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## G = C42.222D4order 128 = 27

### 204th 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 — C42.222D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C2×C4×D4 — C42.222D4
 Lower central C1 — C2 — C2×C4 — C42.222D4
 Upper central C1 — C2×C4 — C2×C42 — C42.222D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.222D4

Generators and relations for C42.222D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 436 in 224 conjugacy classes, 98 normal (28 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×9], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×19], D4 [×4], D4 [×6], Q8 [×6], C23, C23 [×10], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], SD16 [×8], C22×C4 [×3], C22×C4 [×9], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8, C24, C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4×D4 [×2], C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C2×SD16 [×4], C23×C4, C22×D4, C42.12C4, C4×SD16 [×4], C22⋊SD16 [×2], D4.D4 [×2], D42Q8 [×2], C23.47D4 [×2], C2×C4×D4, C23.37C23, C42.222D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C2×SD16 [×6], C22×D4, C2×C4○D4 [×2], C22.19C24, C22×SD16, D8⋊C22, C42.222D4

Smallest permutation representation of C42.222D4
On 32 points
Generators in S32
```(1 16 5 12)(2 4 6 8)(3 10 7 14)(9 11 13 15)(17 19 21 23)(18 25 22 29)(20 27 24 31)(26 28 30 32)
(1 25 10 24)(2 26 11 17)(3 27 12 18)(4 28 13 19)(5 29 14 20)(6 30 15 21)(7 31 16 22)(8 32 9 23)
(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 9 5 13)(2 12 6 16)(3 15 7 11)(4 10 8 14)(17 31 21 27)(18 26 22 30)(19 29 23 25)(20 32 24 28)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4J 4K ··· 4P 4Q 4R 4S 4T 8A ··· 8H order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 SD16 C4○D4 D8⋊C22 kernel C42.222D4 C42.12C4 C4×SD16 C22⋊SD16 D4.D4 D4⋊2Q8 C23.47D4 C2×C4×D4 C23.37C23 C42 C22×C4 C2×C4 D4 C2 # reps 1 1 4 2 2 2 2 1 1 2 2 8 8 2

Matrix representation of C42.222D4 in GL4(𝔽17) generated by

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

C42.222D4 in GAP, Magma, Sage, TeX

`C_4^2._{222}D_4`
`% in TeX`

`G:=Group("C4^2.222D4");`
`// GroupNames label`

`G:=SmallGroup(128,1833);`
`// by ID`

`G=gap.SmallGroup(128,1833);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,80,4037,1027,124]);`
`// Polycyclic`

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

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