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## G = C32⋊C9.10S3order 486 = 2·35

### 10th non-split extension by C32⋊C9 of S3 acting faithfully

Aliases: C32⋊C9.10S3, C33.7(C3⋊S3), C33.7C322C2, C3.5(He3.3S3), C32.19(He3⋊C2), SmallGroup(486,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33.7C32 — C32⋊C9.10S3
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C33.7C32 — C32⋊C9.10S3
 Lower central C33.7C32 — C32⋊C9.10S3
 Upper central C1

Generators and relations for C32⋊C9.10S3
G = < a,b,c,d,e | a3=b3=c9=e2=1, d3=b, ab=ba, cac-1=ab-1, dad-1=ac3, eae=ab-1c3, bc=cb, bd=db, ebe=b-1, dcd-1=ab-1c4, ece=c-1, ede=b-1d2 >

81C2
9C3
27S3
27S3
27S3
27S3
81C6
3C32
3C32
3C32
3C32
9C9
9C9
9C9
9C9
27D9
27D9
27C3×S3
27C3×S3
27D9
27C3×S3
27C3×S3
27D9

Character table of C32⋊C9.10S3

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L size 1 81 2 2 2 2 9 9 81 81 18 18 18 18 18 18 18 18 18 18 18 18 ρ1 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 linear of order 2 ρ3 2 0 2 2 2 2 2 2 0 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 0 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 orthogonal lifted from S3 ρ5 2 0 2 2 2 2 2 2 0 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 0 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 orthogonal lifted from S3 ρ7 3 1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 ζ3 ζ32 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ8 3 -1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ9 3 1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 ζ32 ζ3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ10 3 -1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ11 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ12 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ13 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 0 orthogonal lifted from He3.3S3 ρ14 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 orthogonal lifted from He3.3S3 ρ15 6 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 orthogonal lifted from He3.3S3 ρ16 6 0 6 -3 -3 -3 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ17 6 0 6 -3 -3 -3 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ18 6 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 orthogonal lifted from He3.3S3 ρ19 6 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 orthogonal lifted from He3.3S3 ρ20 6 0 6 -3 -3 -3 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ21 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 0 orthogonal lifted from He3.3S3 ρ22 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 orthogonal lifted from He3.3S3

