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## G = C4.4D4⋊13C4order 128 = 27

### 7th semidirect product of C4.4D4 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.4D4⋊13C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C23.32C23 — C4.4D4⋊13C4
 Lower central C1 — C2 — C2×C4 — C4.4D4⋊13C4
 Upper central C1 — C22 — C22×C4 — C4.4D4⋊13C4
 Jennings C1 — C2 — C2 — C22×C4 — C4.4D4⋊13C4

Generators and relations for C4.4D413C4
G = < a,b,c,d | a4=b4=d4=1, c2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a-1b, cd=dc >

Subgroups: 372 in 172 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×12], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×6], Q8 [×8], C23, C23 [×8], C42 [×2], C42 [×7], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8 [×4], C2×Q8 [×4], C24, C4.10D4 [×2], D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×Q8 [×4], C4.4D4 [×4], C4.4D4 [×2], C2×M4(2) [×2], C22×D4, C22×Q8, C426C4 [×2], C2×C4.10D4, C23.37D4 [×2], C23.32C23, C2×C4.4D4, C4.4D413C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, D4.8D4 [×2], C4.4D413C4

Smallest permutation representation of C4.4D413C4
On 32 points
Generators in S32
```(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)
(1 16 9 19)(2 13 10 20)(3 14 11 17)(4 15 12 18)(5 27 29 21)(6 28 30 22)(7 25 31 23)(8 26 32 24)
(1 17 3 19)(2 20 4 18)(5 28 7 26)(6 27 8 25)(9 14 11 16)(10 13 12 15)(21 32 23 30)(22 31 24 29)
(1 21 11 27)(2 24 12 26)(3 23 9 25)(4 22 10 28)(5 20 29 15)(6 19 30 14)(7 18 31 13)(8 17 32 16)```

`G:=sub<Sym(32)| (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), (1,16,9,19)(2,13,10,20)(3,14,11,17)(4,15,12,18)(5,27,29,21)(6,28,30,22)(7,25,31,23)(8,26,32,24), (1,17,3,19)(2,20,4,18)(5,28,7,26)(6,27,8,25)(9,14,11,16)(10,13,12,15)(21,32,23,30)(22,31,24,29), (1,21,11,27)(2,24,12,26)(3,23,9,25)(4,22,10,28)(5,20,29,15)(6,19,30,14)(7,18,31,13)(8,17,32,16)>;`

`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)(29,30,31,32), (1,16,9,19)(2,13,10,20)(3,14,11,17)(4,15,12,18)(5,27,29,21)(6,28,30,22)(7,25,31,23)(8,26,32,24), (1,17,3,19)(2,20,4,18)(5,28,7,26)(6,27,8,25)(9,14,11,16)(10,13,12,15)(21,32,23,30)(22,31,24,29), (1,21,11,27)(2,24,12,26)(3,23,9,25)(4,22,10,28)(5,20,29,15)(6,19,30,14)(7,18,31,13)(8,17,32,16) );`

`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),(29,30,31,32)], [(1,16,9,19),(2,13,10,20),(3,14,11,17),(4,15,12,18),(5,27,29,21),(6,28,30,22),(7,25,31,23),(8,26,32,24)], [(1,17,3,19),(2,20,4,18),(5,28,7,26),(6,27,8,25),(9,14,11,16),(10,13,12,15),(21,32,23,30),(22,31,24,29)], [(1,21,11,27),(2,24,12,26),(3,23,9,25),(4,22,10,28),(5,20,29,15),(6,19,30,14),(7,18,31,13),(8,17,32,16)])`

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4T 8A 8B 8C 8D order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 8 8 8 8 size 1 1 1 1 2 2 8 8 2 2 2 2 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 D4.8D4 kernel C4.4D4⋊13C4 C42⋊6C4 C2×C4.10D4 C23.37D4 C23.32C23 C2×C4.4D4 C4.4D4 C4⋊C4 C22×C4 C2×Q8 C2×C4 C2 # reps 1 2 1 2 1 1 8 4 2 2 4 4

Matrix representation of C4.4D413C4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 13 0
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 16 0 0 0 0 0 0 16 0 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C4.4D413C4 in GAP, Magma, Sage, TeX

`C_4._4D_4\rtimes_{13}C_4`
`% in TeX`

`G:=Group("C4.4D4:13C4");`
`// GroupNames label`

`G:=SmallGroup(128,620);`
`// by ID`

`G=gap.SmallGroup(128,620);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,352,1018,521,248,2804,1411,2028,1027,124]);`
`// Polycyclic`

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

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