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## G = C42.232D4order 128 = 27

### 214th 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 — C42.232D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C2×C4×D4 — C42.232D4
 Lower central C1 — C2 — C2×C4 — C42.232D4
 Upper central C1 — C22 — C2×C42 — C42.232D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.232D4

Generators and relations for C42.232D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=a2b, dcd=a2c3 >

Subgroups: 460 in 225 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×21], C8 [×4], C2×C4 [×6], C2×C4 [×22], D4 [×4], D4 [×10], Q8 [×2], C23, C23 [×11], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×10], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C24, C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×4], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×D8 [×2], C2×SD16 [×2], C23×C4, C22×D4, C42.6C4, SD16⋊C4 [×2], D8⋊C4 [×2], C22⋊D8, C22⋊SD16, D4.2D4 [×2], D4.Q8 [×2], C22.D8, C23.47D4, C2×C4×D4, C23.36C23, C42.232D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8⋊C22, D8⋊C22, C42.232D4

Smallest permutation representation of C42.232D4
On 32 points
Generators in S32
```(1 29 5 25)(2 8 6 4)(3 31 7 27)(9 15 13 11)(10 21 14 17)(12 23 16 19)(18 24 22 20)(26 32 30 28)
(1 16 31 21)(2 13 32 18)(3 10 25 23)(4 15 26 20)(5 12 27 17)(6 9 28 22)(7 14 29 19)(8 11 30 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 19)(10 18)(11 17)(12 24)(13 23)(14 22)(15 21)(16 20)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A ··· 4H 4I ··· 4N 4O 4P 4Q 8A 8B 8C 8D order 1 2 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 8 8 8 8 size 1 1 1 1 2 2 4 4 4 4 8 2 ··· 2 4 ··· 4 8 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 C8⋊C22 D8⋊C22 kernel C42.232D4 C42.6C4 SD16⋊C4 D8⋊C4 C22⋊D8 C22⋊SD16 D4.2D4 D4.Q8 C22.D8 C23.47D4 C2×C4×D4 C23.36C23 C42 C22×C4 D4 C22 C2 # reps 1 1 2 2 1 1 2 2 1 1 1 1 2 2 8 2 2

Matrix representation of C42.232D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 14 14 0 16 0 0 14 3 1 0
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 5 12 4 0 0 0 12 12 0 4
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 14 3 2 0 0 0 14 14 0 15 0 0 0 0 3 3 0 0 0 0 14 3
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 3 14 15 0 0 0 14 14 0 15 0 0 0 0 14 3 0 0 0 0 3 3

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,14,14,0,0,16,0,14,3,0,0,0,0,0,1,0,0,0,0,16,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,5,12,0,0,0,13,12,12,0,0,0,0,4,0,0,0,0,0,0,4],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,2,0,3,14,0,0,0,15,3,3],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,15,0,14,3,0,0,0,15,3,3] >;`

C42.232D4 in GAP, Magma, Sage, TeX

`C_4^2._{232}D_4`
`% in TeX`

`G:=Group("C4^2.232D4");`
`// GroupNames label`

`G:=SmallGroup(128,1846);`
`// by ID`

`G=gap.SmallGroup(128,1846);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,521,304,4037,1027,124]);`
`// 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*b^2,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*c^3>;`
`// generators/relations`

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