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## G = C24.9Q8order 128 = 27

### 8th non-split extension by C24 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 — C24.9Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C24.9Q8
 Lower central C1 — C2 — C2×C4 — C24.9Q8
 Upper central C1 — C2×C4 — C23×C4 — C24.9Q8
 Jennings C1 — C2 — C2 — C22×C4 — C24.9Q8

Generators and relations for C24.9Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=bde2, ab=ba, faf-1=ac=ca, eae-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 236 in 136 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×12], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×2], C23 [×4], C2×C8 [×8], C2×C8 [×6], M4(2) [×16], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C22⋊C8 [×4], C8.C4 [×4], C22×C8 [×2], C2×M4(2) [×8], C2×M4(2) [×4], C23×C4, C4.C42 [×2], C24.4C4 [×2], C2×C8.C4 [×2], C22×M4(2), C24.9Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, M4(2).C4 [×2], C24.9Q8

Smallest permutation representation of C24.9Q8
On 32 points
Generators in S32
```(2 6)(4 8)(9 25)(10 30)(11 27)(12 32)(13 29)(14 26)(15 31)(16 28)(17 21)(19 23)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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 30 18 12 5 26 22 16)(2 25 19 15 6 29 23 11)(3 28 20 10 7 32 24 14)(4 31 21 13 8 27 17 9)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8H 8I ··· 8P order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + - - image C1 C2 C2 C2 C2 C4 D4 D4 Q8 Q8 C4○D4 M4(2).C4 kernel C24.9Q8 C4.C42 C24.4C4 C2×C8.C4 C22×M4(2) C2×M4(2) C2×C8 C22×C4 C22×C4 C24 C2×C4 C2 # reps 1 2 2 2 1 8 4 2 1 1 4 4

Matrix representation of C24.9Q8 in GL6(𝔽17)

 16 0 0 0 0 0 2 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 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
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 13 0 0 0 0 0 8 4 0 0 0 0 0 0 0 1 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 16 0
,
 13 13 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 4 0 0 0 0 0 0 4 0 0

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

C24.9Q8 in GAP, Magma, Sage, TeX

`C_2^4._9Q_8`
`% in TeX`

`G:=Group("C2^4.9Q8");`
`// GroupNames label`

`G:=SmallGroup(128,543);`
`// by ID`

`G=gap.SmallGroup(128,543);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,2019,248,124]);`
`// Polycyclic`

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

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