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## G = C42.56D4order 128 = 27

### 38th 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 — C2×C4 — C42.56D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.37C23 — C42.56D4
 Lower central C1 — C22 — C2×C4 — C42.56D4
 Upper central C1 — C22 — C2×C42 — C42.56D4
 Jennings C1 — C2 — C22 — C22×C4 — C42.56D4

Generators and relations for C42.56D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b-1, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >

Subgroups: 220 in 114 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×2], C22 [×2], C8 [×2], C2×C4 [×6], C2×C4 [×17], Q8 [×4], C23, C42 [×4], C42 [×6], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8, C2.C42 [×2], C2.C42, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C23.31D4 [×4], C424C4, C42.6C4, C23.37C23, C42.56D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C8.C22 [×2], C23.C23, C23.38D4, C2×C4≀C2, C42.56D4

Smallest permutation representation of C42.56D4
On 32 points
Generators in S32
```(1 12 25 21)(2 22 26 13)(3 14 27 23)(4 24 28 15)(5 16 29 17)(6 18 30 9)(7 10 31 19)(8 20 32 11)
(1 31 5 27)(2 32 6 28)(3 25 7 29)(4 26 8 30)(9 24 13 20)(10 17 14 21)(11 18 15 22)(12 19 16 23)
(1 17 25 16)(2 11 6 15)(3 10 27 19)(4 22 8 18)(5 21 29 12)(7 14 31 23)(9 28 13 32)(20 30 24 26)
(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)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A 8B 8C 8D order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 8 8 8 8 size 1 1 1 1 2 2 2 ··· 2 4 ··· 4 8 8 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 D4 C4≀C2 C8.C22 C23.C23 kernel C42.56D4 C23.31D4 C42⋊4C4 C42.6C4 C23.37C23 C4×Q8 C4⋊Q8 C42 C22×C4 C4 C22 C2 # reps 1 4 1 1 1 4 4 2 2 8 2 2

Matrix representation of C42.56D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 16 0 0 0 0 0 1 4 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 13 0 0 0 0 4 0
,
 5 8 0 0 0 0 7 12 0 0 0 0 0 0 0 0 0 13 0 0 0 0 4 0 0 0 13 0 0 0 0 0 0 4 0 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[16,1,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[5,7,0,0,0,0,8,12,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,13,0,0,0] >;`

C42.56D4 in GAP, Magma, Sage, TeX

`C_4^2._{56}D_4`
`% in TeX`

`G:=Group("C4^2.56D4");`
`// GroupNames label`

`G:=SmallGroup(128,238);`
`// by ID`

`G=gap.SmallGroup(128,238);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,387,184,1123,1018,248,1971]);`
`// Polycyclic`

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

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