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## G = C22⋊F11order 440 = 23·5·11

### The semidirect product of C22 and F11 acting via F11/C11⋊C5=C2

Aliases: C22⋊F11, D222C10, Dic11⋊C10, C11⋊C20⋊C2, C11⋊C52D4, C11⋊D4⋊C5, C112(C5×D4), (C2×C22)⋊1C10, (C2×F11)⋊2C2, C2.5(C2×F11), C22.5(C2×C10), (C22×C11⋊C5)⋊1C2, (C2×C11⋊C5).5C22, SmallGroup(440,11)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22⋊F11
G = < a,b,c,d | a2=b2=c11=d10=1, dad-1=ab=ba, ac=ca, bc=cb, bd=db, dcd-1=c6 >

Character table of C22⋊F11

 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 22A 22B 22C size 1 1 2 22 22 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 ζ54 ζ5 ζ53 ζ52 ζ53 ζ5 ζ54 ζ52 -ζ53 -ζ52 -ζ5 -ζ54 -ζ53 -ζ52 -ζ5 -ζ54 1 ζ5 ζ53 ζ52 ζ54 -1 1 -1 linear of order 10 ρ6 1 1 1 1 1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ52 ζ53 ζ54 ζ5 ζ54 ζ52 ζ53 ζ5 ζ54 ζ52 ζ53 1 ζ52 ζ5 ζ54 ζ53 1 1 1 linear of order 5 ρ7 1 1 -1 -1 1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ54 ζ5 ζ53 -ζ52 -ζ53 -ζ54 -ζ5 -ζ52 -ζ53 -ζ54 -ζ5 1 ζ54 ζ52 ζ53 ζ5 -1 1 -1 linear of order 10 ρ8 1 1 1 1 1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ5 ζ54 ζ52 ζ53 ζ52 ζ5 ζ54 ζ53 ζ52 ζ5 ζ54 1 ζ5 ζ53 ζ52 ζ54 1 1 1 linear of order 5 ρ9 1 1 -1 1 -1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ5 ζ54 ζ52 -ζ53 -ζ52 -ζ5 ζ54 ζ53 ζ52 ζ5 -ζ54 1 -ζ5 -ζ53 -ζ52 -ζ54 -1 1 -1 linear of order 10 ρ10 1 1 1 1 1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ53 ζ52 ζ5 ζ54 ζ5 ζ53 ζ52 ζ54 ζ5 ζ53 ζ52 1 ζ53 ζ54 ζ5 ζ52 1 1 1 linear of order 5 ρ11 1 1 -1 -1 1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ53 ζ52 ζ5 -ζ54 -ζ5 -ζ53 -ζ52 -ζ54 -ζ5 -ζ53 -ζ52 1 ζ53 ζ54 ζ5 ζ52 -1 1 -1 linear of order 10 ρ12 1 1 1 -1 -1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ5 ζ54 ζ52 ζ53 ζ52 ζ5 -ζ54 -ζ53 -ζ52 -ζ5 ζ54 1 -ζ5 -ζ53 -ζ52 -ζ54 1 1 1 linear of order 10 ρ13 1 1 -1 1 -1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ54 ζ5 ζ53 -ζ52 -ζ53 -ζ54 ζ5 ζ52 ζ53 ζ54 -ζ5 1 -ζ54 -ζ52 -ζ53 -ζ5 -1 1 -1 linear of order 10 ρ14 1 1 -1 -1 1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ52 ζ53 ζ54 -ζ5 -ζ54 -ζ52 -ζ53 -ζ5 -ζ54 -ζ52 -ζ53 1 ζ52 ζ5 ζ54 ζ53 -1 1 -1 linear of order 10 ρ15 1 1 1 -1 -1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ54 ζ5 ζ53 ζ52 ζ53 ζ54 -ζ5 -ζ52 -ζ53 -ζ54 ζ5 1 -ζ54 -ζ52 -ζ53 -ζ5 1 1 1 linear of order 10 ρ16 1 1 1 -1 -1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ52 ζ53 ζ54 ζ5 ζ54 ζ52 -ζ53 -ζ5 -ζ54 -ζ52 ζ53 1 -ζ52 -ζ5 -ζ54 -ζ53 1 1 1 linear of order 10 ρ17 1 1 -1 1 -1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ52 ζ53 ζ54 -ζ5 -ζ54 -ζ52 ζ53 ζ5 ζ54 ζ52 -ζ53 1 -ζ52 -ζ5 -ζ54 -ζ53 -1 1 -1 linear of order 10 ρ18 1 1 1 -1 -1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ53 ζ52 ζ5 ζ54 ζ5 ζ53 -ζ52 -ζ54 -ζ5 -ζ53 ζ52 1 -ζ53 -ζ54 -ζ5 -ζ52 1 1 1 linear of order 10 ρ19 1 1 1 1 1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ54 ζ5 ζ53 ζ52 ζ53 ζ54 ζ5 ζ52 ζ53 ζ54 ζ5 1 ζ54 ζ52 ζ53 ζ5 1 1 1 linear of order 5 ρ20 1 1 -1 1 -1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ53 ζ52 ζ5 -ζ54 -ζ5 -ζ53 ζ52 ζ54 ζ5 ζ53 -ζ52 1 -ζ53 -ζ54 -ζ5 -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 0 -2 0 orthogonal lifted from D4 ρ22 2 -2 0 0 0 2ζ52 2ζ53 2ζ54 2ζ5 -2ζ54 -2ζ53 -2ζ52 -2ζ5 0 0 0 0 0 0 0 0 2 0 0 0 0 0 -2 0 complex lifted from C5×D4 ρ23 2 -2 0 0 0 2ζ53 2ζ52 2ζ5 2ζ54 -2ζ5 -2ζ52 -2ζ53 -2ζ54 0 0 0 0 0 0 0 0 2 0 0 0 0 0 -2 0 complex lifted from C5×D4 ρ24 2 -2 0 0 0 2ζ54 2ζ5 2ζ53 2ζ52 -2ζ53 -2ζ5 -2ζ54 -2ζ52 0 0 0 0 0 0 0 0 2 0 0 0 0 0 -2 0 complex lifted from C5×D4 ρ25 2 -2 0 0 0 2ζ5 2ζ54 2ζ52 2ζ53 -2ζ52 -2ζ54 -2ζ5 -2ζ53 0 0 0 0 0 0 0 0 2 0 0 0 0 0 -2 0 complex lifted from C5×D4 ρ26 10 10 10 0 0 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 -10 0 0 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 √-11 1 -√-11 complex 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 -√-11 1 √-11 complex faithful

