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## G = C42.71D4order 128 = 27

### 53rd 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 — C42.71D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C42.C2 — C42.71D4
 Lower central C1 — C22 — C2×C4 — C42.71D4
 Upper central C1 — C22 — C2×C42 — C42.71D4
 Jennings C1 — C22 — C22 — C42 — C42.71D4

Generators and relations for C42.71D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >

Subgroups: 180 in 98 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], C23, C42 [×2], C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C8⋊C4 [×4], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42.C2 [×4], C42.C2 [×2], C42.2C22 [×4], C42.6C4 [×2], C2×C42.C2, C42.71D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C2×C4.10D4, C42⋊C22 [×2], C42.71D4

Character table of C42.71D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 2 2 2 2 4 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 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ6 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 ρ7 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 ρ8 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 ρ9 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 i -i i -i -i i -i i linear of order 4 ρ10 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 i i -i -i -i -i i i linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 -i -i -i -i i i i i linear of order 4 ρ12 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 -i i i -i i -i -i i linear of order 4 ρ13 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -i -i i i i i -i -i linear of order 4 ρ14 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 -i i -i i i -i i -i linear of order 4 ρ15 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 i -i -i i -i i i -i linear of order 4 ρ16 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 i i i i -i -i -i -i 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 orthogonal lifted from D4 ρ21 4 -4 4 -4 4 -4 0 0 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 ρ22 4 -4 4 -4 -4 4 0 0 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 ρ23 4 4 -4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22 ρ24 4 -4 -4 4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22 ρ25 4 4 -4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22 ρ26 4 -4 -4 4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22

Smallest permutation representation of C42.71D4
On 64 points
Generators in S64
```(1 62 49 38)(2 35 50 59)(3 64 51 40)(4 37 52 61)(5 58 53 34)(6 39 54 63)(7 60 55 36)(8 33 56 57)(9 48 29 20)(10 17 30 45)(11 42 31 22)(12 19 32 47)(13 44 25 24)(14 21 26 41)(15 46 27 18)(16 23 28 43)
(1 36 53 64)(2 33 54 61)(3 38 55 58)(4 35 56 63)(5 40 49 60)(6 37 50 57)(7 34 51 62)(8 39 52 59)(9 46 25 22)(10 43 26 19)(11 48 27 24)(12 45 28 21)(13 42 29 18)(14 47 30 23)(15 44 31 20)(16 41 32 17)
(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 20 36 15 53 44 64 31)(2 26 33 19 54 10 61 43)(3 46 38 25 55 22 58 9)(4 12 35 45 56 28 63 21)(5 24 40 11 49 48 60 27)(6 30 37 23 50 14 57 47)(7 42 34 29 51 18 62 13)(8 16 39 41 52 32 59 17)```

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

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

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

Matrix representation of C42.71D4 in GL8(𝔽17)

 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 16 16 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 16 16 0 0 0 0 0 0 2 1
,
 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 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 0 0 4 7 0 0 0 0 0 0 10 13 0 0 0 0 1 6 0 0 0 0 0 0 11 16 0 0 0 0 0 0 0 0 0 0 0 0 7 12 0 0 0 0 0 0 10 10 0 0 0 0 0 14 0 0 0 0 0 0 11 0 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0

`G:=sub<GL(8,GF(17))| [0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,1,11,0,0,0,0,0,0,6,16,0,0,0,0,4,10,0,0,0,0,0,0,7,13,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,14,0,0,0,0,0,7,10,0,0,0,0,0,0,12,10,0,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C42.71D4 in GAP, Magma, Sage, TeX

`C_4^2._{71}D_4`
`% in TeX`

`G:=Group("C4^2.71D4");`
`// GroupNames label`

`G:=SmallGroup(128,266);`
`// by ID`

`G=gap.SmallGroup(128,266);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1123,1018,248,1971,102]);`
`// Polycyclic`

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

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