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## G = C26⋊C3order 192 = 26·3

### 3rd semidirect product of C26 and C3 acting faithfully

Aliases: C263C3, C248A4, C22⋊(C22⋊A4), SmallGroup(192,1541)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C26⋊C3
 Chief series C1 — C22 — C24 — C26 — C26⋊C3
 Lower central C26 — C26⋊C3
 Upper central C1

Generators and relations for C26⋊C3
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g3=1, gag-1=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, gbg-1=a, gcg-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, gdg-1=c, geg-1=ef=fe, gfg-1=e >

Subgroups: 3310 in 1015 conjugacy classes, 45 normal (3 characteristic)
C1, C2, C3, C22, C22, C23, A4, C24, C24, C25, C22⋊A4, C26, C26⋊C3
Quotients: C1, C3, A4, C22⋊A4, C26⋊C3

Character table of C26⋊C3

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 2P 2Q 2R 2S 2T 2U 3A 3B size 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 64 64 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 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 linear of order 3 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 linear of order 3 ρ4 3 -1 3 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 3 -1 -1 -1 3 -1 -1 -1 3 0 0 orthogonal lifted from A4 ρ5 3 3 -1 -1 -1 3 -1 3 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 3 0 0 orthogonal lifted from A4 ρ6 3 -1 -1 -1 -1 3 -1 -1 3 -1 -1 3 -1 -1 -1 -1 -1 3 3 -1 -1 -1 0 0 orthogonal lifted from A4 ρ7 3 -1 -1 3 -1 -1 -1 -1 3 -1 -1 -1 -1 3 3 -1 -1 -1 -1 3 -1 -1 0 0 orthogonal lifted from A4 ρ8 3 3 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 3 -1 0 0 orthogonal lifted from A4 ρ9 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 3 3 3 3 0 0 orthogonal lifted from A4 ρ10 3 3 -1 -1 3 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 3 -1 -1 0 0 orthogonal lifted from A4 ρ11 3 -1 3 -1 -1 -1 3 -1 -1 -1 3 -1 -1 -1 3 -1 -1 -1 3 -1 -1 -1 0 0 orthogonal lifted from A4 ρ12 3 -1 -1 -1 3 -1 -1 -1 3 -1 3 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 3 0 0 orthogonal lifted from A4 ρ13 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 3 3 3 3 -1 -1 -1 -1 0 0 orthogonal lifted from A4 ρ14 3 3 -1 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 3 -1 -1 3 -1 -1 -1 0 0 orthogonal lifted from A4 ρ15 3 -1 3 -1 -1 -1 -1 3 -1 -1 -1 3 -1 -1 -1 3 -1 -1 -1 3 -1 -1 0 0 orthogonal lifted from A4 ρ16 3 -1 -1 -1 -1 3 3 -1 -1 -1 -1 -1 -1 3 -1 3 -1 -1 -1 -1 3 -1 0 0 orthogonal lifted from A4 ρ17 3 -1 -1 3 -1 -1 -1 3 -1 -1 3 -1 -1 -1 -1 -1 -1 3 -1 -1 3 -1 0 0 orthogonal lifted from A4 ρ18 3 -1 -1 -1 3 -1 -1 -1 -1 3 -1 3 -1 -1 3 -1 -1 -1 -1 -1 3 -1 0 0 orthogonal lifted from A4 ρ19 3 -1 -1 -1 -1 -1 3 3 3 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 orthogonal lifted from A4 ρ20 3 -1 -1 3 -1 -1 3 -1 -1 -1 -1 3 -1 -1 -1 -1 3 -1 -1 -1 -1 3 0 0 orthogonal lifted from A4 ρ21 3 -1 -1 -1 3 -1 -1 3 -1 -1 -1 -1 -1 3 -1 -1 3 -1 3 -1 -1 -1 0 0 orthogonal lifted from A4 ρ22 3 -1 -1 -1 -1 3 -1 -1 -1 3 3 -1 -1 -1 -1 -1 3 -1 -1 3 -1 -1 0 0 orthogonal lifted from A4 ρ23 3 -1 3 3 3 3 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 orthogonal lifted from A4 ρ24 3 3 -1 -1 -1 -1 -1 -1 -1 -1 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 0 0 orthogonal lifted from A4

Permutation representations of C26⋊C3
On 24 points - transitive group 24T390
Generators in S24
```(1 14)(2 17)(3 7)(4 19)(5 10)(6 22)(8 16)(9 15)(11 21)(12 23)(13 18)(20 24)
(1 8)(2 15)(3 18)(4 23)(5 20)(6 11)(7 13)(9 17)(10 24)(12 19)(14 16)(21 22)
(1 14)(2 24)(3 11)(4 19)(5 9)(6 18)(7 21)(8 16)(10 15)(12 23)(13 22)(17 20)
(1 12)(2 15)(3 22)(4 16)(5 20)(6 7)(8 19)(9 17)(10 24)(11 13)(14 23)(18 21)
(1 23)(3 22)(4 8)(6 7)(11 13)(12 14)(16 19)(18 21)
(1 23)(2 24)(4 8)(5 9)(10 15)(12 14)(16 19)(17 20)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)```

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

`G:=Group( (1,14)(2,17)(3,7)(4,19)(5,10)(6,22)(8,16)(9,15)(11,21)(12,23)(13,18)(20,24), (1,8)(2,15)(3,18)(4,23)(5,20)(6,11)(7,13)(9,17)(10,24)(12,19)(14,16)(21,22), (1,14)(2,24)(3,11)(4,19)(5,9)(6,18)(7,21)(8,16)(10,15)(12,23)(13,22)(17,20), (1,12)(2,15)(3,22)(4,16)(5,20)(6,7)(8,19)(9,17)(10,24)(11,13)(14,23)(18,21), (1,23)(3,22)(4,8)(6,7)(11,13)(12,14)(16,19)(18,21), (1,23)(2,24)(4,8)(5,9)(10,15)(12,14)(16,19)(17,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24) );`

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

`G:=TransitiveGroup(24,390);`

Matrix representation of C26⋊C3 in GL9(𝔽7)

 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 6 0 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 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 6
,
 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 3 0 6 0 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 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 3 0 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6
,
 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 4 2 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 6
,
 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 4 2 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6
,
 0 1 0 0 0 0 0 0 0 3 5 5 0 0 0 0 0 0 0 0 2 0 0 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 1 0 0 0 0 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 1 0 0

`G:=sub<GL(9,GF(7))| [6,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6],[1,0,3,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1],[1,0,3,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6],[6,0,4,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6],[6,0,4,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6],[0,3,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;`

C26⋊C3 in GAP, Magma, Sage, TeX

`C_2^6\rtimes C_3`
`% in TeX`

`G:=Group("C2^6:C3");`
`// GroupNames label`

`G:=SmallGroup(192,1541);`
`// by ID`

`G=gap.SmallGroup(192,1541);`
`# by ID`

`G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,191,675,1264,4037,7062]);`
`// Polycyclic`

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

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