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## G = C42.397D4order 128 = 27

### 30th non-split extension by C42 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C42.397D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C4×C4○D4 — C42.397D4
 Lower central C1 — C2 — C4 — C42.397D4
 Upper central C1 — C2×C4 — C2×C42 — C42.397D4
 Jennings C1 — C22 — C22 — C42 — C42.397D4

Generators and relations for C42.397D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 236 in 132 conjugacy classes, 62 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×14], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×6], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4×C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C22×C8, C2×C4○D4, D4⋊C8 [×2], Q8⋊C8 [×2], C2×C4⋊C8, C42.12C4, C4×C4○D4, C42.397D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C8⋊C22, C8.C22, C2×C22⋊C8, C23.36D4, C42⋊C22, C42.397D4

Smallest permutation representation of C42.397D4
On 64 points
Generators in S64
```(1 13 49 27)(2 28 50 14)(3 15 51 29)(4 30 52 16)(5 9 53 31)(6 32 54 10)(7 11 55 25)(8 26 56 12)(17 47 63 39)(18 40 64 48)(19 41 57 33)(20 34 58 42)(21 43 59 35)(22 36 60 44)(23 45 61 37)(24 38 62 46)
(1 19 53 61)(2 20 54 62)(3 21 55 63)(4 22 56 64)(5 23 49 57)(6 24 50 58)(7 17 51 59)(8 18 52 60)(9 45 27 33)(10 46 28 34)(11 47 29 35)(12 48 30 36)(13 41 31 37)(14 42 32 38)(15 43 25 39)(16 44 26 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 18 19 52 53 60 61 8)(2 51 20 59 54 7 62 17)(3 58 21 6 55 24 63 50)(4 5 22 23 56 49 64 57)(9 44 45 26 27 40 33 16)(10 25 46 39 28 15 34 43)(11 38 47 14 29 42 35 32)(12 13 48 41 30 31 36 37)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O ··· 4T 8A ··· 8P order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8 D4 D4 M4(2) C8⋊C22 C8.C22 C42⋊C22 kernel C42.397D4 D4⋊C8 Q8⋊C8 C2×C4⋊C8 C42.12C4 C4×C4○D4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C4○D4 C42 C22×C4 C2×C4 C4 C4 C2 # reps 1 2 2 1 1 1 2 2 2 2 16 2 2 4 1 1 2

Matrix representation of C42.397D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 16 15 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1 0 16 16
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 1 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 4 0 0 0 0 4 4 0 0 4 4 13 13 0 0 0 13 4 4
,
 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 13 0 4 0 0 0 0 13 13 0 0 13 13 4 4 0 0 4 0 13 13

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,0,0,16,16,0,16,0,0,15,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,13,0,4,13,0,0,0,4,13,4,0,0,4,4,13,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,4,0,0,13,0,13,0,0,0,0,13,4,13,0,0,4,13,4,13] >;`

C42.397D4 in GAP, Magma, Sage, TeX

`C_4^2._{397}D_4`
`% in TeX`

`G:=Group("C4^2.397D4");`
`// GroupNames label`

`G:=SmallGroup(128,209);`
`// by ID`

`G=gap.SmallGroup(128,209);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,1123,570,136,172]);`
`// 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=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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