Copied to
clipboard

## G = D7⋊A4order 168 = 23·3·7

### The semidirect product of D7 and A4 acting via A4/C22=C3

Aliases: D7⋊A4, C22⋊F7, C7⋊A4⋊C2, C7⋊(C2×A4), (C2×C14)⋊2C6, (C22×D7)⋊2C3, SmallGroup(168,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D7⋊A4
 Chief series C1 — C7 — C2×C14 — C7⋊A4 — D7⋊A4
 Lower central C2×C14 — D7⋊A4
 Upper central C1

Generators and relations for D7⋊A4
G = < a,b,c,d,e | a7=b2=c2=d2=e3=1, bab=a-1, ac=ca, ad=da, eae-1=a2, bc=cb, bd=db, ebe-1=ab, ece-1=cd=dc, ede-1=c >

3C2
7C2
21C2
28C3
21C22
21C22
28C6
3C14
3D7
7C23
7A4
3D14
3D14
4F7

Character table of D7⋊A4

 class 1 2A 2B 2C 3A 3B 6A 6B 7 14A 14B 14C size 1 3 7 21 28 28 28 28 6 6 6 6 ρ1 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 linear of order 2 ρ3 1 1 -1 -1 ζ3 ζ32 ζ6 ζ65 1 1 1 1 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 linear of order 3 ρ5 1 1 -1 -1 ζ32 ζ3 ζ65 ζ6 1 1 1 1 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 linear of order 3 ρ7 3 -1 -3 1 0 0 0 0 3 -1 -1 -1 orthogonal lifted from C2×A4 ρ8 3 -1 3 -1 0 0 0 0 3 -1 -1 -1 orthogonal lifted from A4 ρ9 6 6 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F7 ρ10 6 -2 0 0 0 0 0 0 -1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 orthogonal faithful ρ11 6 -2 0 0 0 0 0 0 -1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 orthogonal faithful ρ12 6 -2 0 0 0 0 0 0 -1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 orthogonal faithful

Permutation representations of D7⋊A4
On 28 points - transitive group 28T28
Generators in S28
```(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)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(2 5 3)(4 6 7)(8 28 17)(9 25 19)(10 22 21)(11 26 16)(12 23 18)(13 27 20)(14 24 15)```

`G:=sub<Sym(28)| (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (2,5,3)(4,6,7)(8,28,17)(9,25,19)(10,22,21)(11,26,16)(12,23,18)(13,27,20)(14,24,15)>;`

`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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (2,5,3)(4,6,7)(8,28,17)(9,25,19)(10,22,21)(11,26,16)(12,23,18)(13,27,20)(14,24,15) );`

`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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(2,5,3),(4,6,7),(8,28,17),(9,25,19),(10,22,21),(11,26,16),(12,23,18),(13,27,20),(14,24,15)]])`

`G:=TransitiveGroup(28,28);`

D7⋊A4 is a maximal quotient of   Q8.F7  Q8⋊F7  Dic7⋊A4

Matrix representation of D7⋊A4 in GL6(𝔽43)

 42 1 0 0 0 0 42 0 1 0 0 0 42 0 0 1 0 0 42 0 0 0 1 0 42 0 0 0 0 1 42 0 0 0 0 0
,
 42 0 0 0 0 0 42 0 0 0 0 1 42 0 0 0 1 0 42 0 0 1 0 0 42 0 1 0 0 0 42 1 0 0 0 0
,
 28 3 38 0 5 40 0 31 41 38 5 2 40 3 26 41 0 2 2 0 41 26 3 40 2 5 38 41 31 0 40 5 0 38 3 28
,
 31 38 2 0 41 5 0 26 40 2 41 3 5 38 28 40 0 3 3 0 40 28 38 5 3 41 2 40 26 0 5 41 0 2 38 31
,
 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 1 0

`G:=sub<GL(6,GF(43))| [42,42,42,42,42,42,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[42,42,42,42,42,42,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0],[28,0,40,2,2,40,3,31,3,0,5,5,38,41,26,41,38,0,0,38,41,26,41,38,5,5,0,3,31,3,40,2,2,40,0,28],[31,0,5,3,3,5,38,26,38,0,41,41,2,40,28,40,2,0,0,2,40,28,40,2,41,41,0,38,26,38,5,3,3,5,0,31],[0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0] >;`

D7⋊A4 in GAP, Magma, Sage, TeX

`D_7\rtimes A_4`
`% in TeX`

`G:=Group("D7:A4");`
`// GroupNames label`

`G:=SmallGroup(168,49);`
`// by ID`

`G=gap.SmallGroup(168,49);`
`# by ID`

`G:=PCGroup([5,-2,-3,-2,2,-7,97,188,3604,1209]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^2=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^2,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;`
`// generators/relations`

Export

׿
×
𝔽