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## G = C42.2Q8order 128 = 27

### 2nd non-split extension by C42 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.2Q8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C8⋊C4 — C42.2Q8
 Lower central C1 — C2 — C22 — C42.2Q8
 Upper central C1 — C2×C4 — C2×C42 — C42.2Q8
 Jennings C1 — C22 — C22 — C2×C42 — C42.2Q8

Generators and relations for C42.2Q8
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2bc2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=ac3 >

Subgroups: 120 in 76 conjugacy classes, 44 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×2], C23, C42 [×2], C42 [×2], C2×C8 [×4], C2×C8 [×8], C22×C4, C22×C4 [×2], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×4], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C2×C8⋊C4, C42.12C4 [×2], C42.2Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C22.7C42, M4(2)⋊4C4 [×2], C42.2Q8

Smallest permutation representation of C42.2Q8
On 64 points
Generators in S64
```(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 33 27 15)(2 38 28 12)(3 35 29 9)(4 40 30 14)(5 37 31 11)(6 34 32 16)(7 39 25 13)(8 36 26 10)(17 64 53 44)(18 61 54 41)(19 58 55 46)(20 63 56 43)(21 60 49 48)(22 57 50 45)(23 62 51 42)(24 59 52 47)
(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 22 39 63 31 54 9 47)(2 19 36 64 32 51 14 48)(3 24 33 57 25 56 11 41)(4 21 38 58 26 53 16 42)(5 18 35 59 27 50 13 43)(6 23 40 60 28 55 10 44)(7 20 37 61 29 52 15 45)(8 17 34 62 30 49 12 46)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 8A ··· 8X order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 type + + + + - image C1 C2 C2 C4 C4 C8 D4 Q8 M4(2) M4(2)⋊4C4 kernel C42.2Q8 C2×C8⋊C4 C42.12C4 C22⋊C8 C22×C8 C2×C8 C42 C42 C2×C4 C2 # reps 1 1 2 8 4 16 3 1 4 4

Matrix representation of C42.2Q8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 4 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 8 0 0 0 0 9 0 0 0 0 0 0 0 0 8 0 0 0 0 9 0
,
 0 8 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,9,0,0,0,0,8,0,0,0,0,0,0,0,0,9,0,0,0,0,8,0],[0,9,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C42.2Q8 in GAP, Magma, Sage, TeX

`C_4^2._2Q_8`
`% in TeX`

`G:=Group("C4^2.2Q8");`
`// GroupNames label`

`G:=SmallGroup(128,13);`
`// by ID`

`G=gap.SmallGroup(128,13);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,570,248,102]);`
`// Polycyclic`

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

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