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## G = C8.(C4⋊C4)  order 128 = 27

### 4th non-split extension by C8 of C4⋊C4 acting via C4⋊C4/C22=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8.(C4⋊C4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C42⋊C2 — C8○2M4(2) — C8.(C4⋊C4)
 Lower central C1 — C2 — C2×C4 — C8.(C4⋊C4)
 Upper central C1 — C4 — C42⋊C2 — C8.(C4⋊C4)
 Jennings C1 — C2 — C2 — C22×C4 — C8.(C4⋊C4)

Generators and relations for C8.(C4⋊C4)
G = < a,b,c | a8=c4=1, b4=a4, bab-1=a5, cac-1=a-1, cbc-1=a6b3 >

Subgroups: 148 in 86 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×4], C22 [×3], C22, C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×4], C23, C42 [×2], C42 [×2], C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×6], C2×C8 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C4×C8 [×2], C8⋊C4 [×2], C4.Q8, C2.D8, C8.C4 [×2], C42⋊C2, C42⋊C2 [×2], C22×C8, C2×M4(2), C2×M4(2) [×2], C4.9C42 [×2], M4(2)⋊4C4 [×2], C82M4(2), C23.25D4, C2×C8.C4, C8.(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C428C4, C8.(C4⋊C4)

Smallest permutation representation of C8.(C4⋊C4)
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 14 25 22 5 10 29 18)(2 11 26 19 6 15 30 23)(3 16 27 24 7 12 31 20)(4 13 28 21 8 9 32 17)
(2 8)(3 7)(4 6)(9 21 13 17)(10 20 14 24)(11 19 15 23)(12 18 16 22)(26 32)(27 31)(28 30)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 Q8 C4○D4 C4○D4 C8.(C4⋊C4) kernel C8.(C4⋊C4) C4.9C42 M4(2)⋊4C4 C8○2M4(2) C23.25D4 C2×C8.C4 C4×C8 C8⋊C4 C2×C8 C2×C8 C2×C4 C23 C1 # reps 1 2 2 1 1 1 4 4 2 2 6 2 4

Matrix representation of C8.(C4⋊C4) in GL4(𝔽17) generated by

 14 12 0 0 12 14 0 0 0 0 3 5 0 0 5 3
,
 0 0 14 5 0 0 5 14 5 14 0 0 14 5 0 0
,
 0 16 0 0 1 0 0 0 0 0 0 4 0 0 13 0
`G:=sub<GL(4,GF(17))| [14,12,0,0,12,14,0,0,0,0,3,5,0,0,5,3],[0,0,5,14,0,0,14,5,14,5,0,0,5,14,0,0],[0,1,0,0,16,0,0,0,0,0,0,13,0,0,4,0] >;`

C8.(C4⋊C4) in GAP, Magma, Sage, TeX

`C_8.(C_4\rtimes C_4)`
`% in TeX`

`G:=Group("C8.(C4:C4)");`
`// GroupNames label`

`G:=SmallGroup(128,565);`
`// by ID`

`G=gap.SmallGroup(128,565);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,2019,172,2028,1027]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=c^4=1,b^4=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^3>;`
`// generators/relations`

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