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## G = C42.425D4order 128 = 27

### 58th non-split extension by C42 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.425D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C42 — C42.425D4
 Lower central C1 — C22 — C42.425D4
 Upper central C1 — C22×C4 — C42.425D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.425D4

Generators and relations for C42.425D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 316 in 200 conjugacy classes, 92 normal (34 characteristic)
C1, C2 [×7], C2 [×4], C4 [×8], C4 [×8], C22 [×7], C22 [×4], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×18], C2×C4 [×32], C23, C23 [×6], C23 [×4], C42 [×4], C42 [×6], C2×C8 [×12], C22×C4 [×6], C22×C4 [×8], C22×C4 [×14], C24, C22⋊C8 [×4], C4⋊C8 [×4], C2×C42 [×4], C2×C42 [×4], C22×C8 [×4], C23×C4 [×3], C22.7C42 [×2], C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C22×C42, C42.425D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×6], M4(2) [×6], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C22⋊C8 [×4], C4⋊C8 [×4], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22×C8, C2×M4(2) [×3], C23.7Q8, C2×C22⋊C8, C24.4C4, C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.6C4, C42.425D4

Smallest permutation representation of C42.425D4
On 64 points
Generators in S64
```(1 63 55 47)(2 48 56 64)(3 57 49 41)(4 42 50 58)(5 59 51 43)(6 44 52 60)(7 61 53 45)(8 46 54 62)(9 37 21 31)(10 32 22 38)(11 39 23 25)(12 26 24 40)(13 33 17 27)(14 28 18 34)(15 35 19 29)(16 30 20 36)
(1 37 5 33)(2 38 6 34)(3 39 7 35)(4 40 8 36)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 63 21 59)(18 64 22 60)(19 57 23 61)(20 58 24 62)(25 53 29 49)(26 54 30 50)(27 55 31 51)(28 56 32 52)
(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 30 37 50 5 26 33 54)(2 49 38 25 6 53 34 29)(3 32 39 52 7 28 35 56)(4 51 40 27 8 55 36 31)(9 58 43 24 13 62 47 20)(10 23 44 61 14 19 48 57)(11 60 45 18 15 64 41 22)(12 17 46 63 16 21 42 59)```

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C8 D4 D4 Q8 M4(2) C4○D4 M4(2) kernel C42.425D4 C22.7C42 C2×C22⋊C8 C2×C4⋊C8 C22×C42 C2×C42 C23×C4 C22×C4 C42 C22×C4 C22×C4 C2×C4 C2×C4 C23 # reps 1 2 2 2 1 4 4 16 4 2 2 8 4 4

Matrix representation of C42.425D4 in GL5(𝔽17)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 13 0 0 0 0 0 4
,
 13 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 9 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 16 0
,
 8 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 16 0

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,0,4],[13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[9,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,16,0],[8,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0] >;`

C42.425D4 in GAP, Magma, Sage, TeX

`C_4^2._{425}D_4`
`% in TeX`

`G:=Group("C4^2.425D4");`
`// GroupNames label`

`G:=SmallGroup(128,529);`
`// by ID`

`G=gap.SmallGroup(128,529);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,124]);`
`// 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*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;`
`// generators/relations`

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