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## G = C42.221D4order 128 = 27

### 203rd 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 — C42.221D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C2×C4×D4 — C42.221D4
 Lower central C1 — C2 — C2×C4 — C42.221D4
 Upper central C1 — C2×C4 — C2×C42 — C42.221D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.221D4

Generators and relations for C42.221D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1b2, dad=ab2, bc=cb, dbd=a2b, dcd=a2c3 >

Subgroups: 532 in 246 conjugacy classes, 98 normal (28 characteristic)
C1, C2 [×3], C2 [×8], C4 [×6], C4 [×7], C22, C22 [×2], C22 [×24], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×21], D4 [×4], D4 [×18], Q8 [×2], C23, C23 [×12], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], D8 [×8], C22×C4 [×3], C22×C4 [×11], C2×D4 [×4], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24, C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C2.D8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×6], C4×D4 [×3], C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×D8 [×4], C23×C4, C22×D4, C2×C4○D4, C42.12C4, C4×D8 [×4], C22⋊D8 [×2], C4⋊D8 [×2], D4⋊Q8 [×2], C22.D8 [×2], C2×C4×D4, C22.26C24, C42.221D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C4○D4 [×4], C24, C2×D8 [×6], C22×D4, C2×C4○D4 [×2], C22.19C24, C22×D8, D8⋊C22, C42.221D4

Smallest permutation representation of C42.221D4
On 32 points
Generators in S32
```(1 29 5 25)(2 8 6 4)(3 31 7 27)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 24 22 20)(26 32 30 28)
(1 11 27 17)(2 12 28 18)(3 13 29 19)(4 14 30 20)(5 15 31 21)(6 16 32 22)(7 9 25 23)(8 10 26 24)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 32)(8 31)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4J 4K ··· 4P 4Q 4R 8A ··· 8H order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 4 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 8 8 1 1 1 1 2 ··· 2 4 ··· 4 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 D8 C4○D4 D8⋊C22 kernel C42.221D4 C42.12C4 C4×D8 C22⋊D8 C4⋊D8 D4⋊Q8 C22.D8 C2×C4×D4 C22.26C24 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 C42.221D4 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 0 1 0 0 16 0
,
 4 0 0 0 0 4 0 0 0 0 0 1 0 0 16 0
,
 0 16 0 0 1 0 0 0 0 0 3 14 0 0 3 3
,
 0 1 0 0 1 0 0 0 0 0 3 14 0 0 14 14
`G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,16,0,0,1,0],[0,1,0,0,16,0,0,0,0,0,3,3,0,0,14,3],[0,1,0,0,1,0,0,0,0,0,3,14,0,0,14,14] >;`

C42.221D4 in GAP, Magma, Sage, TeX

`C_4^2._{221}D_4`
`% in TeX`

`G:=Group("C4^2.221D4");`
`// GroupNames label`

`G:=SmallGroup(128,1832);`
`// by ID`

`G=gap.SmallGroup(128,1832);`
`# by ID`

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

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

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