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## G = C72⋊3C9order 441 = 32·72

### 3rd semidirect product of C72 and C9 acting via C9/C3=C3

Aliases: C723C9, C72(C7⋊C9), C21.4(C7⋊C3), (C7×C21).3C3, C3.(C723C3), SmallGroup(441,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C72⋊3C9
 Chief series C1 — C7 — C72 — C7×C21 — C72⋊3C9
 Lower central C72 — C72⋊3C9
 Upper central C1 — C3

Generators and relations for C723C9
G = < a,b,c | a7=b7=c9=1, ab=ba, cac-1=a2, cbc-1=b4 >

Smallest permutation representation of C723C9
On 63 points
Generators in S63
```(1 27 29 49 11 44 56)(2 30 12 57 19 50 45)(3 13 20 37 31 58 51)(4 21 32 52 14 38 59)(5 33 15 60 22 53 39)(6 16 23 40 34 61 54)(7 24 35 46 17 41 62)(8 36 18 63 25 47 42)(9 10 26 43 28 55 48)
(1 56 44 11 49 29 27)(2 50 57 30 45 19 12)(3 37 51 20 58 13 31)(4 59 38 14 52 32 21)(5 53 60 33 39 22 15)(6 40 54 23 61 16 34)(7 62 41 17 46 35 24)(8 47 63 36 42 25 18)(9 43 48 26 55 10 28)
(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)```

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

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

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

57 conjugacy classes

 class 1 3A 3B 7A ··· 7P 9A ··· 9F 21A ··· 21AF order 1 3 3 7 ··· 7 9 ··· 9 21 ··· 21 size 1 1 1 3 ··· 3 49 ··· 49 3 ··· 3

57 irreducible representations

 dim 1 1 1 3 3 3 3 type + image C1 C3 C9 C7⋊C3 C7⋊C9 C72⋊3C3 C72⋊3C9 kernel C72⋊3C9 C7×C21 C72 C21 C7 C3 C1 # reps 1 2 6 4 8 12 24

Matrix representation of C723C9 in GL4(𝔽127) generated by

 1 0 0 0 0 32 0 0 0 0 64 0 0 0 0 8
,
 1 0 0 0 0 4 0 0 0 0 16 0 0 0 0 2
,
 99 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
`G:=sub<GL(4,GF(127))| [1,0,0,0,0,32,0,0,0,0,64,0,0,0,0,8],[1,0,0,0,0,4,0,0,0,0,16,0,0,0,0,2],[99,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;`

C723C9 in GAP, Magma, Sage, TeX

`C_7^2\rtimes_3C_9`
`% in TeX`

`G:=Group("C7^2:3C9");`
`// GroupNames label`

`G:=SmallGroup(441,7);`
`// by ID`

`G=gap.SmallGroup(441,7);`
`# by ID`

`G:=PCGroup([4,-3,-3,-7,-7,12,434,2019]);`
`// Polycyclic`

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

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