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## G = C42.20D4order 128 = 27

### 2nd non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C42.20D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C8⋊C4 — C42.20D4
 Lower central C1 — C4 — C42.20D4
 Upper central C1 — C2×C4 — C42.20D4
 Jennings C1 — C22 — C22 — C2×C42 — C42.20D4

Generators and relations for C42.20D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1bc3 >

Subgroups: 120 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×2], C23, C42 [×4], C2×C8 [×4], C2×C8 [×8], C22×C4 [×3], C4×C8 [×4], C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C2×C8⋊C4, C42.12C4 [×2], C42.20D4
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C22.7C42, C4.9C42, C4.10C42, C42.20D4

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 8A ··· 8X order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + - image C1 C2 C2 C4 C4 C8 D4 D4 Q8 M4(2) C4.9C42 C4.10C42 kernel C42.20D4 C2×C8⋊C4 C42.12C4 C4×C8 C22×C8 C2×C8 C42 C22×C4 C22×C4 C2×C4 C2 C2 # reps 1 1 2 8 4 16 2 1 1 4 2 2

Matrix representation of C42.20D4 in GL6(𝔽17)

 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 1 14 0 0 0 1 0 3 0 0 0 0 0 1 0 0 0 0 0 16 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 3 16 0 0 0 4 14 1 0 0 0 0 13 0 0 0 0 0 0 13
,
 0 15 0 0 0 0 15 0 0 0 0 0 0 0 10 15 13 11 0 0 7 2 8 14 0 0 13 4 14 1 0 0 0 13 10 8
,
 8 0 0 0 0 0 0 9 0 0 0 0 0 0 16 12 2 0 0 0 1 5 7 0 0 0 2 2 0 0 0 0 15 0 12 13

`G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,14,3,1,16,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,3,14,13,0,0,0,16,1,0,13],[0,15,0,0,0,0,15,0,0,0,0,0,0,0,10,7,13,0,0,0,15,2,4,13,0,0,13,8,14,10,0,0,11,14,1,8],[8,0,0,0,0,0,0,9,0,0,0,0,0,0,16,1,2,15,0,0,12,5,2,0,0,0,2,7,0,12,0,0,0,0,0,13] >;`

C42.20D4 in GAP, Magma, Sage, TeX

`C_4^2._{20}D_4`
`% in TeX`

`G:=Group("C4^2.20D4");`
`// GroupNames label`

`G:=SmallGroup(128,7);`
`// by ID`

`G=gap.SmallGroup(128,7);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,184,570,248,1684,102]);`
`// Polycyclic`

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

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