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## G = (C32×C9)⋊C6order 486 = 2·35

### 12nd semidirect product of C32×C9 and C6 acting faithfully

Aliases: C32⋊C910S3, (C32×C9)⋊12C6, C324D94C3, C33.9(C3⋊S3), (C3×He3).14S3, C33.64(C3×S3), C32.23C332C2, C3.5(He34S3), C32.16(C32⋊C6), C3.5(He3.4S3), (C3×C9).31(C3×S3), C32.41(C3×C3⋊S3), SmallGroup(486,151)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — (C32×C9)⋊C6
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32.23C33 — (C32×C9)⋊C6
 Lower central C32×C9 — (C32×C9)⋊C6
 Upper central C1

Generators and relations for (C32×C9)⋊C6
G = < a,b,c,d | a3=b3=c9=d6=1, ab=ba, ac=ca, dad-1=a-1c6, bc=cb, dbd-1=b-1, dcd-1=bc5 >

Subgroups: 1088 in 99 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, 3- 1+2, C33, C33, C32⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C32⋊C9, C32×C9, C3×He3, C3×3- 1+2, C32⋊D9, He34S3, C324D9, C32.23C33, (C32×C9)⋊C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He34S3, He3.4S3, (C32×C9)⋊C6

Character table of (C32×C9)⋊C6

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 9M 9N 9O size 1 81 2 2 2 2 6 6 6 9 9 18 18 81 81 6 6 6 6 6 6 6 6 6 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 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 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 6 ρ4 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ5 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ6 1 -1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 6 ρ7 2 0 2 2 2 2 -1 -1 -1 2 2 -1 -1 0 0 -1 -1 2 2 2 -1 -1 -1 -1 2 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ8 2 0 2 2 2 2 2 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ9 2 0 2 2 2 2 -1 -1 -1 2 2 -1 -1 0 0 -1 -1 -1 -1 -1 2 2 2 -1 -1 2 -1 -1 -1 2 orthogonal lifted from S3 ρ10 2 0 2 2 2 2 -1 -1 -1 2 2 -1 -1 0 0 2 2 -1 -1 -1 -1 -1 -1 2 -1 -1 2 2 -1 -1 orthogonal lifted from S3 ρ11 2 0 2 2 2 2 -1 -1 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 -1 -1 2 2 2 -1 -1 -1 -1 -1+√-3 ζ65 ζ65 ζ6 -1-√-3 ζ6 complex lifted from C3×S3 ρ12 2 0 2 2 2 2 -1 -1 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 -1 -1 2 2 2 -1 -1 -1 -1 -1-√-3 ζ6 ζ6 ζ65 -1+√-3 ζ65 complex lifted from C3×S3 ρ13 2 0 2 2 2 2 -1 -1 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 -1 -1 -1 -1 -1 2 2 2 -1 ζ6 -1-√-3 ζ6 ζ65 ζ65 -1+√-3 complex lifted from C3×S3 ρ14 2 0 2 2 2 2 -1 -1 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 2 2 -1 -1 -1 -1 -1 -1 2 ζ6 ζ6 -1-√-3 -1+√-3 ζ65 ζ65 complex lifted from C3×S3 ρ15 2 0 2 2 2 2 -1 -1 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 2 2 -1 -1 -1 -1 -1 -1 2 ζ65 ζ65 -1+√-3 -1-√-3 ζ6 ζ6 complex lifted from C3×S3 ρ16 2 0 2 2 2 2 2 2 2 -1-√-3 -1+√-3 -1-√-3 -1+√-3 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 complex lifted from C3×S3 ρ17 2 0 2 2 2 2 2 2 2 -1+√-3 -1-√-3 -1+√-3 -1-√-3 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 complex lifted from C3×S3 ρ18 2 0 2 2 2 2 -1 -1 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 -1 -1 -1 -1 -1 2 2 2 -1 ζ65 -1+√-3 ζ65 ζ6 ζ6 -1-√-3 complex lifted from C3×S3 ρ19 6 0 -3 -3 -3 6 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ20 6 0 -3 -3 -3 6 -3 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ21 6 0 -3 -3 -3 6 -3 -3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ22 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.4S3 ρ23 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 0 0 0 0 0 0 orthogonal lifted from He3.4S3 ρ24 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.4S3 ρ25 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 0 0 0 0 0 0 3ζ95+3ζ94 0 0 0 0 0 0 orthogonal lifted from He3.4S3 ρ26 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 0 0 0 0 0 0 3ζ97+3ζ92 0 0 0 0 0 0 orthogonal lifted from He3.4S3 ρ27 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 0 0 0 0 0 0 3ζ98+3ζ9 0 0 0 0 0 0 orthogonal lifted from He3.4S3 ρ28 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.4S3 ρ29 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 0 0 0 0 0 0 orthogonal lifted from He3.4S3 ρ30 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 0 0 0 0 0 0 orthogonal lifted from He3.4S3

