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## G = C42.275D4order 128 = 27

### 257th 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 — C42.275D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4⋊1D4 — C42.275D4
 Lower central C1 — C2 — C2×C4 — C42.275D4
 Upper central C1 — C22 — C2×C42 — C42.275D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.275D4

Generators and relations for C42.275D4
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=b2c3 >

Subgroups: 628 in 245 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×7], C4 [×4], C4 [×7], C22, C22 [×2], C22 [×25], C8 [×4], C2×C4 [×6], C2×C4 [×11], D4 [×32], Q8 [×2], C23, C23 [×17], C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4 [×4], C2×D4 [×23], C2×Q8, C24 [×2], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C41D4 [×4], C41D4 [×2], C2×D8 [×4], C2×SD16 [×4], C22×D4 [×2], C22×D4 [×2], C42.6C4, C22⋊D8 [×2], C22⋊SD16 [×2], C4⋊D8 [×2], C4⋊SD16 [×2], C42.29C22 [×2], C83D4 [×2], C23.36C23, C2×C41D4, C42.275D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×4], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C8⋊C22 [×2], C42.275D4

Character table of C42.275D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D size 1 1 1 1 2 2 8 8 8 8 8 2 2 2 2 4 4 4 4 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 0 0 0 0 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 0 0 0 0 0 -2 -2 -2 -2 2 -2 2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 2 0 0 0 0 0 2 -2 2 -2 2 -2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 -4 0 0 0 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ22 4 -4 4 -4 0 0 0 0 0 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ23 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 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ25 4 -4 4 -4 0 0 0 0 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ26 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

Smallest permutation representation of C42.275D4
On 32 points
Generators in S32
```(1 10 30 19)(2 15 31 24)(3 12 32 21)(4 9 25 18)(5 14 26 23)(6 11 27 20)(7 16 28 17)(8 13 29 22)
(1 7 5 3)(2 29 6 25)(4 31 8 27)(9 24 13 20)(10 16 14 12)(11 18 15 22)(17 23 21 19)(26 32 30 28)
(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)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 16)(7 15)(8 14)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)```

`G:=sub<Sym(32)| (1,10,30,19)(2,15,31,24)(3,12,32,21)(4,9,25,18)(5,14,26,23)(6,11,27,20)(7,16,28,17)(8,13,29,22), (1,7,5,3)(2,29,6,25)(4,31,8,27)(9,24,13,20)(10,16,14,12)(11,18,15,22)(17,23,21,19)(26,32,30,28), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28)>;`

`G:=Group( (1,10,30,19)(2,15,31,24)(3,12,32,21)(4,9,25,18)(5,14,26,23)(6,11,27,20)(7,16,28,17)(8,13,29,22), (1,7,5,3)(2,29,6,25)(4,31,8,27)(9,24,13,20)(10,16,14,12)(11,18,15,22)(17,23,21,19)(26,32,30,28), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28) );`

`G=PermutationGroup([(1,10,30,19),(2,15,31,24),(3,12,32,21),(4,9,25,18),(5,14,26,23),(6,11,27,20),(7,16,28,17),(8,13,29,22)], [(1,7,5,3),(2,29,6,25),(4,31,8,27),(9,24,13,20),(10,16,14,12),(11,18,15,22),(17,23,21,19),(26,32,30,28)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,16),(7,15),(8,14),(17,27),(18,26),(19,25),(20,32),(21,31),(22,30),(23,29),(24,28)])`

Matrix representation of C42.275D4 in GL8(𝔽17)

 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 16 1 0 0 0 0 0 0 15 1 0 0 0 0 0 0 16 9 1 9 0 0 0 0 2 0 13 16
,
 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 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 15 16 0 0 0 0 0 0 0 16 0 0 0 0 1 8 16 8 0 0 0 0 0 1 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0

`G:=sub<GL(8,GF(17))| [0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,15,16,2,0,0,0,0,1,1,9,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,9,16],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,8,1,0,0,0,0,15,0,16,0,0,0,0,0,16,16,8,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,0,16,0,0,0,0,0,0,1,0,0] >;`

C42.275D4 in GAP, Magma, Sage, TeX

`C_4^2._{275}D_4`
`% in TeX`

`G:=Group("C4^2.275D4");`
`// GroupNames label`

`G:=SmallGroup(128,1949);`
`// by ID`

`G=gap.SmallGroup(128,1949);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,675,4037,1027,124]);`
`// 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=b^2*c^3>;`
`// generators/relations`

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