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## G = C42.290D4order 128 = 27

### 272nd non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.290D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C4⋊Q8 — C42.290D4
 Lower central C1 — C2 — C2×C4 — C42.290D4
 Upper central C1 — C22 — C2×C42 — C42.290D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.290D4

Generators and relations for C42.290D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=a2b2c3 >

Subgroups: 340 in 181 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×11], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×19], D4 [×4], Q8 [×10], C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×8], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C42.6C4, D42Q8 [×2], C4.Q16 [×2], C22.D8 [×2], C23.47D4 [×2], C42.28C22 [×2], C8⋊Q8 [×2], C23.36C23, C2×C4⋊Q8, C42.290D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C2×C8⋊C22, C2×C8.C22, C42.290D4

Character table of C42.290D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D size 1 1 1 1 2 2 8 2 2 2 2 4 4 4 4 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 1 1 -1 1 1 -1 1 -1 linear of order 2 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 2 2 2 -2 -2 0 -2 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 2 0 -2 -2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 2 0 2 2 -2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 -2 0 2 2 -2 -2 2 -2 2 -2 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 orthogonal lifted from C8⋊C22 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 orthogonal lifted from C8⋊C22 ρ23 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 2- 1+4, Schur index 2 ρ24 4 4 -4 -4 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, 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 2- 1+4, Schur index 2 ρ26 4 4 -4 -4 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.290D4
On 64 points
Generators in S64
```(1 52 29 59)(2 49 30 64)(3 54 31 61)(4 51 32 58)(5 56 25 63)(6 53 26 60)(7 50 27 57)(8 55 28 62)(9 41 34 24)(10 46 35 21)(11 43 36 18)(12 48 37 23)(13 45 38 20)(14 42 39 17)(15 47 40 22)(16 44 33 19)
(1 15 5 11)(2 33 6 37)(3 9 7 13)(4 35 8 39)(10 28 14 32)(12 30 16 26)(17 51 21 55)(18 59 22 63)(19 53 23 49)(20 61 24 57)(25 36 29 40)(27 38 31 34)(41 50 45 54)(42 58 46 62)(43 52 47 56)(44 60 48 64)
(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)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 34)(10 16)(11 40)(12 14)(13 38)(15 36)(17 44)(19 42)(20 24)(21 48)(23 46)(27 31)(33 35)(37 39)(41 45)(49 51)(50 57)(52 63)(53 55)(54 61)(56 59)(58 64)(60 62)```

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

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

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

Matrix representation of C42.290D4 in GL8(𝔽17)

 4 8 0 0 0 0 0 0 0 13 0 0 0 0 0 0 8 8 13 8 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 1 16 0 0 0 0 16 0 16 16 0 0 0 0 1 16 0 1 0 0 0 0 16 16 16 0
,
 16 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 15 16 2 0 0 0 0 0 0 0 1 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 16 0 0 0 0 0 0 1 0
,
 1 0 16 0 0 0 0 0 16 0 1 16 0 0 0 0 2 0 16 0 0 0 0 0 1 1 0 0 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
,
 1 0 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 16 0 1 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(17))| [4,0,8,0,0,0,0,0,8,13,8,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,0,16,1,16,0,0,0,0,1,0,16,16,0,0,0,0,1,16,0,16,0,0,0,0,16,16,1,0],[16,0,0,0,0,0,0,0,15,1,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[1,16,2,1,0,0,0,0,0,0,0,1,0,0,0,0,16,1,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,16,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C42.290D4 in GAP, Magma, Sage, TeX

`C_4^2._{290}D_4`
`% in TeX`

`G:=Group("C4^2.290D4");`
`// GroupNames label`

`G:=SmallGroup(128,1970);`
`// by ID`

`G=gap.SmallGroup(128,1970);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,100,675,4037,1027,124]);`
`// Polycyclic`

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

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