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G = C24.19Q8order 128 = 27

3rd non-split extension by C24 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 — C24.19Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C24.19Q8
 Lower central C1 — C2 — C2×C4 — C24.19Q8
 Upper central C1 — C2×C4 — C23×C4 — C24.19Q8
 Jennings C1 — C2 — C2 — C22×C4 — C24.19Q8

Generators and relations for C24.19Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=bde2, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 236 in 146 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×14], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×2], C23 [×6], C2×C8 [×2], C2×C8 [×6], C2×C8 [×14], M4(2) [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C22⋊C8 [×4], C8.C4 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×M4(2) [×4], C23×C4, C4.C42 [×2], C24.4C4 [×2], C2×C8.C4 [×2], C23×C8, C24.19Q8
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], C8.C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, C2×C8.C4 [×2], C24.19Q8

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 8A ··· 8P 8Q ··· 8X order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 ··· 2 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + - - image C1 C2 C2 C2 C2 C4 D4 D4 Q8 Q8 C4○D4 C8.C4 kernel C24.19Q8 C4.C42 C24.4C4 C2×C8.C4 C23×C8 C22×C8 C2×C8 C22×C4 C22×C4 C24 C2×C4 C22 # reps 1 2 2 2 1 8 4 2 1 1 4 16

Matrix representation of C24.19Q8 in GL4(𝔽17) generated by

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

C24.19Q8 in GAP, Magma, Sage, TeX

`C_2^4._{19}Q_8`
`% in TeX`

`G:=Group("C2^4.19Q8");`
`// GroupNames label`

`G:=SmallGroup(128,542);`
`// by ID`

`G=gap.SmallGroup(128,542);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,124]);`
`// 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,a*d=d*a,a*e=e*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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