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## G = C42.374D4order 128 = 27

### 7th 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 — C42.374D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C4×C4○D4 — C42.374D4
 Lower central C1 — C2 — C2×C4 — C42.374D4
 Upper central C1 — C2×C4 — C2×C42 — C42.374D4
 Jennings C1 — C22 — C22 — C42 — C42.374D4

Generators and relations for C42.374D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 228 in 123 conjugacy classes, 50 normal (36 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C2×C4○D4, D4⋊C8 [×2], Q8⋊C8 [×2], C42.12C4, C42.6C4, C4×C4○D4, C42.374D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C2×M4(2) [×2], C4○D8 [×2], C24.4C4, C23.24D4, C42⋊C22, C42.374D4

Smallest permutation representation of C42.374D4
On 64 points
Generators in S64
```(1 42 37 49)(2 54 38 47)(3 44 39 51)(4 56 40 41)(5 46 33 53)(6 50 34 43)(7 48 35 55)(8 52 36 45)(9 24 31 62)(10 59 32 21)(11 18 25 64)(12 61 26 23)(13 20 27 58)(14 63 28 17)(15 22 29 60)(16 57 30 19)
(1 61 33 19)(2 62 34 20)(3 63 35 21)(4 64 36 22)(5 57 37 23)(6 58 38 24)(7 59 39 17)(8 60 40 18)(9 43 27 54)(10 44 28 55)(11 45 29 56)(12 46 30 49)(13 47 31 50)(14 48 32 51)(15 41 25 52)(16 42 26 53)
(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 61 8 33 60 19 40)(2 7 62 59 34 39 20 17)(3 58 63 38 35 24 21 6)(4 37 64 23 36 5 22 57)(9 14 43 48 27 32 54 51)(10 47 44 31 28 50 55 13)(11 30 45 49 29 12 56 46)(15 26 41 53 25 16 52 42)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E ··· 4L 4M ··· 4S 8A ··· 8H 8I 8J 8K 8L order 1 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 8 8 8 8 size 1 1 1 1 4 4 4 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 M4(2) M4(2) C4○D8 C42⋊C22 kernel C42.374D4 D4⋊C8 Q8⋊C8 C42.12C4 C42.6C4 C4×C4○D4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C42 C22×C4 D4 Q8 C4 C2 # reps 1 2 2 1 1 1 2 2 2 2 2 2 4 4 8 2

Matrix representation of C42.374D4 in GL4(𝔽17) generated by

 13 0 0 0 0 13 0 0 0 0 16 0 0 0 16 1
,
 1 0 0 0 0 1 0 0 0 0 13 0 0 0 0 13
,
 3 14 0 0 3 3 0 0 0 0 1 15 0 0 7 16
,
 3 14 0 0 14 14 0 0 0 0 16 2 0 0 6 1
`G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,16,16,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[3,3,0,0,14,3,0,0,0,0,1,7,0,0,15,16],[3,14,0,0,14,14,0,0,0,0,16,6,0,0,2,1] >;`

C42.374D4 in GAP, Magma, Sage, TeX

`C_4^2._{374}D_4`
`% in TeX`

`G:=Group("C4^2.374D4");`
`// GroupNames label`

`G:=SmallGroup(128,220);`
`// by ID`

`G=gap.SmallGroup(128,220);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,570,136,172]);`
`// 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=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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