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## G = C26⋊C7order 448 = 26·7

### 2nd semidirect product of C26 and C7 acting faithfully

Aliases: C262C7, C231F8, SmallGroup(448,1393)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C26⋊C7
 Chief series C1 — C23 — C26 — C26⋊C7
 Lower central C26 — C26⋊C7
 Upper central C1

Generators and relations for C26⋊C7
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g7=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, gag-1=cb=bc, bd=db, be=eb, bf=fb, gbg-1=a, cd=dc, ce=ec, cf=fc, gcg-1=b, de=ed, df=fd, gdg-1=fe=ef, geg-1=d, gfg-1=e >

Subgroups: 2962 in 424 conjugacy classes, 12 normal (3 characteristic)
C1, C2, C22, C7, C23, C23, C24, C25, F8, C26, C26⋊C7
Quotients: C1, C7, F8, C26⋊C7

Character table of C26⋊C7

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 7A 7B 7C 7D 7E 7F size 1 7 7 7 7 7 7 7 7 7 64 64 64 64 64 64 ρ1 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 ζ73 ζ76 ζ72 ζ75 ζ7 ζ74 linear of order 7 ρ3 1 1 1 1 1 1 1 1 1 1 ζ75 ζ73 ζ7 ζ76 ζ74 ζ72 linear of order 7 ρ4 1 1 1 1 1 1 1 1 1 1 ζ72 ζ74 ζ76 ζ7 ζ73 ζ75 linear of order 7 ρ5 1 1 1 1 1 1 1 1 1 1 ζ74 ζ7 ζ75 ζ72 ζ76 ζ73 linear of order 7 ρ6 1 1 1 1 1 1 1 1 1 1 ζ7 ζ72 ζ73 ζ74 ζ75 ζ76 linear of order 7 ρ7 1 1 1 1 1 1 1 1 1 1 ζ76 ζ75 ζ74 ζ73 ζ72 ζ7 linear of order 7 ρ8 7 -1 -1 -1 7 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from F8 ρ9 7 -1 -1 -1 -1 -1 -1 7 -1 -1 0 0 0 0 0 0 orthogonal lifted from F8 ρ10 7 -1 -1 -1 -1 -1 7 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from F8 ρ11 7 -1 -1 7 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from F8 ρ12 7 -1 -1 -1 -1 7 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from F8 ρ13 7 -1 -1 -1 -1 -1 -1 -1 -1 7 0 0 0 0 0 0 orthogonal lifted from F8 ρ14 7 -1 7 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from F8 ρ15 7 -1 -1 -1 -1 -1 -1 -1 7 -1 0 0 0 0 0 0 orthogonal lifted from F8 ρ16 7 7 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from F8

