 | -ζ5 | -ζ53 | -ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ54 | 1 | ζ5 | ζ54 | ζ53 | ζ52 | 1 | 1 | 1 | linear of order 10 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | ζ5 | ζ52 | ζ54 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | ζ52 | -ζ54 | -ζ52 | -ζ53 | ζ53 | ζ5 | ζ54 | -ζ5 | 1 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | 1 | -1 | -1 | linear of order 10 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | ζ52 | ζ54 | ζ53 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | -ζ54 | -ζ53 | -ζ54 | -ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ52 | 1 | ζ53 | ζ52 | ζ54 | ζ5 | 1 | 1 | 1 | linear of order 10 |
| ρ11 | 1 | 1 | -1 | 1 | -1 | ζ53 | ζ5 | ζ52 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | -ζ5 | ζ52 | ζ5 | ζ54 | -ζ54 | -ζ53 | -ζ52 | ζ53 | 1 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | 1 | -1 | -1 | linear of order 10 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | ζ5 | ζ52 | ζ54 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | ζ52 | ζ54 | ζ52 | ζ53 | ζ53 | ζ5 | ζ54 | ζ5 | 1 | ζ54 | ζ5 | ζ52 | ζ53 | 1 | 1 | 1 | linear of order 5 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | ζ52 | ζ54 | ζ53 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | -ζ54 | ζ53 | ζ54 | ζ5 | -ζ5 | -ζ52 | -ζ53 | ζ52 | 1 | -ζ53 | -ζ52 | -ζ54 | -ζ5 | 1 | -1 | -1 | linear of order 10 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | ζ53 | ζ5 | ζ52 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | -ζ5 | -ζ52 | -ζ5 | -ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ53 | 1 | ζ52 | ζ53 | ζ5 | ζ54 | 1 | 1 | 1 | linear of order 10 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | ζ52 | ζ54 | ζ53 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | ζ54 | ζ53 | ζ54 | ζ5 | ζ5 | ζ52 | ζ53 | ζ52 | 1 | ζ53 | ζ52 | ζ54 | ζ5 | 1 | 1 | 1 | linear of order 5 |
| ρ16 | 1 | 1 | -1 | 1 | -1 | ζ5 | ζ52 | ζ54 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | -ζ52 | ζ54 | ζ52 | ζ53 | -ζ53 | -ζ5 | -ζ54 | ζ5 | 1 | -ζ54 | -ζ5 | -ζ52 | -ζ53 | 1 | -1 | -1 | linear of order 10 |
| ρ17 | 1 | 1 | 1 | -1 | -1 | ζ54 | ζ53 | ζ5 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | ζ53 | -ζ5 | -ζ53 | -ζ52 | ζ52 | ζ54 | ζ5 | -ζ54 | 1 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | 1 | -1 | -1 | linear of order 10 |
| ρ18 | 1 | 1 | 1 | -1 | -1 | ζ53 | ζ5 | ζ52 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | ζ5 | -ζ52 | -ζ5 | -ζ54 | ζ54 | ζ53 | ζ52 | -ζ53 | 1 | -ζ52 | -ζ53 | -ζ5 | -ζ54 | 1 | -1 | -1 | linear of order 10 |
| ρ19 | 1 | 1 | -1 | -1 | 1 | ζ5 | ζ52 | ζ54 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | -ζ52 | -ζ54 | -ζ52 | -ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ5 | 1 | ζ54 | ζ5 | ζ52 | ζ53 | 1 | 1 | 1 | linear of order 10 |
| ρ20 | 1 | 1 | -1 | 1 | -1 | ζ54 | ζ53 | ζ5 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | -ζ53 | ζ5 | ζ53 | ζ52 | -ζ52 | -ζ54 | -ζ5 | ζ54 | 1 | -ζ5 | -ζ54 | -ζ53 | -ζ52 | 1 | -1 | -1 | linear of order 10 |
| ρ21 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | -2 | 0 | 0 | 0 | 2ζ5 | 2ζ52 | 2ζ54 | 2ζ53 | -2ζ52 | -2ζ54 | -2ζ5 | -2ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | complex lifted from C5×D4 |
| ρ23 | 2 | -2 | 0 | 0 | 0 | 2ζ54 | 2ζ53 | 2ζ5 | 2ζ52 | -2ζ53 | -2ζ5 | -2ζ54 | -2ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | complex lifted from C5×D4 |
| ρ24 | 2 | -2 | 0 | 0 | 0 | 2ζ53 | 2ζ5 | 2ζ52 | 2ζ54 | -2ζ5 | -2ζ52 | -2ζ53 | -2ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | complex lifted from C5×D4 |
