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## G = Q8.C42order 128 = 27

### 2nd non-split extension by Q8 of C42 acting via C42/C2×C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8.C42
G = < a,b,c,d | a4=c4=1, b2=d4=a2, bab-1=dad-1=a-1, dcd-1=ac=ca, cbc-1=a-1b, bd=db >

Subgroups: 236 in 150 conjugacy classes, 76 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×6], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×13], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], C2×C8 [×8], M4(2) [×4], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4×C8 [×3], C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×4], C2×C42, C42⋊C2, C22×C8 [×2], C22×C8, C2×M4(2) [×2], C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×C4○D4, C426C4, C4.C42, C2×C4×C8, C82M4(2), C23.24D4, C2×C4≀C2, C2×C8○D4, Q8.C42
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], C4×C22⋊C4, C8○D8 [×2], Q8.C42

Smallest permutation representation of Q8.C42
On 32 points
Generators in S32
```(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 18 13 22)(10 23 14 19)(11 20 15 24)(12 17 16 21)
(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 28 21 32)(18 29 22 25)(19 30 23 26)(20 31 24 27)
(1 7 5 3)(2 26)(4 28)(6 30)(8 32)(9 20)(10 16 14 12)(11 22)(13 24)(15 18)(17 23 21 19)(25 31 29 27)
(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)```

`G:=sub<Sym(32)| (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,18,13,22)(10,23,14,19)(11,20,15,24)(12,17,16,21), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27), (1,7,5,3)(2,26)(4,28)(6,30)(8,32)(9,20)(10,16,14,12)(11,22)(13,24)(15,18)(17,23,21,19)(25,31,29,27), (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)>;`

`G:=Group( (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,18,13,22)(10,23,14,19)(11,20,15,24)(12,17,16,21), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27), (1,7,5,3)(2,26)(4,28)(6,30)(8,32)(9,20)(10,16,14,12)(11,22)(13,24)(15,18)(17,23,21,19)(25,31,29,27), (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) );`

`G=PermutationGroup([(1,27,5,31),(2,32,6,28),(3,29,7,25),(4,26,8,30),(9,18,13,22),(10,23,14,19),(11,20,15,24),(12,17,16,21)], [(1,15,5,11),(2,16,6,12),(3,9,7,13),(4,10,8,14),(17,28,21,32),(18,29,22,25),(19,30,23,26),(20,31,24,27)], [(1,7,5,3),(2,26),(4,28),(6,30),(8,32),(9,20),(10,16,14,12),(11,22),(13,24),(15,18),(17,23,21,19),(25,31,29,27)], [(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)])`

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O ··· 4T 8A ··· 8H 8I ··· 8T 8U ··· 8AB order 1 2 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 2 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 C4○D4 C8○D8 kernel Q8.C42 C42⋊6C4 C4.C42 C2×C4×C8 C8○2M4(2) C23.24D4 C2×C4≀C2 C2×C8○D4 D4⋊C4 Q8⋊C4 C4≀C2 C8○D4 C2×C8 C2×C4 C23 C2 # reps 1 1 1 1 1 1 1 1 4 4 8 8 4 2 2 16

Matrix representation of Q8.C42 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 4 0 0 0 13 13
,
 0 1 0 0 1 0 0 0 0 0 8 16 0 0 14 9
,
 1 0 0 0 0 16 0 0 0 0 4 0 0 0 6 16
,
 0 1 0 0 1 0 0 0 0 0 1 2 0 0 6 16
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,13,0,0,0,13],[0,1,0,0,1,0,0,0,0,0,8,14,0,0,16,9],[1,0,0,0,0,16,0,0,0,0,4,6,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,1,6,0,0,2,16] >;`

Q8.C42 in GAP, Magma, Sage, TeX

`Q_8.C_4^2`
`% in TeX`

`G:=Group("Q8.C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,496);`
`// by ID`

`G=gap.SmallGroup(128,496);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,2019,248,172,4037]);`
`// Polycyclic`

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

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