Smallest permutation representation of C32⋊C9.10S3
On 81 points
Generators in S81
```(1 41 31)(2 29 45)(3 6 9)(4 44 34)(5 32 39)(7 38 28)(8 35 42)(11 21 78)(12 79 22)(14 24 81)(15 73 25)(17 27 75)(18 76 19)(30 33 36)(37 40 43)(46 59 65)(47 69 57)(48 54 51)(49 62 68)(50 72 60)(52 56 71)(53 66 63)(55 61 58)(64 70 67)
(1 34 38)(2 35 39)(3 36 40)(4 28 41)(5 29 42)(6 30 43)(7 31 44)(8 32 45)(9 33 37)(10 77 20)(11 78 21)(12 79 22)(13 80 23)(14 81 24)(15 73 25)(16 74 26)(17 75 27)(18 76 19)(46 71 62)(47 72 63)(48 64 55)(49 65 56)(50 66 57)(51 67 58)(52 68 59)(53 69 60)(54 70 61)
(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)(73 74 75 76 77 78 79 80 81)
(1 46 21 34 71 11 38 62 78)(2 57 15 35 50 73 39 66 25)(3 61 10 36 54 77 40 70 20)(4 49 24 28 65 14 41 56 81)(5 60 18 29 53 76 42 69 19)(6 55 13 30 48 80 43 64 23)(7 52 27 31 68 17 44 59 75)(8 63 12 32 47 79 45 72 22)(9 58 16 33 51 74 37 67 26)
(2 9)(3 8)(4 7)(5 6)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 55)(20 63)(21 62)(22 61)(23 60)(24 59)(25 58)(26 57)(27 56)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 45)(46 78)(47 77)(48 76)(49 75)(50 74)(51 73)(52 81)(53 80)(54 79)```

`G:=sub<Sym(81)| (1,41,31)(2,29,45)(3,6,9)(4,44,34)(5,32,39)(7,38,28)(8,35,42)(11,21,78)(12,79,22)(14,24,81)(15,73,25)(17,27,75)(18,76,19)(30,33,36)(37,40,43)(46,59,65)(47,69,57)(48,54,51)(49,62,68)(50,72,60)(52,56,71)(53,66,63)(55,61,58)(64,70,67), (1,34,38)(2,35,39)(3,36,40)(4,28,41)(5,29,42)(6,30,43)(7,31,44)(8,32,45)(9,33,37)(10,77,20)(11,78,21)(12,79,22)(13,80,23)(14,81,24)(15,73,25)(16,74,26)(17,75,27)(18,76,19)(46,71,62)(47,72,63)(48,64,55)(49,65,56)(50,66,57)(51,67,58)(52,68,59)(53,69,60)(54,70,61), (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)(73,74,75,76,77,78,79,80,81), (1,46,21,34,71,11,38,62,78)(2,57,15,35,50,73,39,66,25)(3,61,10,36,54,77,40,70,20)(4,49,24,28,65,14,41,56,81)(5,60,18,29,53,76,42,69,19)(6,55,13,30,48,80,43,64,23)(7,52,27,31,68,17,44,59,75)(8,63,12,32,47,79,45,72,22)(9,58,16,33,51,74,37,67,26), (2,9)(3,8)(4,7)(5,6)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,55)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,45)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,81)(53,80)(54,79)>;`

`G:=Group( (1,41,31)(2,29,45)(3,6,9)(4,44,34)(5,32,39)(7,38,28)(8,35,42)(11,21,78)(12,79,22)(14,24,81)(15,73,25)(17,27,75)(18,76,19)(30,33,36)(37,40,43)(46,59,65)(47,69,57)(48,54,51)(49,62,68)(50,72,60)(52,56,71)(53,66,63)(55,61,58)(64,70,67), (1,34,38)(2,35,39)(3,36,40)(4,28,41)(5,29,42)(6,30,43)(7,31,44)(8,32,45)(9,33,37)(10,77,20)(11,78,21)(12,79,22)(13,80,23)(14,81,24)(15,73,25)(16,74,26)(17,75,27)(18,76,19)(46,71,62)(47,72,63)(48,64,55)(49,65,56)(50,66,57)(51,67,58)(52,68,59)(53,69,60)(54,70,61), (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)(73,74,75,76,77,78,79,80,81), (1,46,21,34,71,11,38,62,78)(2,57,15,35,50,73,39,66,25)(3,61,10,36,54,77,40,70,20)(4,49,24,28,65,14,41,56,81)(5,60,18,29,53,76,42,69,19)(6,55,13,30,48,80,43,64,23)(7,52,27,31,68,17,44,59,75)(8,63,12,32,47,79,45,72,22)(9,58,16,33,51,74,37,67,26), (2,9)(3,8)(4,7)(5,6)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,55)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,45)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,81)(53,80)(54,79) );`