| ρ25 | 2 | -2 | 0 | 0 | 0 | 2ζ52 | 2ζ54 | 2ζ53 | 2ζ5 | -2ζ54 | -2ζ53 | -2ζ52 | -2ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | complex lifted from C5×D4 |
| ρ26 | 10 | 10 | 0 | 0 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F11 |
| ρ27 | 10 | 10 | 0 | 0 | -10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C2×F11 |
| ρ28 | 10 | -10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | √11 | -√11 | orthogonal faithful |
| ρ29 | 10 | -10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -√11 | √11 | orthogonal faithful |
►On 44 points
Generators in S44
(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 11)(2 10)(3 9)(4 8)(5 7)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)
(2 10 38 26 6)(3 19 31 7 11)(4 28 24 32 16)(5 37 17 13 21)(8 20 40 44 36)(9 29 33 25 41)(14 30 42 18 22)(15 39 35 43 27)
G:=sub<Sym(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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29), (2,10,38,26,6)(3,19,31,7,11)(4,28,24,32,16)(5,37,17,13,21)(8,20,40,44,36)(9,29,33,25,41)(14,30,42,18,22)(15,39,35,43,27)>;
G:=Group( (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29), (2,10,38,26,6)(3,19,31,7,11)(4,28,24,32,16)(5,37,17,13,21)(8,20,40,44,36)(9,29,33,25,41)(14,30,42,18,22)(15,39,35,43,27) );
G=PermutationGroup([[(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29)], [(2,10,38,26,6),(3,19,31,7,11),(4,28,24,32,16),(5,37,17,13,21),(8,20,40,44,36),(9,29,33,25,41),(14,30,42,18,22),(15,39,35,43,27)]])
Matrix representation of D44⋊C5 ►in GL10(𝔽661)
| 0 | 0 | 448 | 0 | 0 | 213 | 448 | 437 | 437 | 448 |
| 213 | 0 | 448 | 448 | 0 | 213 | 0 | 224 | 213 | 224 |
| 0 | 213 | 448 | 448 | 448 | 213 | 0 | 437 | 0 | 0 |
| 437 | 0 | 0 | 448 | 448 | 0 | 0 | 437 | 213 | 448 |
| 213 | 437 | 448 | 0 | 448 | 0 | 448 | 437 | 213 | 0 |
| 0 | 213 | 224 | 448 | 0 | 0 | 448 | 224 | 213 | 0 |
| 213 | 0 | 0 | 224 | 448 | 213 | 448 | 224 | 0 | 0 |
| 213 | 213 | 448 | 0 | 224 | 0 | 0 | 224 | 0 | 448 |
| 213 | 213 | 0 | 448 | 0 | 437 | 448 | 437 | 0 | 448 |
| 0 | 213 | 0 | 0 | 448 | 213 | 224 | 224 | 213 | 448 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 660 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 660 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 660 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 660 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 660 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 660 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 660 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 660 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 660 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 660 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(10,GF(661))| [0,213,0,437,213,0,213,213,213,0,0,0,213,0,437,213,0,213,213,213,448,448,448,0,448,224,0,448,0,0,0,448,448,448,0,448,224,0,448,0,0,0,448,448,448,0,448,224,0,448,213,213,213,0,0,0,213,0,437,213,448,0,0,0,448,448,448,0,448,224,437,224,437,437,437,224,224,224,437,224,437,213,0,213,213,213,0,0,0,213,448,224,0,448,0,0,0,448,448,448],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,660,660,660,660,660,660,660,660,660,660,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0] >;
D44⋊C5 in GAP, Magma, Sage, TeX
D_{44}\rtimes C_5 % in TeX
G:=Group("D44:C5"); // GroupNames label
G:=SmallGroup(440,9);
// by ID
G=gap.SmallGroup(440,9);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-11,221,106,10004,2264]);
// Polycyclic
G:=Group<a,b,c|a^44=b^2=c^5=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b>;
// generators/relations
Export
Subgroup lattice of D44⋊C5 in TeX
Character table of D44⋊C5 in TeX