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## G = C42.62Q8order 128 = 27

### 22nd non-split extension by C42 of Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.62Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C42.62Q8
 Lower central C1 — C2 — C2×C4 — C42.62Q8
 Upper central C1 — C2×C4 — C22×C8 — C42.62Q8
 Jennings C1 — C2 — C2 — C22×C4 — C42.62Q8

Generators and relations for C42.62Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b2c2, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=bc3 >

Subgroups: 164 in 104 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×2], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×10], C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4, C4×C8 [×2], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C8.C4 [×2], C2×C42, C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×2], C426C4 [×2], C2×C4×C8, C42.6C22 [×2], C23.25D4, C2×C8.C4, C42.62Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, C8○D8 [×2], C42.62Q8

Smallest permutation representation of C42.62Q8
On 32 points
Generators in S32
```(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 11 13 15)(10 12 14 16)(25 27 29 31)(26 28 30 32)
(1 20 5 24)(2 21 6 17)(3 22 7 18)(4 23 8 19)(9 30 13 26)(10 31 14 27)(11 32 15 28)(12 25 16 29)
(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)
(1 28 7 26 5 32 3 30)(2 14 8 12 6 10 4 16)(9 24 15 22 13 20 11 18)(17 31 23 29 21 27 19 25)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 8A ··· 8P 8Q 8R 8S 8T order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 8 8 8 8 2 ··· 2 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 C4 Q8 D4 Q8 C4○D4 C4○D4 C8○D8 kernel C42.62Q8 C42⋊6C4 C2×C4×C8 C42.6C22 C23.25D4 C2×C8.C4 C4.Q8 C2.D8 C42 C2×C8 C2×C8 C2×C4 C23 C2 # reps 1 2 1 2 1 1 4 4 2 4 2 2 2 16

Matrix representation of C42.62Q8 in GL4(𝔽17) generated by

 16 6 0 0 0 4 0 0 0 0 1 0 0 0 11 13
,
 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 2 2 0 0 0 15 0 0 0 0 8 0 0 0 0 8
,
 13 6 0 0 8 4 0 0 0 0 1 15 0 0 7 16
`G:=sub<GL(4,GF(17))| [16,0,0,0,6,4,0,0,0,0,1,11,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,2,15,0,0,0,0,8,0,0,0,0,8],[13,8,0,0,6,4,0,0,0,0,1,7,0,0,15,16] >;`

C42.62Q8 in GAP, Magma, Sage, TeX

`C_4^2._{62}Q_8`
`% in TeX`

`G:=Group("C4^2.62Q8");`
`// GroupNames label`

`G:=SmallGroup(128,677);`
`// by ID`

`G=gap.SmallGroup(128,677);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2804,172,124]);`
`// Polycyclic`

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

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