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## G = C4⋊C4.106D4order 128 = 27

### 61st non-split extension by C4⋊C4 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 — C22×C4 — C4⋊C4.106D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×D4⋊C4 — C4⋊C4.106D4
 Lower central C1 — C2 — C22×C4 — C4⋊C4.106D4
 Upper central C1 — C23 — C2×C42 — C4⋊C4.106D4
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊C4.106D4

Generators and relations for C4⋊C4.106D4
G = < a,b,c,d | a4=b4=c4=1, d2=a, bab-1=cac-1=a-1, ad=da, cbc-1=a2b-1, dbd-1=a-1b, dcd-1=ac-1 >

Subgroups: 344 in 142 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×7], C22 [×7], C22 [×10], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×15], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, C2.C42, D4⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×D4, C22.4Q16, C23.65C23, C24.3C22, C2×D4⋊C4 [×2], C2×C4⋊C8, C2×C4.Q8, C4⋊C4.106D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, SD16 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C2×SD16, C4○D8, C8⋊C22 [×2], C23.Q8, C4⋊SD16, D4.2D4, C88D4, C82D4, D42Q8, D4.Q8, C4⋊C4.106D4

Smallest permutation representation of C4⋊C4.106D4
On 64 points
Generators in S64
```(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 11 47 62)(2 14 48 57)(3 9 41 60)(4 12 42 63)(5 15 43 58)(6 10 44 61)(7 13 45 64)(8 16 46 59)(17 52 29 33)(18 55 30 36)(19 50 31 39)(20 53 32 34)(21 56 25 37)(22 51 26 40)(23 54 27 35)(24 49 28 38)
(1 49 50 8)(2 7 51 56)(3 55 52 6)(4 5 53 54)(9 22 29 57)(10 64 30 21)(11 20 31 63)(12 62 32 19)(13 18 25 61)(14 60 26 17)(15 24 27 59)(16 58 28 23)(33 44 41 36)(34 35 42 43)(37 48 45 40)(38 39 46 47)
(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)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 Q8 SD16 C4○D4 C4○D8 C8⋊C22 kernel C4⋊C4.106D4 C22.4Q16 C23.65C23 C24.3C22 C2×D4⋊C4 C2×C4⋊C8 C2×C4.Q8 C4⋊C4 C2×C8 C22×C4 C2×D4 C2×C4 C2×C4 C22 C22 # reps 1 1 1 1 2 1 1 2 2 2 2 4 6 4 2

Matrix representation of C4⋊C4.106D4 in GL6(𝔽17)

 0 16 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 14 0 0 0 0 14 14 0 0 0 0 0 0 9 15 0 0 0 0 7 8 0 0 0 0 0 0 0 13 0 0 0 0 13 0
,
 12 12 0 0 0 0 12 5 0 0 0 0 0 0 1 0 0 0 0 0 9 16 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 12 12 0 0 0 0 5 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,9,7,0,0,0,0,15,8,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[12,12,0,0,0,0,12,5,0,0,0,0,0,0,1,9,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[12,5,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C4⋊C4.106D4 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{106}D_4`
`% in TeX`

`G:=Group("C4:C4.106D4");`
`// GroupNames label`

`G:=SmallGroup(128,797);`
`// by ID`

`G=gap.SmallGroup(128,797);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,2019,521,248,2804,718,172,4037,1027,124]);`
`// Polycyclic`

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

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