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## G = C24.7Q8order 128 = 27

### 6th 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 — C4 — C24.7Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C24.7Q8
 Lower central C1 — C2 — C4 — C24.7Q8
 Upper central C1 — C2×C4 — C23×C4 — C24.7Q8
 Jennings C1 — C2 — C2 — C22×C4 — C24.7Q8

Generators and relations for C24.7Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=cde2, ab=ba, ac=ca, eae-1=ad=da, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=bde3 >

Subgroups: 276 in 186 conjugacy classes, 108 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×10], C8 [×12], C2×C4 [×2], C2×C4 [×26], C23, C23 [×6], C23 [×2], C2×C8 [×4], C2×C8 [×16], M4(2) [×8], M4(2) [×20], C22×C4 [×2], C22×C4 [×12], C24, C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×16], C2×M4(2) [×10], C23×C4, C4.C42 [×4], C22×M4(2), C22×M4(2) [×2], C24.7Q8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×C2.C42, M4(2).C4 [×2], C24.7Q8

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 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 2 2 2 4 type + + + + - - image C1 C2 C2 C4 D4 Q8 Q8 M4(2).C4 kernel C24.7Q8 C4.C42 C22×M4(2) C2×M4(2) C22×C4 C22×C4 C24 C2 # reps 1 4 3 24 6 1 1 4

Matrix representation of C24.7Q8 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 13 16 0 0 0 0 0 0 1 0 0 0 1 0 0 16
,
 16 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 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 4 0 16 0 0 0 1 0 0 16
,
 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 13 15 0 0 0 0 6 4 0 0 0 0 7 13 0 4 0 0 0 16 16 0
,
 5 3 0 0 0 0 14 12 0 0 0 0 0 0 4 0 15 0 0 0 0 0 4 1 0 0 10 0 13 0 0 0 11 13 16 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,4,1,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,6,7,0,0,0,15,4,13,16,0,0,0,0,0,16,0,0,0,0,4,0],[5,14,0,0,0,0,3,12,0,0,0,0,0,0,4,0,10,11,0,0,0,0,0,13,0,0,15,4,13,16,0,0,0,1,0,0] >;`

C24.7Q8 in GAP, Magma, Sage, TeX

`C_2^4._7Q_8`
`% in TeX`

`G:=Group("C2^4.7Q8");`
`// GroupNames label`

`G:=SmallGroup(128,470);`
`// by ID`

`G=gap.SmallGroup(128,470);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,2019,248,172,4037,124]);`
`// Polycyclic`

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

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