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## G = C4.D4⋊3C4order 128 = 27

### 2nd semidirect product of C4.D4 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.D4⋊3C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — C8○2M4(2) — C4.D4⋊3C4
 Lower central C1 — C2 — C2×C4 — C4.D4⋊3C4
 Upper central C1 — C22 — C22×C4 — C4.D4⋊3C4
 Jennings C1 — C2 — C2 — C22×C4 — C4.D4⋊3C4

Generators and relations for C4.D43C4
G = < a,b,c,d | a4=d4=1, b4=a2, c2=a, bab-1=dad-1=a-1, ac=ca, cbc-1=a-1b3, dbd-1=b3, dcd-1=ab2c >

Subgroups: 276 in 116 conjugacy classes, 48 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×3], C22 [×3], C22 [×10], C8 [×7], C2×C4 [×6], C2×C4 [×4], D4 [×6], C23, C23 [×6], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×4], M4(2) [×4], C22×C4, C22×C4, C2×D4 [×2], C2×D4 [×5], C24, C4×C8, C8⋊C4, C4.D4 [×4], D4⋊C4 [×4], C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2) [×3], C22×D4, C4.C42, C22.C42, C82M4(2), C2×C4.D4, C2×D4⋊C4, C23.37D4, M4(2)⋊C4, C4.D43C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C24.C22, D4.3D4, D4.4D4, C4.D43C4

Smallest permutation representation of C4.D43C4
On 32 points
Generators in S32
```(1 21 5 17)(2 18 6 22)(3 23 7 19)(4 20 8 24)(9 25 13 29)(10 30 14 26)(11 27 15 31)(12 32 16 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 29 30 31 32)
(1 6 21 22 5 2 17 18)(3 4 23 20 7 8 19 24)(9 12 25 32 13 16 29 28)(10 27 30 15 14 31 26 11)
(1 30 19 12)(2 25 20 15)(3 28 21 10)(4 31 22 13)(5 26 23 16)(6 29 24 11)(7 32 17 14)(8 27 18 9)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 8 8 2 2 2 2 4 4 4 4 8 8 2 2 2 2 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C4○D4 D4.3D4 D4.4D4 kernel C4.D4⋊3C4 C4.C42 C22.C42 C8○2M4(2) C2×C4.D4 C2×D4⋊C4 C23.37D4 M4(2)⋊C4 C4.D4 C4⋊C4 C2×C8 C2×C4 C23 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 6 2 2 2

Matrix representation of C4.D43C4 in GL6(𝔽17)

 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 1 0 0 0 0 16 0
,
 13 0 0 0 0 0 13 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 1 0 0 0 0 1 0 0 0
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 16 0 0 0
,
 13 8 0 0 0 0 0 4 0 0 0 0 0 0 14 3 0 0 0 0 3 3 0 0 0 0 0 0 14 3 0 0 0 0 3 3

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,0,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,14,3,0,0,0,0,3,3,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;`

C4.D43C4 in GAP, Magma, Sage, TeX

`C_4.D_4\rtimes_3C_4`
`% in TeX`

`G:=Group("C4.D4:3C4");`
`// GroupNames label`

`G:=SmallGroup(128,663);`
`// by ID`

`G=gap.SmallGroup(128,663);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,58,2019,248,718]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=d^4=1,b^4=a^2,c^2=a,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^3,d*b*d^-1=b^3,d*c*d^-1=a*b^2*c>;`
`// generators/relations`

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