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## G = C42.443D4order 128 = 27

### 76th 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 — C2×C4 — C42.443D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C4×C4○D4 — C42.443D4
 Lower central C1 — C2 — C2×C4 — C42.443D4
 Upper central C1 — C2×C4 — C2×C42 — C42.443D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.443D4

Generators and relations for C42.443D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 468 in 248 conjugacy classes, 102 normal (44 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×12], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×20], D4 [×2], D4 [×17], Q8 [×2], Q8 [×5], C23, C23 [×3], C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4○D4 [×10], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×4], C4×Q8 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8 [×2], C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C4○D8 [×8], C2×C4○D4, C2×C4○D4 [×2], C23.24D4 [×2], C2×C4⋊C8, C4⋊D8, C4⋊SD16, D4.D4, C42Q16, D4.2D4 [×2], Q8.D4 [×2], C4×C4○D4, C22.26C24, C2×C4○D8 [×2], C42.443D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C4○D8, D8⋊C22, C42.443D4

Smallest permutation representation of C42.443D4
On 64 points
Generators in S64
```(1 50 29 61)(2 62 30 51)(3 52 31 63)(4 64 32 53)(5 54 25 57)(6 58 26 55)(7 56 27 59)(8 60 28 49)(9 21 38 43)(10 44 39 22)(11 23 40 45)(12 46 33 24)(13 17 34 47)(14 48 35 18)(15 19 36 41)(16 42 37 20)
(1 14 5 10)(2 15 6 11)(3 16 7 12)(4 9 8 13)(17 64 21 60)(18 57 22 61)(19 58 23 62)(20 59 24 63)(25 39 29 35)(26 40 30 36)(27 33 31 37)(28 34 32 38)(41 55 45 51)(42 56 46 52)(43 49 47 53)(44 50 48 54)
(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 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 41)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(26 32)(27 31)(28 30)(33 37)(34 36)(38 40)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4J 4K ··· 4R 4S 4T 8A ··· 8H order 1 2 2 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 4 4 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 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 C4○D8 D8⋊C22 kernel C42.443D4 C23.24D4 C2×C4⋊C8 C4⋊D8 C4⋊SD16 D4.D4 C4⋊2Q16 D4.2D4 Q8.D4 C4×C4○D4 C22.26C24 C2×C4○D8 C42 C22×C4 C4○D4 C2×C4 C4 C2 # reps 1 2 1 1 1 1 1 2 2 1 1 2 2 2 4 4 8 2

Matrix representation of C42.443D4 in GL4(𝔽17) generated by

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

C42.443D4 in GAP, Magma, Sage, TeX

`C_4^2._{443}D_4`
`% in TeX`

`G:=Group("C4^2.443D4");`
`// GroupNames label`

`G:=SmallGroup(128,1767);`
`// by ID`

`G=gap.SmallGroup(128,1767);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,248,4037,1027,124]);`
`// Polycyclic`

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

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