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## G = C42.399D4order 128 = 27

### 32nd 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 — C42.399D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×Q8 — C42.399D4
 Lower central C1 — C2 — C4 — C42.399D4
 Upper central C1 — C2×C4 — C2×C42 — C42.399D4
 Jennings C1 — C22 — C22 — C42 — C42.399D4

Generators and relations for C42.399D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 204 in 124 conjugacy classes, 62 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×13], Q8 [×4], Q8 [×6], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×3], C4×C8 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C22×Q8, Q8⋊C8 [×4], C42.12C4 [×2], C2×C4×Q8, C42.399D4
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 [×2], C2×C22⋊C8, C23.38D4, C42⋊C22, C42.399D4

Smallest permutation representation of C42.399D4
On 64 points
Generators in S64
```(1 34 59 51)(2 52 60 35)(3 36 61 53)(4 54 62 37)(5 38 63 55)(6 56 64 39)(7 40 57 49)(8 50 58 33)(9 41 30 20)(10 21 31 42)(11 43 32 22)(12 23 25 44)(13 45 26 24)(14 17 27 46)(15 47 28 18)(16 19 29 48)
(1 36 63 49)(2 37 64 50)(3 38 57 51)(4 39 58 52)(5 40 59 53)(6 33 60 54)(7 34 61 55)(8 35 62 56)(9 43 26 18)(10 44 27 19)(11 45 28 20)(12 46 29 21)(13 47 30 22)(14 48 31 23)(15 41 32 24)(16 42 25 17)
(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 19 36 10 63 44 49 27)(2 30 37 22 64 13 50 47)(3 42 38 25 57 17 51 16)(4 11 39 45 58 28 52 20)(5 23 40 14 59 48 53 31)(6 26 33 18 60 9 54 43)(7 46 34 29 61 21 55 12)(8 15 35 41 62 32 56 24)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + - image C1 C2 C2 C2 C4 C4 C4 C8 D4 D4 M4(2) C8.C22 C42⋊C22 kernel C42.399D4 Q8⋊C8 C42.12C4 C2×C4×Q8 C2×C4⋊C4 C4×Q8 C22×Q8 C2×Q8 C42 C22×C4 C2×C4 C4 C2 # reps 1 4 2 1 2 4 2 16 2 2 4 2 2

Matrix representation of C42.399D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 2 0 0 0 0 16 1 0 0 0 0 0 0 16 2 0 0 0 0 16 1
,
 13 0 0 0 0 0 0 13 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 9 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 5 0 0 0 10 7 0 0 0 0 5 7 0 0
,
 9 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 16 0 0 0 0 0 0 16 0 0

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

C42.399D4 in GAP, Magma, Sage, TeX

`C_4^2._{399}D_4`
`% in TeX`

`G:=Group("C4^2.399D4");`
`// GroupNames label`

`G:=SmallGroup(128,211);`
`// by ID`

`G=gap.SmallGroup(128,211);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,387,184,1123,570,136,172]);`
`// 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^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;`
`// generators/relations`

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