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## G = C42.32D4order 128 = 27

### 14th 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 — C23 — C42.32D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C4⋊M4(2) — C42.32D4
 Lower central C1 — C22 — C23 — C42.32D4
 Upper central C1 — C22 — C2×C42 — C42.32D4
 Jennings C1 — C22 — C22 — C2×C42 — C42.32D4

Generators and relations for C42.32D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1b-1c3 >

Subgroups: 136 in 73 conjugacy classes, 36 normal (6 characteristic)
C1, C2 [×3], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×6], C2×C4 [×6], C2×C4 [×6], C23, C42, C42 [×3], C2×C8 [×6], M4(2) [×6], C22×C4 [×3], C4⋊C8 [×6], C2×C42, C2×M4(2) [×6], C4⋊M4(2) [×3], C42.32D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4.D4 [×3], C4.10D4 [×3], C22.C42 [×3], C42.32D4

Character table of C42.32D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 1 1 4 2 2 2 2 2 2 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -i -i i -i -i i -1 i -1 i 1 linear of order 4 ρ6 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -i i -i -1 -1 1 -i -i 1 i i i linear of order 4 ρ7 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 -i -i -i i i i 1 -i 1 i -1 linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -i -1 1 i -i i -1 i -i -i 1 i linear of order 4 ρ9 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -i -i i 1 1 -1 i -i -1 i -i i linear of order 4 ρ10 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 i -1 1 -i i -i -1 -i i i 1 -i linear of order 4 ρ11 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 i 1 -1 i -i i 1 -i -i i -1 -i linear of order 4 ρ12 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -i 1 -1 -i i -i 1 i i -i -1 i linear of order 4 ρ13 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 i i -i 1 1 -1 -i i -1 -i i -i linear of order 4 ρ14 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 i i i -i -i -i 1 i 1 -i -1 linear of order 4 ρ15 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 i -i i -1 -1 1 i i 1 -i -i -i linear of order 4 ρ16 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 i i -i i i -i -1 -i -1 -i 1 linear of order 4 ρ17 2 2 2 2 -2 2 -2 2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 2 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 -2 2 -2 2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 -2 -2 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 -4 -4 4 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ22 4 -4 4 -4 0 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ23 4 4 -4 -4 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ24 4 4 -4 -4 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ25 4 -4 4 -4 0 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ26 4 -4 -4 4 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2

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

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

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

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

Matrix representation of C42.32D4 in GL8(𝔽17)

 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 0 1 0 0 0 0 11 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 10 0 0 1 0 0 0 0 0 7 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0
,
 12 5 0 0 0 0 0 0 5 5 0 0 0 0 0 0 14 4 5 12 0 0 0 0 4 5 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0
,
 5 5 7 7 0 0 0 0 3 14 7 10 0 0 0 0 15 6 12 14 0 0 0 0 6 7 12 3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(17))| [0,1,0,11,0,0,0,0,16,0,6,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,16,10,0,0,0,0,0,1,0,0,7,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[12,5,14,4,0,0,0,0,5,5,4,5,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[5,3,15,6,0,0,0,0,5,14,6,7,0,0,0,0,7,7,12,12,0,0,0,0,7,10,14,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C42.32D4 in GAP, Magma, Sage, TeX

`C_4^2._{32}D_4`
`% in TeX`

`G:=Group("C4^2.32D4");`
`// GroupNames label`

`G:=SmallGroup(128,41);`
`// by ID`

`G=gap.SmallGroup(128,41);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,136,2804,172]);`
`// Polycyclic`

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

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