Copied to
clipboard

## G = C24.11Q8order 128 = 27

### 10th 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 — C22×C4 — C24.11Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C24.4C4 — C24.11Q8
 Lower central C1 — C2 — C22×C4 — C24.11Q8
 Upper central C1 — C4 — C22×C4 — C24.11Q8
 Jennings C1 — C2 — C2 — C22×C4 — C24.11Q8

Generators and relations for C24.11Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=be2, faf-1=ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=be3 >

Subgroups: 216 in 111 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4, C22 [×3], C22 [×11], C8 [×9], C2×C4 [×6], C2×C4 [×7], C23, C23 [×5], C2×C8 [×6], C2×C8 [×3], M4(2) [×12], C22×C4 [×2], C22×C4 [×6], C24, C22⋊C8 [×6], C8.C4 [×6], C2×M4(2) [×6], C23×C4, C4.10C42, C24.4C4 [×3], M4(2).C4 [×3], C24.11Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C23.4Q8, C24.11Q8

Character table of C24.11Q8

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 8 8 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 -1 -1 1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 1 -1 linear of order 2 ρ8 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ9 2 2 2 -2 -2 0 0 2 2 2 -2 -2 0 0 0 0 0 0 2 0 0 0 0 0 0 -2 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 0 0 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 2 0 -2 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 2 0 0 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 -2 0 2 0 0 orthogonal lifted from D4 ρ12 2 2 2 -2 -2 0 0 2 2 2 -2 -2 0 0 0 0 0 0 -2 0 0 0 0 0 0 2 orthogonal lifted from D4 ρ13 2 2 -2 2 -2 0 0 2 2 -2 -2 2 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 2 -2 0 0 2 2 -2 -2 2 0 0 2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 2 2 2 -2 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 2 2 2 2 -2 -2 -2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 2 -2 2 -2 0 0 -2 -2 2 2 -2 0 0 0 0 2i 0 0 0 0 0 0 0 -2i 0 complex lifted from C4○D4 ρ18 2 2 -2 -2 2 0 0 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 -2i 0 2i 0 0 0 complex lifted from C4○D4 ρ19 2 2 -2 2 -2 0 0 -2 -2 2 2 -2 0 0 0 0 -2i 0 0 0 0 0 0 0 2i 0 complex lifted from C4○D4 ρ20 2 2 -2 -2 2 0 0 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 2i 0 -2i 0 0 0 complex lifted from C4○D4 ρ21 2 2 2 -2 -2 0 0 -2 -2 -2 2 2 0 0 0 -2i 0 0 0 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 -2 -2 0 0 -2 -2 -2 2 2 0 0 0 2i 0 0 0 -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 0 0 0 2 -2 4i -4i 0 0 0 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 2 -2 -4i 4i 0 0 0 -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 -2 2 4i -4i 0 0 0 -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 -2 2 -4i 4i 0 0 0 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C24.11Q8
On 16 points - transitive group 16T403
Generators in S16
```(10 14)(12 16)
(2 6)(4 8)(10 14)(12 16)
(9 13)(10 14)(11 15)(12 16)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 15 3 9 5 11 7 13)(2 14 8 12 6 10 4 16)```

`G:=sub<Sym(16)| (10,14)(12,16), (2,6)(4,8)(10,14)(12,16), (9,13)(10,14)(11,15)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15,3,9,5,11,7,13)(2,14,8,12,6,10,4,16)>;`

`G:=Group( (10,14)(12,16), (2,6)(4,8)(10,14)(12,16), (9,13)(10,14)(11,15)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15,3,9,5,11,7,13)(2,14,8,12,6,10,4,16) );`

`G=PermutationGroup([(10,14),(12,16)], [(2,6),(4,8),(10,14),(12,16)], [(9,13),(10,14),(11,15),(12,16)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,15,3,9,5,11,7,13),(2,14,8,12,6,10,4,16)])`

`G:=TransitiveGroup(16,403);`

Matrix representation of C24.11Q8 in GL4(𝔽5) generated by

 4 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 1 0 0 0 0 4 0 0 0 0 1
,
 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 0 0 2 0 0 0 0 3 4 0 0 0 0 1 0 0
,
 0 2 0 0 1 0 0 0 0 0 0 4 0 0 2 0
`G:=sub<GL(4,GF(5))| [4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,0,4,0,0,0,0,1,2,0,0,0,0,3,0,0],[0,1,0,0,2,0,0,0,0,0,0,2,0,0,4,0] >;`

C24.11Q8 in GAP, Magma, Sage, TeX

`C_2^4._{11}Q_8`
`% in TeX`

`G:=Group("C2^4.11Q8");`
`// GroupNames label`

`G:=SmallGroup(128,823);`
`// by ID`

`G=gap.SmallGroup(128,823);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,1018,248,1411,4037]);`
`// Polycyclic`

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

Export

׿
×
𝔽