Permutation representations of C26⋊C7
On 28 points - transitive group 28T60
Generators in S28
```(1 20)(2 25)(4 11)(5 28)(6 13)(7 19)(8 24)(9 21)(12 17)(14 23)(16 27)(18 22)
(1 20)(2 21)(3 26)(5 12)(6 22)(7 14)(8 24)(9 25)(10 15)(13 18)(17 28)(19 23)
(1 8)(2 21)(3 15)(4 27)(6 13)(7 23)(9 25)(10 26)(11 16)(14 19)(18 22)(20 24)
(1 20)(4 16)(6 18)(7 19)(8 24)(11 27)(13 22)(14 23)
(1 20)(2 21)(5 17)(7 19)(8 24)(9 25)(12 28)(14 23)
(1 20)(2 21)(3 15)(6 18)(8 24)(9 25)(10 26)(13 22)
(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)```

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

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

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

`G:=TransitiveGroup(28,60);`

Matrix representation of C26⋊C7 in GL14(𝔽29)

 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 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 8 28 0 0 0 0 0 0 0 0 0 8 28 28 0 0 1 0 0 0 0 0 0 0 0 7 28 21 0 0 0 1 0 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 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 27 15 0 0 28 0 0 0 0 0 0 0 0 0 0 0 27 15 0 1 0 0 0 0 0 0 0 0 15 2 0 0 0 0 28
,
 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 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 0 28 0 0 0 0 0 0 0 0 0 0 1 7 0 21 1 0 0 0 0 0 0 0 0 0 8 28 0 22 0 1 0 0 0 0 0 0 0 0 0 0 8 0 0 0 28 0 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 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 14 27 13 1 0 0 0 0 0 0 0 0 0 13 0 0 0 0 28 0 0 0 0 0 0 0 0 0 27 13 2 0 0 1
,
 28 0 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 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 28 0 1 0 0 0 0 0 0 0 0 0 0 1 0 7 0 28 0 0 0 0 0 0 0 0 0 1 0 28 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 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 16 28 0 0 0 0 0 0 0 0 0 16 27 27 0 0 1 0 0 0 0 0 0 0 0 14 27 13 0 0 0 1
,
 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 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 8 28 0 0 0 0 0 0 0 0 0 8 28 28 0 0 1 0 0 0 0 0 0 0 0 7 28 21 0 0 0 1 0 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 0 1 0 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 0 1 0 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 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 0 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 0 28 0 0 0 0 0 0 0 0 0 0 1 7 0 21 1 0 0 0 0 0 0 0 0 0 8 28 0 22 0 1 0 0 0 0 0 0 0 0 0 0 8 0 0 0 28 0 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 0 1 0 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 0 1 0 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 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 28 0 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 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 28 0 1 0 0 0 0 0 0 0 0 0 0 1 0 7 0 28 0 0 0 0 0 0 0 0 0 1 0 28 0 0 28 0 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 0 1 0 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 0 1 0 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 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 0 1 0 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 0 1 0 0 0 0 0 0 0 0 0 0 28 22 1 8 27 0 0 0 0 0 0 0 0 0 0 0 0 0 21 1 0 0 0 0 0 0 0 0 0 0 0 0 22 0 1 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 1 0 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 0 1 0 0 0 0 0 0 0 0 0 0 28 22 1 8 28 0 0 0 0 0 0 0 0 0 0 0 0 0 21 1 0 0 0 0 0 0 0 0 0 0 0 0 22 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(14,GF(29))| [28,0,0,0,0,8,7,0,0,0,0,0,0,0,0,28,0,0,0,28,28,0,0,0,0,0,0,0,0,0,28,0,0,28,21,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,27,0,15,0,0,0,0,0,0,0,0,1,0,0,15,0,2,0,0,0,0,0,0,0,0,0,28,0,0,27,0,0,0,0,0,0,0,0,0,0,0,28,0,15,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[28,0,0,0,1,8,0,0,0,0,0,0,0,0,0,28,0,0,7,28,0,0,0,0,0,0,0,0,0,0,1,0,0,0,8,0,0,0,0,0,0,0,0,0,0,28,21,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,13,0,0,0,0,0,0,0,0,0,28,0,0,14,0,27,0,0,0,0,0,0,0,0,0,28,0,27,0,13,0,0,0,0,0,0,0,0,0,0,28,13,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,0,0,0,0,0,0,0,0,28,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,16,14,0,0,0,0,0,0,0,0,28,0,0,0,27,27,0,0,0,0,0,0,0,0,0,28,0,0,27,13,0,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,8,7,0,0,0,0,0,0,0,0,28,0,0,0,28,28,0,0,0,0,0,0,0,0,0,28,0,0,28,21,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,1,8,0,0,0,0,0,0,0,0,0,28,0,0,7,28,0,0,0,0,0,0,0,0,0,0,1,0,0,0,8,0,0,0,0,0,0,0,0,0,0,28,21,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,0,0,0,0,0,0,0,0,28,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,27,21,22,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,28,21,22,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0] >;`

C26⋊C7 in GAP, Magma, Sage, TeX

`C_2^6\rtimes C_7`
`% in TeX`

`G:=Group("C2^6:C7");`
`// GroupNames label`

`G:=SmallGroup(448,1393);`
`// by ID`

`G=gap.SmallGroup(448,1393);`
`# by ID`

`G:=PCGroup([7,-7,-2,2,2,-2,2,2,197,590,983,3924,9413,13726]);`
`// Polycyclic`

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

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