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G = D44⋊C5order 440 = 23·5·11

The semidirect product of D44 and C5 acting faithfully

Aliases: D44⋊C5, C4⋊F11, C441C10, D221C10, C11⋊C51D4, C111(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

 Derived series C1 — C22 — D44⋊C5
 Chief series C1 — C11 — C22 — C2×C11⋊C5 — C2×F11 — D44⋊C5
 Lower central C11 — C22 — D44⋊C5
 Upper central C1 — C2 — C4

Generators and relations for D44⋊C5
G = < a,b,c | a44=b2=c5=1, bab=a-1, cac-1=a5, cbc-1=a4b >

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

Smallest permutation representation of D44⋊C5
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`

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