Copied to
clipboard

## G = C4.F11order 440 = 23·5·11

### The non-split extension by C4 of F11 acting via F11/C11⋊C5=C2

Aliases: C4.F11, Dic22⋊C5, C44.1C10, Dic11.C10, C11⋊C5⋊Q8, C11⋊(C5×Q8), C11⋊C20.C2, C2.3(C2×F11), C22.1(C2×C10), (C4×C11⋊C5).1C2, (C2×C11⋊C5).1C22, SmallGroup(440,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C4.F11
 Chief series C1 — C11 — C22 — C2×C11⋊C5 — C11⋊C20 — C4.F11
 Lower central C11 — C22 — C4.F11
 Upper central C1 — C2 — C4

Generators and relations for C4.F11
G = < a,b,c | a4=b11=1, c10=a2, ab=ba, cac-1=a-1, cbc-1=b6 >

Character table of C4.F11

 class 1 2 4A 4B 4C 5A 5B 5C 5D 10A 10B 10C 10D 11 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 20K 20L 22 44A 44B size 1 1 2 22 22 11 11 11 11 11 11 11 11 10 22 22 22 22 22 22 22 22 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 ζ5 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ53 1 ζ52 ζ52 ζ53 ζ53 ζ5 ζ54 ζ52 ζ5 ζ53 ζ5 ζ54 ζ54 1 1 1 linear of order 5 ρ6 1 1 1 1 1 ζ54 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ52 1 ζ53 ζ53 ζ52 ζ52 ζ54 ζ5 ζ53 ζ54 ζ52 ζ54 ζ5 ζ5 1 1 1 linear of order 5 ρ7 1 1 1 -1 -1 ζ5 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ53 1 -ζ52 ζ52 ζ53 -ζ53 -ζ5 -ζ54 -ζ52 ζ5 -ζ53 -ζ5 -ζ54 ζ54 1 1 1 linear of order 10 ρ8 1 1 1 1 1 ζ53 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ54 1 ζ5 ζ5 ζ54 ζ54 ζ53 ζ52 ζ5 ζ53 ζ54 ζ53 ζ52 ζ52 1 1 1 linear of order 5 ρ9 1 1 -1 1 -1 ζ53 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ54 1 ζ5 -ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ5 -ζ53 ζ54 ζ53 ζ52 -ζ52 1 -1 -1 linear of order 10 ρ10 1 1 -1 -1 1 ζ54 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ52 1 -ζ53 -ζ53 -ζ52 ζ52 ζ54 ζ5 ζ53 -ζ54 -ζ52 -ζ54 -ζ5 -ζ5 1 -1 -1 linear of order 10 ρ11 1 1 -1 -1 1 ζ52 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ5 1 -ζ54 -ζ54 -ζ5 ζ5 ζ52 ζ53 ζ54 -ζ52 -ζ5 -ζ52 -ζ53 -ζ53 1 -1 -1 linear of order 10 ρ12 1 1 1 1 1 ζ52 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ5 1 ζ54 ζ54 ζ5 ζ5 ζ52 ζ53 ζ54 ζ52 ζ5 ζ52 ζ53 ζ53 1 1 1 linear of order 5 ρ13 1 1 1 -1 -1 ζ54 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ52 1 -ζ53 ζ53 ζ52 -ζ52 -ζ54 -ζ5 -ζ53 ζ54 -ζ52 -ζ54 -ζ5 ζ5 1 1 1 linear of order 10 ρ14 1 1 1 -1 -1 ζ53 