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## G = C42.96D4order 128 = 27

### 78th non-split extension by C42 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.96D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4×D4 — C42.96D4
 Lower central C1 — C2 — C23 — C42.96D4
 Upper central C1 — C22 — C2×C42 — C42.96D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.96D4

Generators and relations for C42.96D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 356 in 168 conjugacy classes, 62 normal (20 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×5], C22 [×3], C22 [×18], C8 [×4], C2×C4 [×4], C2×C4 [×6], C2×C4 [×15], D4 [×8], C23, C23 [×4], C23 [×8], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×2], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C24 [×2], C4.D4 [×4], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C2×M4(2) [×4], C23×C4 [×2], C22×D4, C22.C42 [×2], C2×C4.D4 [×2], C4⋊M4(2) [×2], C2×C4×D4, C42.96D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4.D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, C2×C4.D4, M4(2).8C22, C42.96D4

Smallest permutation representation of C42.96D4
On 32 points
Generators in S32
```(1 16 25 19)(2 24 26 13)(3 10 27 21)(4 18 28 15)(5 12 29 23)(6 20 30 9)(7 14 31 17)(8 22 32 11)
(1 31 5 27)(2 28 6 32)(3 25 7 29)(4 30 8 26)(9 22 13 18)(10 19 14 23)(11 24 15 20)(12 21 16 17)
(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 4 31 30 5 8 27 26)(2 25 28 7 6 29 32 3)(9 19 22 14 13 23 18 10)(11 17 24 12 15 21 20 16)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + - + image C1 C2 C2 C2 C2 C4 C4 D4 D4 Q8 C4○D4 C4.D4 M4(2).8C22 kernel C42.96D4 C22.C42 C2×C4.D4 C4⋊M4(2) C2×C4×D4 C2×C22⋊C4 C23×C4 C42 C2×D4 C2×D4 C2×C4 C4 C2 # reps 1 2 2 2 1 4 4 4 2 2 4 2 2

Matrix representation of C42.96D4 in GL6(𝔽17)

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

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

C42.96D4 in GAP, Magma, Sage, TeX

`C_4^2._{96}D_4`
`% in TeX`

`G:=Group("C4^2.96D4");`
`// GroupNames label`

`G:=SmallGroup(128,532);`
`// by ID`

`G=gap.SmallGroup(128,532);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2804,1027,124]);`
`// Polycyclic`

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

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