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## G = C42.45D4order 128 = 27

### 27th 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 — C42.45D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C22.26C24 — C42.45D4
 Lower central C1 — C2 — C2×C4 — C42.45D4
 Upper central C1 — C2×C4 — C2×C42 — C42.45D4
 Jennings C1 — C22 — C22 — C42 — C42.45D4

Generators and relations for C42.45D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 284 in 130 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×5], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×9], D4 [×12], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×6], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4×C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C22×C8, C2×C4○D4, D4⋊C8 [×4], C2×C4⋊C8, C42.12C4, C22.26C24, C42.45D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C8○D4 [×2], C2×D8, C2×SD16, (C22×C8)⋊C2, C2×D4⋊C4, C42⋊C22, C42.45D4

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 4K 4L 4M 4N 8A ··· 8P order 1 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 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 2 2 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D8 SD16 C8○D4 C42⋊C22 kernel C42.45D4 D4⋊C8 C2×C4⋊C8 C42.12C4 C22.26C24 C4⋊D4 C4⋊1D4 C4⋊Q8 C42 C22×C4 C2×C4 C2×C4 C4 C2 # reps 1 4 1 1 1 4 2 2 2 2 4 4 8 2

Matrix representation of C42.45D4 in GL4(𝔽17) generated by

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

C42.45D4 in GAP, Magma, Sage, TeX

`C_4^2._{45}D_4`
`% in TeX`

`G:=Group("C4^2.45D4");`
`// GroupNames label`

`G:=SmallGroup(128,212);`
`// by ID`

`G=gap.SmallGroup(128,212);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,1123,570,136,172]);`
`// Polycyclic`

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

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