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ54 1 -ζ5 ζ5 ζ54 -ζ54 -ζ53 -ζ52 -ζ5 ζ53 -ζ54 -ζ53 -ζ52 ζ52 1 1 1 linear of order 10 ρ15 1 1 -1 1 -1 ζ5 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ53 1 ζ52 -ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ52 -ζ5 ζ53 ζ5 ζ54 -ζ54 1 -1 -1 linear of order 10 ρ16 1 1 -1 -1 1 ζ5 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ53 1 -ζ52 -ζ52 -ζ53 ζ53 ζ5 ζ54 ζ52 -ζ5 -ζ53 -ζ5 -ζ54 -ζ54 1 -1 -1 linear of order 10 ρ17 1 1 -1 -1 1 ζ53 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ54 1 -ζ5 -ζ5 -ζ54 ζ54 ζ53 ζ52 ζ5 -ζ53 -ζ54 -ζ53 -ζ52 -ζ52 1 -1 -1 linear of order 10 ρ18 1 1 -1 1 -1 ζ52 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ5 1 ζ54 -ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ54 -ζ52 ζ5 ζ52 ζ53 -ζ53 1 -1 -1 linear of order 10 ρ19 1 1 -1 1 -1 ζ54 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ52 1 ζ53 -ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ53 -ζ54 ζ52 ζ54 ζ5 -ζ5 1 -1 -1 linear of order 10 ρ20 1 1 1 -1 -1 ζ52 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ5 1 -ζ54 ζ54 ζ5 -ζ5 -ζ52 -ζ53 -ζ54 ζ52 -ζ5 -ζ52 -ζ53 ζ53 1 1 1 linear of order 10 ρ21 2 -2 0 0 0 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 symplectic lifted from Q8, Schur index 2 ρ22 2 -2 0 0 0 2ζ54 2ζ5 2ζ53 2ζ52 -2ζ54 -2ζ53 -2ζ5 -2ζ52 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 complex lifted from C5×Q8 ρ23 2 -2 0 0 0 2ζ52 2ζ53 2ζ54 2ζ5 -2ζ52 -2ζ54 -2ζ53 -2ζ5 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 complex lifted from C5×Q8 ρ24 2 -2 0 0 0 2ζ53 2ζ52 2ζ5 2ζ54 -2ζ53 -2ζ5 -2ζ52 -2ζ54 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 complex lifted from C5×Q8 ρ25 2 -2 0 0 0 2ζ5 2ζ54 2ζ52 2ζ53 -2ζ5 -2ζ52 -2ζ54 -2ζ53 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 complex lifted from C5×Q8 ρ26 10 10 -10 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 1 orthogonal lifted from C2×F11 ρ27 10 10 10 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 -1 orthogonal lifted from F11 ρ28 10 -10 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 √11 -√11 symplectic faithful, Schur index 2 ρ29 10 -10 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 -√11 √11 symplectic faithful, Schur index 2

