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## G = C4⋊Q8.C4order 128 = 27

### 5th non-split extension by C4⋊Q8 of C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4⋊Q8.C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C22×Q8 — C23.38C23 — C4⋊Q8.C4
 Lower central C1 — C2 — C22 — C2×C4 — C4⋊Q8.C4
 Upper central C1 — C2 — C23 — C22×Q8 — C4⋊Q8.C4
 Jennings C1 — C2 — C22 — C2×Q8 — C4⋊Q8.C4

Generators and relations for C4⋊Q8.C4
G = < a,b,c,d | a4=b4=1, c2=d4=b2, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a2b, cd=dc >

Subgroups: 252 in 115 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4.10D4, C4.10D4, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C42.C4, C42.3C4, C2×C4.10D4, C23.38C23, C4⋊Q8.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C4⋊Q8.C4

Character table of C4⋊Q8.C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 8 4 4 4 4 4 4 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 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 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 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 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 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 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 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 linear of order 2 ρ9 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 i -i i -i i -i i -i linear of order 4 ρ10 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 -i -i i i -i i i -i linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 i -i -i -i -i i i i linear of order 4 ρ12 1 1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -i -i -i i i -i i i linear of order 4 ρ13 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 -i i -i i -i i -i i linear of order 4 ρ14 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 i i -i -i i -i -i i linear of order 4 ρ15 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -i i i i i -i -i -i linear of order 4 ρ16 1 1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 i i i -i -i i -i -i linear of order 4 ρ17 2 2 -2 2 -2 0 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 0 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 0 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C4⋊Q8.C4
On 32 points
Generators in S32
```(1 25)(2 8 6 4)(3 31)(5 29)(7 27)(9 11 13 15)(10 23)(12 21)(14 19)(16 17)(18 24 22 20)(26 28 30 32)
(1 31 5 27)(2 32 6 28)(3 29 7 25)(4 30 8 26)(9 20 13 24)(10 17 14 21)(11 18 15 22)(12 23 16 19)
(1 10 5 14)(2 11 6 15)(3 12 7 16)(4 13 8 9)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)
(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)```

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

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

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

Matrix representation of C4⋊Q8.C4 in GL8(𝔽17)

 0 0 0 16 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 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 16 16 1 1 16 1 16 2 0 16 1 0 16 0 16 1
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 16 0 16 16 1 1 16 1 16 2 0 0 0 0 1 0 0 0 1 1 0 0 1 16 0 16
,
 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 13 13 4 4 13 4 13 8 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 4
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 16 16 1 16 1 15 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 16 1

`G:=sub<GL(8,GF(17))| [0,0,0,16,0,0,16,0,0,0,1,0,0,0,16,16,0,1,0,0,0,0,1,1,16,0,0,0,0,0,1,0,0,0,0,0,0,1,16,16,0,0,0,0,16,0,1,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1],[0,0,16,0,0,16,0,1,0,0,0,16,0,16,0,1,1,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,1,1,0,0,0,0,0,1,0,16,0,0,0,0,16,16,0,0,0,0,0,0,0,2,0,16],[0,0,0,13,13,0,0,0,0,0,13,0,13,0,0,0,0,13,0,0,4,0,0,0,13,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,0,13,0,0,0,0,0,13,13,0,0,0,0,0,0,8,0,0,4],[0,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,16,0,1,0,0,1,0,0,0,0,0,1,0,16,0,0,0,1,0,0,1,1,0,0,0,16,0,0,0,15,0,0,0,1] >;`

C4⋊Q8.C4 in GAP, Magma, Sage, TeX

`C_4\rtimes Q_8.C_4`
`% in TeX`

`G:=Group("C4:Q8.C4");`
`// GroupNames label`

`G:=SmallGroup(128,865);`
`// by ID`

`G=gap.SmallGroup(128,865);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,1018,248,1971,375,172,4037]);`
`// Polycyclic`

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

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