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## G = Q8⋊C9⋊4C6order 432 = 24·33

### 3rd semidirect product of Q8⋊C9 and C6 acting via C6/C2=C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — Q8⋊C9⋊4C6
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C32 — Q8⋊3- 1+2 — Q8⋊C9⋊4C6
 Lower central Q8 — C3×Q8 — Q8⋊C9⋊4C6
 Upper central C1 — C12 — C3×C12

Generators and relations for Q8⋊C94C6
G = < a,b,c,d | a4=c9=d6=1, b2=a2, bab-1=dad-1=a-1, cac-1=b, cbc-1=ab, bd=db, dcd-1=a-1c7 >

Subgroups: 194 in 70 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C9, C32, C12, C12, C2×C6, C4○D4, C18, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, C3×Q8, 3- 1+2, C36, C3×C12, C3×C12, C62, C3×C4○D4, C3×C4○D4, C2×3- 1+2, Q8⋊C9, C6×C12, D4×C32, Q8×C32, C4×3- 1+2, Q8.C18, C32×C4○D4, Q8⋊3- 1+2, Q8⋊C94C6
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, 3- 1+2, C3×A4, C4.A4, C2×3- 1+2, C6×A4, C32.A4, C3×C4.A4, C2×C32.A4, Q8⋊C94C6

