Copied to
clipboard

## G = C4⋊C4.97D4order 128 = 27

### 52nd non-split extension by C4⋊C4 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C4⋊C4.97D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — C2×C4.D4 — C4⋊C4.97D4
 Lower central C1 — C2 — C22×C4 — C4⋊C4.97D4
 Upper central C1 — C22 — C22×C4 — C4⋊C4.97D4
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊C4.97D4

Generators and relations for C4⋊C4.97D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a-1b, dcd-1=c3 >

Subgroups: 328 in 138 conjugacy classes, 42 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×10], C8 [×5], C2×C4 [×6], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C23 [×6], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×6], SD16 [×8], C22×C4, C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×5], C24, C4.D4 [×2], C4.10D4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8 [×2], C42⋊C2, C22×C8, C2×M4(2) [×3], C2×SD16 [×6], C22×D4, C22×Q8, C4.C42, C2×C4.D4, C2×C4.10D4, C23.37D4, C23.38D4, C42.6C22, C22×SD16, C4⋊C4.97D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, D4.3D4 [×2], C4⋊C4.97D4

Character table of C4⋊C4.97D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 8 8 2 2 2 2 8 8 8 8 4 4 4 4 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 -2 0 0 2 2 -2 -2 0 -2 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 -2 2 0 0 2 -2 2 -2 2 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 -2 2 -2 2 0 0 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 -2 2 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 -2 2 2 -2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 -2 -2 0 0 2 2 -2 -2 0 2 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 -2 -2 2 2 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 2 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 2i -2i 0 0 0 complex lifted from C4○D4 ρ19 2 -2 -2 2 -2 2 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i complex lifted from C4○D4 ρ20 2 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -2i 0 0 2i 0 0 complex lifted from C4○D4 ρ21 2 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 2i 0 0 -2i 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 -2 2 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i complex lifted from C4○D4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 0 2√-2 0 0 0 0 0 0 complex lifted from D4.3D4 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 0 -2√-2 0 0 0 0 0 0 0 complex lifted from D4.3D4 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 0 -2√-2 0 0 0 0 0 0 complex lifted from D4.3D4 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 0 2√-2 0 0 0 0 0 0 0 complex lifted from D4.3D4

Smallest permutation representation of C4⋊C4.97D4
On 32 points
Generators in S32
```(1 12 5 16)(2 13 6 9)(3 14 7 10)(4 15 8 11)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)
(1 32 10 17)(2 18 11 25)(3 26 12 19)(4 20 13 27)(5 28 14 21)(6 22 15 29)(7 30 16 23)(8 24 9 31)
(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 13 5 9)(2 16 6 12)(3 11 7 15)(4 14 8 10)(17 25 21 29)(18 28 22 32)(19 31 23 27)(20 26 24 30)```

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

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

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

Matrix representation of C4⋊C4.97D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 16 0 0 0 0 2 1 0 0 0 0 0 0 1 1 0 0 0 0 15 16
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 0 10 5 0 0 0 0 7 0 0 0 0 5 0 0 0 0 7 7 0 0
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 10 5 0 0 0 0 7 0 0 0 0 0 0 0 10 5 0 0 0 0 7 0
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 10 0 0 0 0 0 0 0 0 5 0 0 0 0 10 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,2,0,0,0,0,16,1,0,0,0,0,0,0,1,15,0,0,0,0,1,16],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,5,7,0,0,10,7,0,0,0,0,5,0,0,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,10,7,0,0,0,0,5,0,0,0,0,0,0,0,10,7,0,0,0,0,5,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,0,0,0,0,5,0,0,0,0,0,0,0,0,10,0,0,0,0,5,0] >;`

C4⋊C4.97D4 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{97}D_4`
`% in TeX`

`G:=Group("C4:C4.97D4");`
`// GroupNames label`

`G:=SmallGroup(128,778);`
`// by ID`

`G=gap.SmallGroup(128,778);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2019,1018,248,718,1027]);`
`// Polycyclic`

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

Export

׿
×
𝔽