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## G = C42.25D4order 128 = 27

### 7th 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 — C42.25D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4⋊C8 — C42.25D4
 Lower central C1 — C2 — C2×C4 — C42.25D4
 Upper central C1 — C22 — C2×C42 — C42.25D4
 Jennings C1 — C22 — C22 — C2×C42 — C42.25D4

Generators and relations for C42.25D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, cac-1=a-1b2, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1bc3 >

Subgroups: 120 in 68 conjugacy classes, 34 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], C23, C42 [×4], C2×C8 [×8], C22×C4 [×3], C8⋊C4 [×4], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C2×C4⋊C8, C42.6C4 [×2], C42.25D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4.D4, C4.10D4, C8.C4 [×2], C4.9C42, C4.C42, C22.C42, C42.25D4

Smallest permutation representation of C42.25D4
On 64 points
Generators in S64
```(1 3 5 7)(2 20 6 24)(4 22 8 18)(9 57 13 61)(10 12 14 16)(11 59 15 63)(17 19 21 23)(25 36 29 40)(26 28 30 32)(27 38 31 34)(33 35 37 39)(41 43 45 47)(42 49 46 53)(44 51 48 55)(50 52 54 56)(58 60 62 64)
(1 62 21 12)(2 13 22 63)(3 64 23 14)(4 15 24 57)(5 58 17 16)(6 9 18 59)(7 60 19 10)(8 11 20 61)(25 48 38 49)(26 50 39 41)(27 42 40 51)(28 52 33 43)(29 44 34 53)(30 54 35 45)(31 46 36 55)(32 56 37 47)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 31 3 34 5 27 7 38)(2 47 20 41 6 43 24 45)(4 54 22 56 8 50 18 52)(9 33 57 35 13 37 61 39)(10 49 12 46 14 53 16 42)(11 26 59 28 15 30 63 32)(17 40 19 25 21 36 23 29)(44 58 51 60 48 62 55 64)```

`G:=sub<Sym(64)| (1,3,5,7)(2,20,6,24)(4,22,8,18)(9,57,13,61)(10,12,14,16)(11,59,15,63)(17,19,21,23)(25,36,29,40)(26,28,30,32)(27,38,31,34)(33,35,37,39)(41,43,45,47)(42,49,46,53)(44,51,48,55)(50,52,54,56)(58,60,62,64), (1,62,21,12)(2,13,22,63)(3,64,23,14)(4,15,24,57)(5,58,17,16)(6,9,18,59)(7,60,19,10)(8,11,20,61)(25,48,38,49)(26,50,39,41)(27,42,40,51)(28,52,33,43)(29,44,34,53)(30,54,35,45)(31,46,36,55)(32,56,37,47), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,3,34,5,27,7,38)(2,47,20,41,6,43,24,45)(4,54,22,56,8,50,18,52)(9,33,57,35,13,37,61,39)(10,49,12,46,14,53,16,42)(11,26,59,28,15,30,63,32)(17,40,19,25,21,36,23,29)(44,58,51,60,48,62,55,64)>;`

`G:=Group( (1,3,5,7)(2,20,6,24)(4,22,8,18)(9,57,13,61)(10,12,14,16)(11,59,15,63)(17,19,21,23)(25,36,29,40)(26,28,30,32)(27,38,31,34)(33,35,37,39)(41,43,45,47)(42,49,46,53)(44,51,48,55)(50,52,54,56)(58,60,62,64), (1,62,21,12)(2,13,22,63)(3,64,23,14)(4,15,24,57)(5,58,17,16)(6,9,18,59)(7,60,19,10)(8,11,20,61)(25,48,38,49)(26,50,39,41)(27,42,40,51)(28,52,33,43)(29,44,34,53)(30,54,35,45)(31,46,36,55)(32,56,37,47), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,31,3,34,5,27,7,38)(2,47,20,41,6,43,24,45)(4,54,22,56,8,50,18,52)(9,33,57,35,13,37,61,39)(10,49,12,46,14,53,16,42)(11,26,59,28,15,30,63,32)(17,40,19,25,21,36,23,29)(44,58,51,60,48,62,55,64) );`

`G=PermutationGroup([(1,3,5,7),(2,20,6,24),(4,22,8,18),(9,57,13,61),(10,12,14,16),(11,59,15,63),(17,19,21,23),(25,36,29,40),(26,28,30,32),(27,38,31,34),(33,35,37,39),(41,43,45,47),(42,49,46,53),(44,51,48,55),(50,52,54,56),(58,60,62,64)], [(1,62,21,12),(2,13,22,63),(3,64,23,14),(4,15,24,57),(5,58,17,16),(6,9,18,59),(7,60,19,10),(8,11,20,61),(25,48,38,49),(26,50,39,41),(27,42,40,51),(28,52,33,43),(29,44,34,53),(30,54,35,45),(31,46,36,55),(32,56,37,47)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,31,3,34,5,27,7,38),(2,47,20,41,6,43,24,45),(4,54,22,56,8,50,18,52),(9,33,57,35,13,37,61,39),(10,49,12,46,14,53,16,42),(11,26,59,28,15,30,63,32),(17,40,19,25,21,36,23,29),(44,58,51,60,48,62,55,64)])`

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4H 4I 4J 8A ··· 8H 8I ··· 8P order 1 2 2 2 2 2 4 ··· 4 4 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 2 ··· 2 4 4 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + - + - image C1 C2 C2 C4 C4 D4 D4 Q8 C8.C4 C4.D4 C4.10D4 C4.9C42 kernel C42.25D4 C2×C4⋊C8 C42.6C4 C8⋊C4 C22×C8 C42 C22×C4 C22×C4 C22 C4 C4 C2 # reps 1 1 2 8 4 2 1 1 8 1 1 2

Matrix representation of C42.25D4 in GL6(𝔽17)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 2 3 0 0 1 0 8 14 0 0 0 0 16 0 0 0 0 0 15 1
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 8 0 0 0 4 0 15 0 0 0 0 13 0 0 0 0 0 0 13
,
 14 2 0 0 0 0 2 3 0 0 0 0 0 0 4 10 9 7 0 0 16 6 3 9 0 0 10 10 0 16 0 0 13 7 0 7
,
 9 12 0 0 0 0 12 8 0 0 0 0 0 0 2 0 1 10 0 0 8 0 0 16 0 0 16 1 0 3 0 0 15 0 0 15

`G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,2,8,16,15,0,0,3,14,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,8,15,0,13],[14,2,0,0,0,0,2,3,0,0,0,0,0,0,4,16,10,13,0,0,10,6,10,7,0,0,9,3,0,0,0,0,7,9,16,7],[9,12,0,0,0,0,12,8,0,0,0,0,0,0,2,8,16,15,0,0,0,0,1,0,0,0,1,0,0,0,0,0,10,16,3,15] >;`

C42.25D4 in GAP, Magma, Sage, TeX

`C_4^2._{25}D_4`
`% in TeX`

`G:=Group("C4^2.25D4");`
`// GroupNames label`

`G:=SmallGroup(128,22);`
`// by ID`

`G=gap.SmallGroup(128,22);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,136,3924,242]);`
`// Polycyclic`

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

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