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G = (C32×C9)⋊C6  order 486 = 2·35

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

metabelian, supersoluble, monomial

Aliases: C32⋊C9⋊10S3, (C32×C9)⋊12C6, C32⋊4D9⋊4C3, C33.9(C3⋊S3), (C3×He3).14S3, C33.64(C3×S3), C32.23C33⋊2C2, C3.5(He3⋊4S3), 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, He3⋊4S3, C32⋊4D9, C32.23C33, (C32×C9)⋊C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He3⋊4S3, He3.4S3, (C32×C9)⋊C6

Character table of (C32×C9)⋊C6

 class 123A3B3C3D3E3F3G3H3I3J3K6A6B9A9B9C9D9E9F9G9H9I9J9K9L9M9N9O
 size 18122226669918188181666666666181818181818
ρ1111111111111111111111111111111    trivial
ρ21-111111111111-1-1111111111111111    linear of order 2
ρ31-11111111ζ32ζ3ζ32ζ3ζ6ζ65111111111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 6
ρ4111111111ζ3ζ32ζ3ζ32ζ3ζ32111111111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ5111111111ζ32ζ3ζ32ζ3ζ32ζ3111111111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ61-11111111ζ3ζ32ζ3ζ32ζ65ζ6111111111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 6
ρ7202222-1-1-122-1-100-1-1222-1-1-1-12-1-1-12-1    orthogonal lifted from S3
ρ8202222222222200-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ9202222-1-1-122-1-100-1-1-1-1-1222-1-12-1-1-12    orthogonal lifted from S3
ρ10202222-1-1-122-1-10022-1-1-1-1-1-12-1-122-1-1    orthogonal lifted from S3
ρ11202222-1-1-1-1+√-3-1-√-3ζ65ζ600-1-1222-1-1-1-1-1+√-3ζ65ζ65ζ6-1-√-3ζ6    complex lifted from C3×S3
ρ12202222-1-1-1-1-√-3-1+√-3ζ6ζ6500-1-1222-1-1-1-1-1-√-3ζ6ζ6ζ65-1+√-3ζ65    complex lifted from C3×S3
ρ13202222-1-1-1-1-√-3-1+√-3ζ6ζ6500-1-1-1-1-1222-1ζ6-1-√-3ζ6ζ65ζ65-1+√-3    complex lifted from C3×S3
ρ14202222-1-1-1-1-√-3-1+√-3ζ6ζ650022-1-1-1-1-1-12ζ6ζ6-1-√-3-1+√-3ζ65ζ65    complex lifted from C3×S3
ρ15202222-1-1-1-1+√-3-1-√-3ζ65ζ60022-1-1-1-1-1-12ζ65ζ65-1+√-3-1-√-3ζ6ζ6    complex lifted from C3×S3
ρ16202222222-1-√-3-1+√-3-1-√-3-1+√-300-1-1-1-1-1-1-1-1-1ζ6ζ6ζ6ζ65ζ65ζ65    complex lifted from C3×S3
ρ17202222222-1+√-3-1-√-3-1+√-3-1-√-300-1-1-1-1-1-1-1-1-1ζ65ζ65ζ65ζ6ζ6ζ6    complex lifted from C3×S3
ρ18202222-1-1-1-1+√-3-1-√-3ζ65ζ600-1-1-1-1-1222-1ζ65-1+√-3ζ65ζ6ζ6-1-√-3    complex lifted from C3×S3
ρ1960-3-3-366-3-3000000000000000000000    orthogonal lifted from C32⋊C6
ρ2060-3-3-36-36-3000000000000000000000    orthogonal lifted from C32⋊C6
ρ2160-3-3-36-3-36000000000000000000000    orthogonal lifted from C32⋊C6
ρ22606-3-3-3000000000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920000000000    orthogonal lifted from He3.4S3
ρ2360-36-3-3000000000000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90000000    orthogonal lifted from He3.4S3
ρ24606-3-3-3000000000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90000000000    orthogonal lifted from He3.4S3
ρ2560-3-36-30000000003ζ98+3ζ93ζ97+3ζ920000003ζ95+3ζ94000000    orthogonal lifted from He3.4S3
ρ2660-3-36-30000000003ζ95+3ζ943ζ98+3ζ90000003ζ97+3ζ92000000    orthogonal lifted from He3.4S3
ρ2760-3-36-30000000003ζ97+3ζ923ζ95+3ζ940000003ζ98+3ζ9000000    orthogonal lifted from He3.4S3
ρ28606-3-3-3000000000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940000000000    orthogonal lifted from He3.4S3
ρ2960-36-3-3000000000000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940000000    orthogonal lifted from He3.4S3
ρ3060-36-3-3000000000000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920000000    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)

000010000000
000001000000
100000000000
010000000000
001000000000
000100000000
000000000010
000000000001
000000100000
000000010000
000000001000
000000000100
,
010000000000
18180000000000
000100000000
00181800000000
000001000000
00001818000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
00121700000000
0021400000000
00001217000000
0000214000000
12170000000000
2140000000000
0000000000214
000000000057
0000002140000
000000570000
0000000021400
000000005700
,
1800000000000
110000000000
000011000000
0000018000000
0001800000000
0018000000000
000000100000
00000018180000
00000000001818
000000000001
000000000100
000000001000

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
 

Export

Character table of (C32×C9)⋊C6 in TeX

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