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## G = C42.63D4order 128 = 27

### 45th 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 — C42.63D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.36C23 — C42.63D4
 Lower central C1 — C22 — C2×C4 — C42.63D4
 Upper central C1 — C22 — C2×C42 — C42.63D4
 Jennings C1 — C2 — C22 — C22×C4 — C42.63D4

Generators and relations for C42.63D4
G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 236 in 109 conjugacy classes, 44 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×11], C22, C22 [×2], C22 [×5], C8 [×2], C2×C4 [×6], C2×C4 [×15], D4 [×3], Q8, C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2.C42 [×2], C2.C42 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C42, 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], C425C4, C42.12C4, C23.36C23, C42.63D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C4○D8 [×2], C23.C23, C23.24D4, C42⋊C22, C42.63D4

Smallest permutation representation of C42.63D4
On 32 points
Generators in S32
```(1 16 29 22)(2 23 30 9)(3 10 31 24)(4 17 32 11)(5 12 25 18)(6 19 26 13)(7 14 27 20)(8 21 28 15)
(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)
(2 28 30 8)(3 27)(4 6 32 26)(7 31)(9 17 23 11)(10 24)(12 16)(13 21 19 15)(14 20)(18 22)
(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)| (1,16,29,22)(2,23,30,9)(3,10,31,24)(4,17,32,11)(5,12,25,18)(6,19,26,13)(7,14,27,20)(8,21,28,15), (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), (2,28,30,8)(3,27)(4,6,32,26)(7,31)(9,17,23,11)(10,24)(12,16)(13,21,19,15)(14,20)(18,22), (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( (1,16,29,22)(2,23,30,9)(3,10,31,24)(4,17,32,11)(5,12,25,18)(6,19,26,13)(7,14,27,20)(8,21,28,15), (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), (2,28,30,8)(3,27)(4,6,32,26)(7,31)(9,17,23,11)(10,24)(12,16)(13,21,19,15)(14,20)(18,22), (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([(1,16,29,22),(2,23,30,9),(3,10,31,24),(4,17,32,11),(5,12,25,18),(6,19,26,13),(7,14,27,20),(8,21,28,15)], [(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)], [(2,28,30,8),(3,27),(4,6,32,26),(7,31),(9,17,23,11),(10,24),(12,16),(13,21,19,15),(14,20),(18,22)], [(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 4J 4K ··· 4Q 8A ··· 8H order 1 2 2 2 2 2 2 4 ··· 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 8 2 ··· 2 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 C4○D8 C23.C23 C42⋊C22 kernel C42.63D4 C22.SD16 C23.31D4 C42⋊5C4 C42.12C4 C23.36C23 C4×D4 C4×Q8 C4.4D4 C42.C2 C42 C22×C4 C22 C2 C2 # reps 1 2 2 1 1 1 2 2 2 2 2 2 8 2 2

Matrix representation of C42.63D4 in GL6(𝔽17)

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

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

C42.63D4 in GAP, Magma, Sage, TeX

`C_4^2._{63}D_4`
`% in TeX`

`G:=Group("C4^2.63D4");`
`// GroupNames label`

`G:=SmallGroup(128,253);`
`// by ID`

`G=gap.SmallGroup(128,253);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,520,1123,1018,248,1971]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,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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