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## G = C42.226D4order 128 = 27

### 208th 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 — C42.226D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — C2×C4×Q8 — C42.226D4
 Lower central C1 — C2 — C2×C4 — C42.226D4
 Upper central C1 — C2×C4 — C2×C42 — C42.226D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.226D4

Generators and relations for C42.226D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=ab2, dad=a-1b2, bc=cb, dbd=a2b, dcd=c3 >

Subgroups: 332 in 195 conjugacy classes, 92 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×13], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×21], D4 [×4], Q8 [×4], Q8 [×8], C23, C23, C42 [×4], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8 [×4], C4×Q8 [×2], C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×SD16 [×2], C2×Q16 [×2], C22×Q8, C42.12C4, C4×SD16 [×2], C4×Q16 [×2], Q8⋊D4, C22⋊Q16, Q8.D4 [×2], Q8.Q8 [×2], C22.D8, C23.47D4, C2×C4×Q8, C23.36C23, C42.226D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4○D8 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C4○D8, C2×C8.C22, C42.226D4

Smallest permutation representation of C42.226D4
On 64 points
Generators in S64
```(1 63 5 59)(2 49 6 53)(3 57 7 61)(4 51 8 55)(9 47 13 43)(10 23 14 19)(11 41 15 45)(12 17 16 21)(18 36 22 40)(20 38 24 34)(25 52 29 56)(26 60 30 64)(27 54 31 50)(28 62 32 58)(33 44 37 48)(35 46 39 42)
(1 37 29 10)(2 38 30 11)(3 39 31 12)(4 40 32 13)(5 33 25 14)(6 34 26 15)(7 35 27 16)(8 36 28 9)(17 57 42 50)(18 58 43 51)(19 59 44 52)(20 60 45 53)(21 61 46 54)(22 62 47 55)(23 63 48 56)(24 64 41 49)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 46)(18 41)(19 44)(20 47)(21 42)(22 45)(23 48)(24 43)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(49 62)(50 57)(51 60)(52 63)(53 58)(54 61)(55 64)(56 59)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E ··· 4J 4K ··· 4T 4U 4V 4W 8A ··· 8H order 1 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 4 4 8 ··· 8 size 1 1 1 1 2 2 8 1 1 1 1 2 ··· 2 4 ··· 4 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 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 C2 C2 C2 D4 D4 C4○D4 C4○D8 C8.C22 kernel C42.226D4 C42.12C4 C4×SD16 C4×Q16 Q8⋊D4 C22⋊Q16 Q8.D4 Q8.Q8 C22.D8 C23.47D4 C2×C4×Q8 C23.36C23 C42 C22×C4 Q8 C22 C4 # reps 1 1 2 2 1 1 2 2 1 1 1 1 2 2 8 8 2

Matrix representation of C42.226D4 in GL4(𝔽17) generated by

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

C42.226D4 in GAP, Magma, Sage, TeX

`C_4^2._{226}D_4`
`% in TeX`

`G:=Group("C4^2.226D4");`
`// GroupNames label`

`G:=SmallGroup(128,1840);`
`// by ID`

`G=gap.SmallGroup(128,1840);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,304,4037,1027,124]);`
`// 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,b*c=c*b,d*b*d=a^2*b,d*c*d=c^3>;`
`// generators/relations`

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