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## G = C42.6Q8order 128 = 27

### 6th 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 — C2×C4 — C42.6Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — C42.6Q8
 Lower central C1 — C2 — C2×C4 — C42.6Q8
 Upper central C1 — C22 — C2×C42 — C42.6Q8
 Jennings C1 — C22 — C22 — C2×C42 — C42.6Q8

Generators and relations for C42.6Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, dad-1=ab2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=ac3 >

Subgroups: 152 in 76 conjugacy classes, 32 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×6], C2×C4 [×10], C23, C42 [×4], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2.C42 [×2], C2.C42 [×2], C4×C8 [×2], C8⋊C4 [×2], C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×M4(2), C425C4, C4×M4(2), C42.6C4, C42.6Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4≀C2 [×2], C4.9C42, C426C4, M4(2)⋊4C4, C42.6Q8

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 C4≀C2 C4.9C42 M4(2)⋊4C4 kernel C42.6Q8 C42⋊5C4 C4×M4(2) C42.6C4 C2.C42 C4×C8 C8⋊C4 C42 C42 C22×C4 C22 C2 C2 # reps 1 1 1 1 4 4 4 1 1 2 8 2 2

Matrix representation of C42.6Q8 in GL6(𝔽17)

 13 0 0 0 0 0 0 13 0 0 0 0 0 0 1 9 0 0 0 0 0 16 0 0 0 0 0 3 1 4 0 0 0 0 0 16
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 4 2 0 0 0 0 0 13 0 0 0 0 14 12 13 1 0 0 0 0 0 4
,
 10 11 0 0 0 0 11 10 0 0 0 0 0 0 12 0 15 0 0 0 0 0 0 1 0 0 2 16 5 0 0 0 0 13 0 0
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 13 16 0 0 0 0 16 0 4 0 0 0 12 0 15 13

`G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,9,16,3,0,0,0,0,0,1,0,0,0,0,0,4,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,14,0,0,0,2,13,12,0,0,0,0,0,13,0,0,0,0,0,1,4],[10,11,0,0,0,0,11,10,0,0,0,0,0,0,12,0,2,0,0,0,0,0,16,13,0,0,15,0,5,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,13,16,12,0,0,0,16,0,0,0,0,0,0,4,15,0,0,0,0,0,13] >;`

C42.6Q8 in GAP, Magma, Sage, TeX

`C_4^2._6Q_8`
`% in TeX`

`G:=Group("C4^2.6Q8");`
`// GroupNames label`

`G:=SmallGroup(128,20);`
`// by ID`

`G=gap.SmallGroup(128,20);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,3924,172]);`
`// Polycyclic`

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

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