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## G = D45⋊C3order 270 = 2·33·5

### The semidirect product of D45 and C3 acting faithfully

Aliases: D45⋊C3, C451C6, C32.D15, 3- 1+2⋊D5, C5⋊(C9⋊C6), C9⋊(C3×D5), C15.3(C3×S3), (C3×C15).1S3, C3.3(C3×D15), (C5×3- 1+2)⋊1C2, SmallGroup(270,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C45 — D45⋊C3
 Chief series C1 — C3 — C15 — C45 — C5×3- 1+2 — D45⋊C3
 Lower central C45 — D45⋊C3
 Upper central C1

Generators and relations for D45⋊C3
G = < a,b,c | a45=b2=c3=1, bab=a-1, cac-1=a31, bc=cb >

Smallest permutation representation of D45⋊C3
On 45 points
Generators in S45
```(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)
(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)
(2 17 32)(3 33 18)(5 20 35)(6 36 21)(8 23 38)(9 39 24)(11 26 41)(12 42 27)(14 29 44)(15 45 30)```

`G:=sub<Sym(45)| (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), (2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24), (2,17,32)(3,33,18)(5,20,35)(6,36,21)(8,23,38)(9,39,24)(11,26,41)(12,42,27)(14,29,44)(15,45,30)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45), (2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24), (2,17,32)(3,33,18)(5,20,35)(6,36,21)(8,23,38)(9,39,24)(11,26,41)(12,42,27)(14,29,44)(15,45,30) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)], [(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24)], [(2,17,32),(3,33,18),(5,20,35),(6,36,21),(8,23,38),(9,39,24),(11,26,41),(12,42,27),(14,29,44),(15,45,30)]])`

32 conjugacy classes

 class 1 2 3A 3B 3C 5A 5B 6A 6B 9A 9B 9C 15A 15B 15C 15D 15E 15F 15G 15H 45A ··· 45L order 1 2 3 3 3 5 5 6 6 9 9 9 15 15 15 15 15 15 15 15 45 ··· 45 size 1 45 2 3 3 2 2 45 45 6 6 6 2 2 2 2 6 6 6 6 6 ··· 6

32 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 6 6 type + + + + + + + image C1 C2 C3 C6 S3 D5 C3×S3 C3×D5 D15 C3×D15 C9⋊C6 D45⋊C3 kernel D45⋊C3 C5×3- 1+2 D45 C45 C3×C15 3- 1+2 C15 C9 C32 C3 C5 C1 # reps 1 1 2 2 1 2 2 4 4 8 1 4

Matrix representation of D45⋊C3 in GL8(𝔽181)

 102 65 0 0 0 0 0 0 116 97 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 180 180 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 180 180 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 0 0 167 180 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 180 180 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
,
 48 0 0 0 0 0 0 0 0 48 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 180 180 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 180 180

`G:=sub<GL(8,GF(181))| [102,116,0,0,0,0,0,0,65,97,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,180,0,0,0,0,0,0,1,180,0,0,0,0,0,0,0,0,0,180,0,0,0,0,0,0,1,180,0,0],[1,167,0,0,0,0,0,0,0,180,0,0,0,0,0,0,0,0,1,180,0,0,0,0,0,0,0,180,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[48,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,180,1,0,0,0,0,0,0,180,0,0,0,0,0,0,0,0,0,0,180,0,0,0,0,0,0,1,180] >;`

D45⋊C3 in GAP, Magma, Sage, TeX

`D_{45}\rtimes C_3`
`% in TeX`

`G:=Group("D45:C3");`
`// GroupNames label`

`G:=SmallGroup(270,15);`
`// by ID`

`G=gap.SmallGroup(270,15);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-5,-3,1532,727,462,1443,4504]);`
`// Polycyclic`

`G:=Group<a,b,c|a^45=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^31,b*c=c*b>;`
`// generators/relations`

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