`G=PermutationGroup([[(1,41,31),(2,29,45),(3,6,9),(4,44,34),(5,32,39),(7,38,28),(8,35,42),(11,21,78),(12,79,22),(14,24,81),(15,73,25),(17,27,75),(18,76,19),(30,33,36),(37,40,43),(46,59,65),(47,69,57),(48,54,51),(49,62,68),(50,72,60),(52,56,71),(53,66,63),(55,61,58),(64,70,67)], [(1,34,38),(2,35,39),(3,36,40),(4,28,41),(5,29,42),(6,30,43),(7,31,44),(8,32,45),(9,33,37),(10,77,20),(11,78,21),(12,79,22),(13,80,23),(14,81,24),(15,73,25),(16,74,26),(17,75,27),(18,76,19),(46,71,62),(47,72,63),(48,64,55),(49,65,56),(50,66,57),(51,67,58),(52,68,59),(53,69,60),(54,70,61)], [(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),(73,74,75,76,77,78,79,80,81)], [(1,46,21,34,71,11,38,62,78),(2,57,15,35,50,73,39,66,25),(3,61,10,36,54,77,40,70,20),(4,49,24,28,65,14,41,56,81),(5,60,18,29,53,76,42,69,19),(6,55,13,30,48,80,43,64,23),(7,52,27,31,68,17,44,59,75),(8,63,12,32,47,79,45,72,22),(9,58,16,33,51,74,37,67,26)], [(2,9),(3,8),(4,7),(5,6),(10,72),(11,71),(12,70),(13,69),(14,68),(15,67),(16,66),(17,65),(18,64),(19,55),(20,63),(21,62),(22,61),(23,60),(24,59),(25,58),(26,57),(27,56),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(36,45),(46,78),(47,77),(48,76),(49,75),(50,74),(51,73),(52,81),(53,80),(54,79)]])`

Matrix representation of C32⋊C9.10S3 in GL12(𝔽19)

 0 1 0 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 17 3 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 0 0 7 9 18 18 0 0 0 0 0 0 0 0 15 11 1 0 0 0 0 0 0 0 0 0 6 3 0 0 1 0 0 0 0 0 0 0 14 11 0 0 0 1
,
 18 18 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 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 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 0 0 0 0 14 12 0 0 0 0 0 0 0 0 0 0 5 18 17 12 0 0 0 0 0 0 0 0 0 10 7 5 0 0 0 0 0 0 0 0 0 18 0 0 17 12 0 0 0 0 0 0 0 0 0 0 7 5
,
 2 14 0 0 0 0 0 0 0 0 0 0 5 7 0 0 0 0 0 0 0 0 0 0 0 0 5 7 0 0 0 0 0 0 0 0 0 0 12 17 0 0 0 0 0 0 0 0 0 0 0 0 5 7 0 0 0 0 0 0 0 0 0 0 12 17 0 0 0 0 0 0 0 0 0 0 0 0 11 10 0 0 16 3 0 0 0 0 0 0 3 11 0 0 16 0 0 0 0 0 0 0 6 0 0 0 16 13 0 0 0 0 0 0 14 14 0 0 6 5 0 0 0 0 0 0 5 3 1 0 5 3 0 0 0 0 0 0 8 6 0 1 7 11
,
 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 6 0 0 1 0 0 0 0 0 0 0 3 18 0 0 18 18 0 0 0 0 0 0 1 10 1 0 0 0 0 0 0 0 0 0 15 17 18 18 0 0

`G:=sub<GL(12,GF(19))| [0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,17,18,7,15,6,14,0,0,0,0,0,0,3,1,9,11,3,11,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,14,5,0,0,0,0,0,0,0,0,0,15,12,18,10,18,0,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,0,0,0,0,12,5],[2,5,0,0,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,0,0,7,17,0,0,0,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,0,0,7,17,0,0,0,0,0,0,0,0,0,0,0,0,11,3,6,14,5,8,0,0,0,0,0,0,10,11,0,14,3,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,16,6,5,7,0,0,0,0,0,0,3,0,13,5,3,11],[0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,1,3,1,15,0,0,0,0,0,0,3,1,6,18,10,17,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0] >;`

C32⋊C9.10S3 in GAP, Magma, Sage, TeX

`C_3^2\rtimes C_9._{10}S_3`
`% in TeX`

`G:=Group("C3^2:C9.10S3");`
`// GroupNames label`

`G:=SmallGroup(486,49);`
`// by ID`

`G=gap.SmallGroup(486,49);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3937,979,2162,224,176,6915,873,1383,3244,11669]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^9=e^2=1,d^3=b,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*c^3,e*a*e=a*b^-1*c^3,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=a*b^-1*c^4,e*c*e=c^-1,e*d*e=b^-1*d^2>;`
`// generators/relations`

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