Smallest permutation representation of (C32×C9)⋊C6
On 81 points
Generators in S81
(1 55 15)(2 56 16)(3 57 17)(4 58 18)(5 59 10)(6 60 11)(7 61 12)(8 62 13)(9 63 14)(19 41 30)(20 42 31)(21 43 32)(22 44 33)(23 45 34)(24 37 35)(25 38 36)(26 39 28)(27 40 29)(46 71 79)(47 72 80)(48 64 81)(49 65 73)(50 66 74)(51 67 75)(52 68 76)(53 69 77)(54 70 78)
(1 39 50)(2 40 51)(3 41 52)(4 42 53)(5 43 54)(6 44 46)(7 45 47)(8 37 48)(9 38 49)(10 21 78)(11 22 79)(12 23 80)(13 24 81)(14 25 73)(15 26 74)(16 27 75)(17 19 76)(18 20 77)(28 66 55)(29 67 56)(30 68 57)(31 69 58)(32 70 59)(33 71 60)(34 72 61)(35 64 62)(36 65 63)
(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)
(2 52 54 9 37 44)(3 43 38 8 46 51)(4 7)(5 49 48 6 40 41)(10 71 78 60 21 33)(11 35 25 59 76 67)(12 55 18 58 15 61)(13 68 81 57 24 30)(14 32 19 56 79 64)(16 65 75 63 27 36)(17 29 22 62 73 70)(20 69 26 72 23 66)(28 77 31 74 34 80)(39 50)(42 47)(45 53)

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

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

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

Matrix representation of (C32×C9)⋊C6 in GL12(𝔽19)

 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 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 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 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 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 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 12 17 0 0 0 0 0 0 0 0 0 0 2 14 0 0 0 0 0 0 0 0 0 0 0 0 12 17 0 0 0 0 0 0 0 0 0 0 2 14 0 0 0 0 0 0 12 17 0 0 0 0 0 0 0 0 0 0 2 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 14 0 0 0 0 0 0 0 0 0 0 5 7 0 0 0 0 0 0 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 2 14 0 0 0 0 0 0 0 0 0 0 5 7 0 0
,
 18 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 18 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 1 0 0 0 0 0 0 0 0 0 0 0 18 18 0 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 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(12,GF(19))| [0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,12,2,0,0,0,0,0,0,0,0,0,0,17,14,0,0,0,0,0,0,12,2,0,0,0,0,0,0,0,0,0,0,17,14,0,0,0,0,0,0,0,0,0,0,0,0,12,2,0,0,0,0,0,0,0,0,0,0,17,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,14,7,0,0,0,0],[18,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0] >;

(C32×C9)⋊C6 in GAP, Magma, Sage, TeX

(C_3^2\times C_9)\rtimes C_6
% in TeX

G:=Group("(C3^2xC9):C6");
// GroupNames label

G:=SmallGroup(486,151);
// by ID

G=gap.SmallGroup(486,151);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,548,986,867,2169,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^9=d^6=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*c^6,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^5>;
// generators/relations

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