Smallest permutation representation of C4.F11
On 88 points
Generators in S88
```(1 7 3 5)(2 6 4 8)(9 72 19 82)(10 83 20 73)(11 74 21 84)(12 85 22 75)(13 76 23 86)(14 87 24 77)(15 78 25 88)(16 69 26 79)(17 80 27 70)(18 71 28 81)(29 65 39 55)(30 56 40 66)(31 67 41 57)(32 58 42 68)(33 49 43 59)(34 60 44 50)(35 51 45 61)(36 62 46 52)(37 53 47 63)(38 64 48 54)
(1 12 55 20 24 16 63 51 67 28 59)(2 64 13 52 56 68 21 9 25 60 17)(3 22 65 10 14 26 53 61 57 18 49)(4 54 23 62 66 58 11 19 15 50 27)(5 75 39 83 87 79 47 35 31 71 43)(6 48 76 36 40 32 84 72 88 44 80)(7 85 29 73 77 69 37 45 41 81 33)(8 38 86 46 30 42 74 82 78 34 70)
(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 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68)(69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88)```

`G:=sub<Sym(88)| (1,7,3,5)(2,6,4,8)(9,72,19,82)(10,83,20,73)(11,74,21,84)(12,85,22,75)(13,76,23,86)(14,87,24,77)(15,78,25,88)(16,69,26,79)(17,80,27,70)(18,71,28,81)(29,65,39,55)(30,56,40,66)(31,67,41,57)(32,58,42,68)(33,49,43,59)(34,60,44,50)(35,51,45,61)(36,62,46,52)(37,53,47,63)(38,64,48,54), (1,12,55,20,24,16,63,51,67,28,59)(2,64,13,52,56,68,21,9,25,60,17)(3,22,65,10,14,26,53,61,57,18,49)(4,54,23,62,66,58,11,19,15,50,27)(5,75,39,83,87,79,47,35,31,71,43)(6,48,76,36,40,32,84,72,88,44,80)(7,85,29,73,77,69,37,45,41,81,33)(8,38,86,46,30,42,74,82,78,34,70), (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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)>;`

`G:=Group( (1,7,3,5)(2,6,4,8)(9,72,19,82)(10,83,20,73)(11,74,21,84)(12,85,22,75)(13,76,23,86)(14,87,24,77)(15,78,25,88)(16,69,26,79)(17,80,27,70)(18,71,28,81)(29,65,39,55)(30,56,40,66)(31,67,41,57)(32,58,42,68)(33,49,43,59)(34,60,44,50)(35,51,45,61)(36,62,46,52)(37,53,47,63)(38,64,48,54), (1,12,55,20,24,16,63,51,67,28,59)(2,64,13,52,56,68,21,9,25,60,17)(3,22,65,10,14,26,53,61,57,18,49)(4,54,23,62,66,58,11,19,15,50,27)(5,75,39,83,87,79,47,35,31,71,43)(6,48,76,36,40,32,84,72,88,44,80)(7,85,29,73,77,69,37,45,41,81,33)(8,38,86,46,30,42,74,82,78,34,70), (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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88) );`

`G=PermutationGroup([[(1,7,3,5),(2,6,4,8),(9,72,19,82),(10,83,20,73),(11,74,21,84),(12,85,22,75),(13,76,23,86),(14,87,24,77),(15,78,25,88),(16,69,26,79),(17,80,27,70),(18,71,28,81),(29,65,39,55),(30,56,40,66),(31,67,41,57),(32,58,42,68),(33,49,43,59),(34,60,44,50),(35,51,45,61),(36,62,46,52),(37,53,47,63),(38,64,48,54)], [(1,12,55,20,24,16,63,51,67,28,59),(2,64,13,52,56,68,21,9,25,60,17),(3,22,65,10,14,26,53,61,57,18,49),(4,54,23,62,66,58,11,19,15,50,27),(5,75,39,83,87,79,47,35,31,71,43),(6,48,76,36,40,32,84,72,88,44,80),(7,85,29,73,77,69,37,45,41,81,33),(8,38,86,46,30,42,74,82,78,34,70)], [(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),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68),(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)]])`

Matrix representation of C4.F11 in GL12(𝔽661)

 1 659 0 0 0 0 0 0 0 0 0 0 1 660 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
,
 515 83 0 0 0 0 0 0 0 0 0 0 226 146 0 0 0 0 0 0 0 0 0 0 0 0 565 0 565 0 565 316 96 0 0 96 0 0 220 96 565 0 0 96 0 0 565 96 0 0 0 0 565 565 0 0 316 96 565 96 0 0 565 316 0 565 0 96 96 0 565 0 0 0 0 96 565 565 565 96 0 316 0 0 0 0 0 0 220 0 565 96 96 96 565 0 0 0 0 96 0 565 565 0 96 0 220 96 0 0 565 96 565 220 0 0 96 96 0 0 0 0 565 96 0 0 565 0 0 96 565 316 0 0 565 0 0 565 220 96 0 96 0 96

`G:=sub<GL(12,GF(661))| [1,1,0,0,0,0,0,0,0,0,0,0,659,660,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[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],[515,226,0,0,0,0,0,0,0,0,0,0,83,146,0,0,0,0,0,0,0,0,0,0,0,0,565,220,0,565,0,0,0,565,565,565,0,0,0,96,0,316,96,0,96,96,96,0,0,0,565,565,565,0,565,220,0,565,0,0,0,0,0,0,565,565,565,0,565,220,0,565,0,0,565,0,0,0,565,565,565,0,565,220,0,0,316,96,0,96,96,96,0,0,0,96,0,0,96,0,316,96,0,96,96,96,0,0,0,0,0,0,96,0,316,96,0,96,96,96,0,0,0,565,565,565,0,565,220,0,565,0,0,0,96,96,96,0,0,0,96,0,316,96] >;`

C4.F11 in GAP, Magma, Sage, TeX

`C_4.F_{11}`
`% in TeX`

`G:=Group("C4.F11");`
`// GroupNames label`

`G:=SmallGroup(440,7);`
`// by ID`

`G=gap.SmallGroup(440,7);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-2,-11,100,221,106,10004,2264]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^11=1,c^10=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^6>;`
`// generators/relations`

Export

׿
×
𝔽