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## G = C42.57D4order 128 = 27

### 39th 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 — C42.57D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.36C23 — C42.57D4
 Lower central C1 — C22 — C2×C4 — C42.57D4
 Upper central C1 — C22 — C2×C42 — C42.57D4
 Jennings C1 — C2 — C22 — C22×C4 — C42.57D4

Generators and relations for C42.57D4
G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 252 in 121 conjugacy classes, 46 normal (36 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22 [×3], C22 [×5], C8 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×3], Q8, C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2.C42 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C22.SD16 [×2], C23.31D4 [×2], C4×C4⋊C4, C42.6C4, C23.36C23, C42.57D4
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, C8.C22, C23.C23, C23.36D4, C2×C4≀C2, C42.57D4

Smallest permutation representation of C42.57D4
On 32 points
Generators in S32
```(1 10 31 21)(2 18 32 15)(3 12 25 23)(4 20 26 9)(5 14 27 17)(6 22 28 11)(7 16 29 19)(8 24 30 13)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(1 17 31 14)(2 24)(3 12 25 23)(4 11)(5 21 27 10)(6 20)(7 16 29 19)(8 15)(9 28)(13 32)(18 30)(22 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)| (1,10,31,21)(2,18,32,15)(3,12,25,23)(4,20,26,9)(5,14,27,17)(6,22,28,11)(7,16,29,19)(8,24,30,13), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,31,14)(2,24)(3,12,25,23)(4,11)(5,21,27,10)(6,20)(7,16,29,19)(8,15)(9,28)(13,32)(18,30)(22,26), (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( (1,10,31,21)(2,18,32,15)(3,12,25,23)(4,20,26,9)(5,14,27,17)(6,22,28,11)(7,16,29,19)(8,24,30,13), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,31,14)(2,24)(3,12,25,23)(4,11)(5,21,27,10)(6,20)(7,16,29,19)(8,15)(9,28)(13,32)(18,30)(22,26), (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([(1,10,31,21),(2,18,32,15),(3,12,25,23),(4,20,26,9),(5,14,27,17),(6,22,28,11),(7,16,29,19),(8,24,30,13)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(1,17,31,14),(2,24),(3,12,25,23),(4,11),(5,21,27,10),(6,20),(7,16,29,19),(8,15),(9,28),(13,32),(18,30),(22,26)], [(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 2F 4A ··· 4H 4I ··· 4R 4S 4T 4U 8A 8B 8C 8D order 1 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 8 8 8 8 size 1 1 1 1 2 2 8 2 ··· 2 4 ··· 4 8 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 C4≀C2 C8⋊C22 C8.C22 C23.C23 kernel C42.57D4 C22.SD16 C23.31D4 C4×C4⋊C4 C42.6C4 C23.36C23 C4×D4 C4×Q8 C4.4D4 C42.C2 C42 C22×C4 C4 C22 C22 C2 # reps 1 2 2 1 1 1 2 2 2 2 2 2 8 1 1 2

Matrix representation of C42.57D4 in GL6(𝔽17)

 13 0 0 0 0 0 0 13 0 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 13
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 16 0 0 0 0 15 13 0 0 0 0 0 0 13 1 0 0 0 0 2 4
,
 13 0 0 0 0 0 0 16 0 0 0 0 0 0 13 1 0 0 0 0 0 4 0 0 0 0 0 0 16 0 0 0 0 0 9 1
,
 0 16 0 0 0 0 13 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 9 1 0 0 13 1 0 0 0 0 0 4 0 0

`G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,15,0,0,0,0,16,13,0,0,0,0,0,0,13,2,0,0,0,0,1,4],[13,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,1,4,0,0,0,0,0,0,16,9,0,0,0,0,0,1],[0,13,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,1,4,0,0,16,9,0,0,0,0,0,1,0,0] >;`

C42.57D4 in GAP, Magma, Sage, TeX

`C_4^2._{57}D_4`
`% in TeX`

`G:=Group("C4^2.57D4");`
`// GroupNames label`

`G:=SmallGroup(128,241);`
`// by ID`

`G=gap.SmallGroup(128,241);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,520,1123,1018,248,1971]);`
`// Polycyclic`

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

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