Smallest permutation representation of Q8⋊C94C6
On 72 points
Generators in S72
```(1 26 46 12)(2 32 47 44)(3 55 48 66)(4 20 49 15)(5 35 50 38)(6 58 51 69)(7 23 52 18)(8 29 53 41)(9 61 54 72)(10 71 24 60)(11 30 25 42)(13 65 27 63)(14 33 19 45)(16 68 21 57)(17 36 22 39)(28 70 40 59)(31 64 43 62)(34 67 37 56)
(1 31 46 43)(2 63 47 65)(3 19 48 14)(4 34 49 37)(5 57 50 68)(6 22 51 17)(7 28 52 40)(8 60 53 71)(9 25 54 11)(10 29 24 41)(12 64 26 62)(13 32 27 44)(15 67 20 56)(16 35 21 38)(18 70 23 59)(30 72 42 61)(33 66 45 55)(36 69 39 58)
(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 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(1 62)(2 38 8 44 5 41)(3 54 6 48 9 51)(4 56)(7 59)(10 63 16 60 13 57)(11 22 14 25 17 19)(12 31)(15 34)(18 28)(20 37)(21 71 27 68 24 65)(23 40)(26 43)(29 47 35 53 32 50)(30 36 33)(39 45 42)(46 64)(49 67)(52 70)(55 61 58)(66 72 69)```

`G:=sub<Sym(72)| (1,26,46,12)(2,32,47,44)(3,55,48,66)(4,20,49,15)(5,35,50,38)(6,58,51,69)(7,23,52,18)(8,29,53,41)(9,61,54,72)(10,71,24,60)(11,30,25,42)(13,65,27,63)(14,33,19,45)(16,68,21,57)(17,36,22,39)(28,70,40,59)(31,64,43,62)(34,67,37,56), (1,31,46,43)(2,63,47,65)(3,19,48,14)(4,34,49,37)(5,57,50,68)(6,22,51,17)(7,28,52,40)(8,60,53,71)(9,25,54,11)(10,29,24,41)(12,64,26,62)(13,32,27,44)(15,67,20,56)(16,35,21,38)(18,70,23,59)(30,72,42,61)(33,66,45,55)(36,69,39,58), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,62)(2,38,8,44,5,41)(3,54,6,48,9,51)(4,56)(7,59)(10,63,16,60,13,57)(11,22,14,25,17,19)(12,31)(15,34)(18,28)(20,37)(21,71,27,68,24,65)(23,40)(26,43)(29,47,35,53,32,50)(30,36,33)(39,45,42)(46,64)(49,67)(52,70)(55,61,58)(66,72,69)>;`

`G:=Group( (1,26,46,12)(2,32,47,44)(3,55,48,66)(4,20,49,15)(5,35,50,38)(6,58,51,69)(7,23,52,18)(8,29,53,41)(9,61,54,72)(10,71,24,60)(11,30,25,42)(13,65,27,63)(14,33,19,45)(16,68,21,57)(17,36,22,39)(28,70,40,59)(31,64,43,62)(34,67,37,56), (1,31,46,43)(2,63,47,65)(3,19,48,14)(4,34,49,37)(5,57,50,68)(6,22,51,17)(7,28,52,40)(8,60,53,71)(9,25,54,11)(10,29,24,41)(12,64,26,62)(13,32,27,44)(15,67,20,56)(16,35,21,38)(18,70,23,59)(30,72,42,61)(33,66,45,55)(36,69,39,58), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,62)(2,38,8,44,5,41)(3,54,6,48,9,51)(4,56)(7,59)(10,63,16,60,13,57)(11,22,14,25,17,19)(12,31)(15,34)(18,28)(20,37)(21,71,27,68,24,65)(23,40)(26,43)(29,47,35,53,32,50)(30,36,33)(39,45,42)(46,64)(49,67)(52,70)(55,61,58)(66,72,69) );`

`G=PermutationGroup([[(1,26,46,12),(2,32,47,44),(3,55,48,66),(4,20,49,15),(5,35,50,38),(6,58,51,69),(7,23,52,18),(8,29,53,41),(9,61,54,72),(10,71,24,60),(11,30,25,42),(13,65,27,63),(14,33,19,45),(16,68,21,57),(17,36,22,39),(28,70,40,59),(31,64,43,62),(34,67,37,56)], [(1,31,46,43),(2,63,47,65),(3,19,48,14),(4,34,49,37),(5,57,50,68),(6,22,51,17),(7,28,52,40),(8,60,53,71),(9,25,54,11),(10,29,24,41),(12,64,26,62),(13,32,27,44),(15,67,20,56),(16,35,21,38),(18,70,23,59),(30,72,42,61),(33,66,45,55),(36,69,39,58)], [(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),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,62),(2,38,8,44,5,41),(3,54,6,48,9,51),(4,56),(7,59),(10,63,16,60,13,57),(11,22,14,25,17,19),(12,31),(15,34),(18,28),(20,37),(21,71,27,68,24,65),(23,40),(26,43),(29,47,35,53,32,50),(30,36,33),(39,45,42),(46,64),(49,67),(52,70),(55,61,58),(66,72,69)]])`

62 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 6E ··· 6L 9A ··· 9F 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12P 18A ··· 18F 36A ··· 36L order 1 2 2 3 3 3 3 4 4 4 6 6 6 6 6 ··· 6 9 ··· 9 12 12 12 12 12 12 12 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 6 1 1 3 3 1 1 6 1 1 3 3 6 ··· 6 12 ··· 12 1 1 1 1 3 3 3 3 6 ··· 6 12 ··· 12 12 ··· 12

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 3 3 3 3 3 3 6 type + + + + image C1 C2 C3 C3 C6 C6 C4.A4 C3×C4.A4 A4 C2×A4 3- 1+2 C3×A4 C2×3- 1+2 C6×A4 C32.A4 C2×C32.A4 Q8⋊C9⋊4C6 kernel Q8⋊C9⋊4C6 Q8⋊3- 1+2 Q8.C18 C32×C4○D4 Q8⋊C9 Q8×C32 C32 C3 C3×C12 C3×C6 C4○D4 C12 Q8 C6 C4 C2 C1 # reps 1 1 6 2 6 2 6 12 1 1 2 2 2 2 6 6 4

Matrix representation of Q8⋊C94C6 in GL5(𝔽37)

 6 0 0 0 0 6 31 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 1
,
 36 2 0 0 0 36 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 36
,
 31 5 0 0 0 16 5 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 10 0 0
,
 14 9 0 0 0 14 23 0 0 0 0 0 36 0 0 0 0 0 26 0 0 0 0 0 10

`G:=sub<GL(5,GF(37))| [6,6,0,0,0,0,31,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1],[36,36,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[31,16,0,0,0,5,5,0,0,0,0,0,0,0,10,0,0,1,0,0,0,0,0,1,0],[14,14,0,0,0,9,23,0,0,0,0,0,36,0,0,0,0,0,26,0,0,0,0,0,10] >;`

Q8⋊C94C6 in GAP, Magma, Sage, TeX

`Q_8\rtimes C_9\rtimes_4C_6`
`% in TeX`

`G:=Group("Q8:C9:4C6");`
`// GroupNames label`

`G:=SmallGroup(432,338);`
`// by ID`

`G=gap.SmallGroup(432,338);`
`# by ID`

`G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,134,261,1901,172,3414,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^9=d^6=1,b^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=b,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=a^-1*c^7>;`
`// generators/relations`

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