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## G = C42.378D4order 128 = 27

### 11st non-split extension by C42 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.378D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.378D4
 Lower central C1 — C22 — C42.378D4
 Upper central C1 — C22×C4 — C42.378D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.378D4

Generators and relations for C42.378D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b-1, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 316 in 206 conjugacy classes, 96 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×8], C22 [×12], C8 [×8], C2×C4 [×16], C2×C4 [×34], C23, C23 [×6], C23 [×4], C42 [×4], C42 [×6], C2×C8 [×8], C2×C8 [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×16], C24, C8⋊C4 [×4], C22⋊C8 [×8], C2×C42 [×2], C2×C42 [×2], C2×C42 [×4], C22×C8 [×4], C23×C4, C23×C4 [×2], C22.7C42 [×2], C2×C8⋊C4 [×2], C2×C22⋊C8 [×2], C22×C42, C42.378D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], M4(2) [×8], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C8⋊C4 [×4], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C2×M4(2) [×4], C4×C22⋊C4, C2×C8⋊C4, C4×M4(2), C24.4C4 [×2], C42.6C4 [×2], C42.378D4

Smallest permutation representation of C42.378D4
On 64 points
Generators in S64
```(1 57 51 16)(2 62 52 13)(3 59 53 10)(4 64 54 15)(5 61 55 12)(6 58 56 9)(7 63 49 14)(8 60 50 11)(17 37 27 44)(18 34 28 41)(19 39 29 46)(20 36 30 43)(21 33 31 48)(22 38 32 45)(23 35 25 42)(24 40 26 47)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 47 13 43)(10 48 14 44)(11 41 15 45)(12 42 16 46)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 63 37 59)(34 64 38 60)(35 57 39 61)(36 58 40 62)
(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 36 25 13 5 40 29 9)(2 57 26 42 6 61 30 46)(3 38 27 15 7 34 31 11)(4 59 28 44 8 63 32 48)(10 18 37 50 14 22 33 54)(12 20 39 52 16 24 35 56)(17 64 49 41 21 60 53 45)(19 58 51 43 23 62 55 47)```

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 M4(2) C4○D4 M4(2) kernel C42.378D4 C22.7C42 C2×C8⋊C4 C2×C22⋊C8 C22×C42 C22⋊C8 C2×C42 C23×C4 C42 C2×C4 C2×C4 C23 # reps 1 2 2 2 1 16 4 4 4 8 4 8

Matrix representation of C42.378D4 in GL5(𝔽17)

 4 0 0 0 0 0 13 0 0 0 0 0 4 0 0 0 0 0 13 0 0 0 0 0 13
,
 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 0 0 15 16 0 0 0 5 2
,
 13 0 0 0 0 0 0 16 0 0 0 13 0 0 0 0 0 0 9 13 0 0 0 12 8

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

C42.378D4 in GAP, Magma, Sage, TeX

`C_4^2._{378}D_4`
`% in TeX`

`G:=Group("C4^2.378D4");`
`// GroupNames label`

`G:=SmallGroup(128,481);`
`// by ID`

`G=gap.SmallGroup(128,481);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,124]);`
`// Polycyclic`

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

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