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## G = C42.379D4order 128 = 27

### 12nd 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 — C42.379D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C42.379D4
 Lower central C1 — C22 — C42.379D4
 Upper central C1 — C22×C4 — C42.379D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.379D4

Generators and relations for C42.379D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 268 in 168 conjugacy classes, 84 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×8], C2×C4 [×2], C2×C4 [×10], C2×C4 [×20], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C22×C8 [×4], C23×C4, C22.7C42 [×2], C2×C4×C8, C2×C8⋊C4, C2×C22⋊C8 [×2], C2×C42⋊C2, C42.379D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C8○D4 [×4], C4×C22⋊C4, C82M4(2) [×2], (C22×C8)⋊C2 [×2], C42.7C22 [×2], C42.379D4

Smallest permutation representation of C42.379D4
On 64 points
Generators in S64
```(1 64 10 19)(2 61 11 24)(3 58 12 21)(4 63 13 18)(5 60 14 23)(6 57 15 20)(7 62 16 17)(8 59 9 22)(25 34 51 44)(26 39 52 41)(27 36 53 46)(28 33 54 43)(29 38 55 48)(30 35 56 45)(31 40 49 42)(32 37 50 47)
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(9 53 13 49)(10 54 14 50)(11 55 15 51)(12 56 16 52)(17 41 21 45)(18 42 22 46)(19 43 23 47)(20 44 24 48)(33 60 37 64)(34 61 38 57)(35 62 39 58)(36 63 40 59)
(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 47 10 37)(2 63 11 18)(3 41 12 39)(4 57 13 20)(5 43 14 33)(6 59 15 22)(7 45 16 35)(8 61 9 24)(17 52 62 26)(19 54 64 28)(21 56 58 30)(23 50 60 32)(25 36 51 46)(27 38 53 48)(29 40 55 42)(31 34 49 44)```

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4P 4Q ··· 4V 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 C4○D4 C8○D4 kernel C42.379D4 C22.7C42 C2×C4×C8 C2×C8⋊C4 C2×C22⋊C8 C2×C42⋊C2 C22⋊C8 C2×C22⋊C4 C2×C4⋊C4 C42 C2×C4 C22 # reps 1 2 1 1 2 1 16 4 4 4 4 16

Matrix representation of C42.379D4 in GL5(𝔽17)

 4 0 0 0 0 0 16 0 0 0 0 15 1 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 13
,
 13 0 0 0 0 0 2 15 0 0 0 0 15 0 0 0 0 0 0 2 0 0 0 15 0
,
 4 0 0 0 0 0 13 4 0 0 0 9 4 0 0 0 0 0 0 13 0 0 0 4 0

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

C42.379D4 in GAP, Magma, Sage, TeX

`C_4^2._{379}D_4`
`% in TeX`

`G:=Group("C4^2.379D4");`
`// GroupNames label`

`G:=SmallGroup(128,482);`
`// by ID`

`G=gap.SmallGroup(128,482);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,100,124]);`
`// Polycyclic`

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

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