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## G = C42.247D4order 128 = 27

### 229th 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 — C42.247D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.247D4
 Lower central C1 — C2 — C2×C4 — C42.247D4
 Upper central C1 — C2×C4 — C2×C42 — C42.247D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.247D4

Generators and relations for C42.247D4
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 564 in 282 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×24], Q8 [×8], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], D8 [×8], SD16 [×16], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×4], C4○D4 [×16], C8⋊C4 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×4], C41D4 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×D8 [×4], C2×SD16 [×8], C2×Q16 [×4], C4○D8 [×16], C2×C4○D4 [×4], C2×C8⋊C4, C83D4 [×4], C8.2D4 [×4], C22.26C24 [×2], C2×C4○D8 [×4], C42.247D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C22×D4 [×3], C2×C41D4, D8⋊C22 [×2], C42.247D4

Smallest permutation representation of C42.247D4
On 64 points
Generators in S64
```(1 63 55 47)(2 60 56 44)(3 57 49 41)(4 62 50 46)(5 59 51 43)(6 64 52 48)(7 61 53 45)(8 58 54 42)(9 22 40 30)(10 19 33 27)(11 24 34 32)(12 21 35 29)(13 18 36 26)(14 23 37 31)(15 20 38 28)(16 17 39 25)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 42 13 46)(10 43 14 47)(11 44 15 48)(12 45 16 41)(25 49 29 53)(26 50 30 54)(27 51 31 55)(28 52 32 56)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 42 5 46)(2 45 6 41)(3 48 7 44)(4 43 8 47)(9 19 13 23)(10 22 14 18)(11 17 15 21)(12 20 16 24)(25 38 29 34)(26 33 30 37)(27 36 31 40)(28 39 32 35)(49 64 53 60)(50 59 54 63)(51 62 55 58)(52 57 56 61)```

`G:=sub<Sym(64)| (1,63,55,47)(2,60,56,44)(3,57,49,41)(4,62,50,46)(5,59,51,43)(6,64,52,48)(7,61,53,45)(8,58,54,42)(9,22,40,30)(10,19,33,27)(11,24,34,32)(12,21,35,29)(13,18,36,26)(14,23,37,31)(15,20,38,28)(16,17,39,25), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,42,13,46)(10,43,14,47)(11,44,15,48)(12,45,16,41)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,42,5,46)(2,45,6,41)(3,48,7,44)(4,43,8,47)(9,19,13,23)(10,22,14,18)(11,17,15,21)(12,20,16,24)(25,38,29,34)(26,33,30,37)(27,36,31,40)(28,39,32,35)(49,64,53,60)(50,59,54,63)(51,62,55,58)(52,57,56,61)>;`

`G:=Group( (1,63,55,47)(2,60,56,44)(3,57,49,41)(4,62,50,46)(5,59,51,43)(6,64,52,48)(7,61,53,45)(8,58,54,42)(9,22,40,30)(10,19,33,27)(11,24,34,32)(12,21,35,29)(13,18,36,26)(14,23,37,31)(15,20,38,28)(16,17,39,25), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,42,13,46)(10,43,14,47)(11,44,15,48)(12,45,16,41)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,42,5,46)(2,45,6,41)(3,48,7,44)(4,43,8,47)(9,19,13,23)(10,22,14,18)(11,17,15,21)(12,20,16,24)(25,38,29,34)(26,33,30,37)(27,36,31,40)(28,39,32,35)(49,64,53,60)(50,59,54,63)(51,62,55,58)(52,57,56,61) );`

`G=PermutationGroup([(1,63,55,47),(2,60,56,44),(3,57,49,41),(4,62,50,46),(5,59,51,43),(6,64,52,48),(7,61,53,45),(8,58,54,42),(9,22,40,30),(10,19,33,27),(11,24,34,32),(12,21,35,29),(13,18,36,26),(14,23,37,31),(15,20,38,28),(16,17,39,25)], [(1,19,5,23),(2,20,6,24),(3,21,7,17),(4,22,8,18),(9,42,13,46),(10,43,14,47),(11,44,15,48),(12,45,16,41),(25,49,29,53),(26,50,30,54),(27,51,31,55),(28,52,32,56),(33,59,37,63),(34,60,38,64),(35,61,39,57),(36,62,40,58)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,42,5,46),(2,45,6,41),(3,48,7,44),(4,43,8,47),(9,19,13,23),(10,22,14,18),(11,17,15,21),(12,20,16,24),(25,38,29,34),(26,33,30,37),(27,36,31,40),(28,39,32,35),(49,64,53,60),(50,59,54,63),(51,62,55,58),(52,57,56,61)])`

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 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 4 4 8 ··· 8 size 1 1 1 1 2 2 8 8 8 8 1 1 1 1 2 2 4 4 4 4 8 8 8 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D8⋊C22 kernel C42.247D4 C2×C8⋊C4 C8⋊3D4 C8.2D4 C22.26C24 C2×C4○D8 C42 C2×C8 C22×C4 C2 # reps 1 1 4 4 2 4 2 8 2 4

Matrix representation of C42.247D4 in GL6(𝔽17)

 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 16 16 14 14 0 0 1 16 3 14 0 0 3 3 1 1 0 0 14 3 16 1
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 3 3 1 1 0 0 3 14 1 16 0 0 16 16 14 14 0 0 16 1 14 3

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

C42.247D4 in GAP, Magma, Sage, TeX

`C_4^2._{247}D_4`
`% in TeX`

`G:=Group("C4^2.247D4");`
`// GroupNames label`

`G:=SmallGroup(128,1882);`
`// by ID`

`G=gap.SmallGroup(128,1882);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,184,248,2804,172]);`
`// Polycyclic`

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

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