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## G = C42.326D4order 128 = 27

### 22nd non-split extension by C42 of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.326D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C42.326D4
 Lower central C1 — C2 — C2×C4 — C42.326D4
 Upper central C1 — C2×C4 — C22×C8 — C42.326D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.326D4

Generators and relations for C42.326D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 292 in 150 conjugacy classes, 56 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×8], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×12], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×2], C42, C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×2], C22⋊C8 [×4], C4≀C2 [×4], C8.C4 [×2], C2×C42, C22×C8 [×2], C2×M4(2) [×2], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×4], C2×C4○D4 [×2], C2×C4×C8, (C22×C8)⋊C2 [×2], C2×C4≀C2 [×2], C2×C8.C4, C2×C4○D8, C42.326D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, C8○D8 [×2], C42.326D4

Smallest permutation representation of C42.326D4
On 32 points
Generators in S32
```(1 3 5 7)(2 4 6 8)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 19 21 23)(18 20 22 24)
(1 23 5 19)(2 24 6 20)(3 17 7 21)(4 18 8 22)(9 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)
(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 10 19 25 5 14 23 29)(2 13 20 28 6 9 24 32)(3 16 21 31 7 12 17 27)(4 11 22 26 8 15 18 30)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O 4P 8A ··· 8P 8Q 8R 8S 8T order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 8 8 1 1 1 1 2 ··· 2 8 8 2 ··· 2 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D4 C4○D4 C4○D4 C8○D8 kernel C42.326D4 C2×C4×C8 (C22×C8)⋊C2 C2×C4≀C2 C2×C8.C4 C2×C4○D8 C2×D8 C2×SD16 C2×Q16 C42 C2×C8 C2×C4 C23 C2 # reps 1 1 2 2 1 1 2 4 2 2 6 2 2 16

Matrix representation of C42.326D4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 13 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 4
,
 4 2 0 0 0 13 0 0 0 0 8 0 0 0 0 2
,
 5 8 0 0 14 12 0 0 0 0 0 2 0 0 15 0
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,2,13,0,0,0,0,8,0,0,0,0,2],[5,14,0,0,8,12,0,0,0,0,0,15,0,0,2,0] >;`

C42.326D4 in GAP, Magma, Sage, TeX

`C_4^2._{326}D_4`
`% in TeX`

`G:=Group("C4^2.326D4");`
`// GroupNames label`

`G:=SmallGroup(128,706);`
`// by ID`

`G=gap.SmallGroup(128,706);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248,2804,172,124]);`
`// Polycyclic`

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

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