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## G = C4⋊Q8⋊15C4order 128 = 27

### 10th semidirect product of C4⋊Q8 and C4 acting via C4/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4⋊Q8⋊15C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C23.32C23 — C4⋊Q8⋊15C4
 Lower central C1 — C2 — C2×C4 — C4⋊Q8⋊15C4
 Upper central C1 — C22 — C22×C4 — C4⋊Q8⋊15C4
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊Q8⋊15C4

Generators and relations for C4⋊Q815C4
G = < a,b,c,d | a4=b4=d4=1, c2=b2, dad-1=ab=ba, cac-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b2c >

Subgroups: 292 in 162 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×14], C22 [×3], C22 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], Q8 [×14], C23, C42 [×2], C42 [×7], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×12], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×9], C4.10D4 [×2], Q8⋊C4 [×4], C2×C42, C2×C4⋊C4 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×Q8 [×4], C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8 [×2], C426C4 [×2], C2×C4.10D4, C23.38D4 [×2], C23.32C23, C2×C4⋊Q8, C4⋊Q815C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, D4.10D4 [×2], C4⋊Q815C4

Smallest permutation representation of C4⋊Q815C4
On 32 points
Generators in S32
```(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 7 12 5)(2 8 11 6)(3 10 14 16)(4 9 13 15)(17 23 19 21)(18 24 20 22)(25 31 27 29)(26 32 28 30)
(1 10 12 16)(2 9 11 15)(3 5 14 7)(4 6 13 8)(17 26 19 28)(18 25 20 27)(21 32 23 30)(22 31 24 29)
(1 17 11 22)(2 24 12 19)(3 30 13 25)(4 27 14 32)(5 23 8 18)(6 20 7 21)(9 31 16 26)(10 28 15 29)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4T 4U 4V 8A 8B 8C 8D order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 8 8 8 8 size 1 1 1 1 2 2 2 2 2 2 4 ··· 4 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 D4.10D4 kernel C4⋊Q8⋊15C4 C42⋊6C4 C2×C4.10D4 C23.38D4 C23.32C23 C2×C4⋊Q8 C4⋊Q8 C4⋊C4 C22×C4 C2×Q8 C2×C4 C2 # reps 1 2 1 2 1 1 8 4 2 2 4 4

Matrix representation of C4⋊Q815C4 in GL6(𝔽17)

 0 16 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 1 7 0 0 0 0 7 16 0 0 0 0 0 0 16 10 0 0 0 0 10 1
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

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

C4⋊Q815C4 in GAP, Magma, Sage, TeX

`C_4\rtimes Q_8\rtimes_{15}C_4`
`% in TeX`

`G:=Group("C4:Q8:15C4");`
`// GroupNames label`

`G:=SmallGroup(128,618);`
`// by ID`

`G=gap.SmallGroup(128,618);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,1018,521,248,2804,1411,2028,1027,124]);`
`// Polycyclic`

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

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