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## G = C42.3Q8order 128 = 27

### 3rd 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 — C42.3Q8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4⋊C8 — C42.3Q8
 Lower central C1 — C2 — C22 — C42.3Q8
 Upper central C1 — C2×C4 — C2×C42 — C42.3Q8
 Jennings C1 — C22 — C22 — C2×C42 — C42.3Q8

Generators and relations for C42.3Q8
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2bc2, ab=ba, ac=ca, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=a-1b2c3 >

Subgroups: 128 in 80 conjugacy classes, 46 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, M4(2), M4(2), C22×C4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C22×C8, C2×M4(2), C4×M4(2), C2×C4⋊C8, C42.12C4, C42.3Q8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4.D4, C4.10D4, C4⋊C8, C22.7C42, C22.C42, M4(2)⋊4C4, C42.3Q8

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

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

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

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

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 1 1 2 2 2 4 4 4 type + + + + + - + - image C1 C2 C2 C2 C4 C4 C4 C8 D4 Q8 M4(2) C4.D4 C4.10D4 M4(2)⋊4C4 kernel C42.3Q8 C4×M4(2) C2×C4⋊C8 C42.12C4 C22⋊C8 C22×C8 C2×M4(2) M4(2) C42 C42 C2×C4 C4 C4 C2 # reps 1 1 1 1 4 4 4 16 3 1 4 1 1 2

Matrix representation of C42.3Q8 in GL6(𝔽17)

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

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

C42.3Q8 in GAP, Magma, Sage, TeX

`C_4^2._3Q_8`
`% in TeX`

`G:=Group("C4^2.3Q8");`
`// GroupNames label`

`G:=SmallGroup(128,15);`
`// by ID`

`G=gap.SmallGroup(128,15);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,570,248,242]);`
`// 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*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^-1*b^2*c^3>;`
`// generators/relations`

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