Smallest permutation representation of C22⋊F11
On 44 points
Generators in S44
```(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(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 3 5 9 6 11 10 8 4 7)(13 14 16 20 17 22 21 19 15 18)(23 34)(24 36 27 42 28 44 32 41 26 40)(25 38 31 39 33 43 30 37 29 35)```

`G:=sub<Sym(44)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(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), (2,3,5,9,6,11,10,8,4,7)(13,14,16,20,17,22,21,19,15,18)(23,34)(24,36,27,42,28,44,32,41,26,40)(25,38,31,39,33,43,30,37,29,35)>;`

`G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(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), (2,3,5,9,6,11,10,8,4,7)(13,14,16,20,17,22,21,19,15,18)(23,34)(24,36,27,42,28,44,32,41,26,40)(25,38,31,39,33,43,30,37,29,35) );`

`G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(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)], [(2,3,5,9,6,11,10,8,4,7),(13,14,16,20,17,22,21,19,15,18),(23,34),(24,36,27,42,28,44,32,41,26,40),(25,38,31,39,33,43,30,37,29,35)]])`

Matrix representation of C22⋊F11 in GL12(𝔽661)

 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 660
,
 660 0 0 0 0 0 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 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 0 0 0 660 0 0 1 0 0 0 0 0 0 0 0 660 0 0 0 1 0 0 0 0 0 0 0 660 0 0 0 0 1 0 0 0 0 0 0 660 0 0 0 0 0 1 0 0 0 0 0 660 0 0 0 0 0 0 1 0 0 0 0 660 0 0 0 0 0 0 0 1 0 0 0 660 0 0 0 0 0 0 0 0 1 0 0 660 0 0 0 0 0 0 0 0 0 1 0 660 0 0 0 0 0 0 0 0 0 0 1 660
,
 471 0 0 0 0 0 0 0 0 0 0 0 0 190 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0

`G:=sub<GL(12,GF(661))| [0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,660],[660,0,0,0,0,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,660,660,660,660,660,660,660,660,660,660],[471,0,0,0,0,0,0,0,0,0,0,0,0,190,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0] >;`

C22⋊F11 in GAP, Magma, Sage, TeX

`C_2^2\rtimes F_{11}`
`% in TeX`

`G:=Group("C2^2:F11");`
`// GroupNames label`

`G:=SmallGroup(440,11);`
`// by ID`

`G=gap.SmallGroup(440,11);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-2,-11,221,10004,2264]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^2=c^11=d^10=1,d*a*d^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^6>;`
`// generators/relations`

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