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G = C32⋊C9⋊6S3  order 486 = 2·35

6th semidirect product of C32⋊C9 and S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C32⋊C9⋊6S3, (C3×He3).5S3, C33.4(C3⋊S3), C3.4(C33⋊S3), C3.2(He3⋊S3), C32.27He3⋊3C2, C3.3(He3.3S3), C32.16(He3⋊C2), SmallGroup(486,46)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C32.27He3 — C32⋊C9⋊6S3
Chief series
C1 — C3 — C32 — C33 — C32⋊C9 — C32.27He3 — C32⋊C9⋊6S3
Lower central
C32.27He3 — C32⋊C9⋊6S3
Upper central
C1

Generators and relations for C32⋊C9⋊6S3
 G = < a,b,c,d,e | a3=b3=c9=d3=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=abc3, bc=cb, bd=db, ebe=b-1, dcd-1=ab-1c4, ece=c-1, ede=d-1 >

Subgroups: 1024 in 72 conjugacy classes, 13 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, He3, C33, C33, C32⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C32⋊C9, C3×He3, C32⋊D9, He3⋊4S3, C32.27He3, C32⋊C9⋊6S3
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊S3, He3.3S3, He3⋊S3, C32⋊C9⋊6S3

Character table of C32⋊C9⋊6S3

 class 123A3B3C3D3E3F3G3H3I6A6B9A9B9C9D9E9F9G9H9I
 size 1812222991818188181181818181818181818
ρ11111111111111111111111    trivial
ρ21-1111111111-1-1111111111    linear of order 2
ρ320222222-1-1-100-1-1-12-1-122-1    orthogonal lifted from S3
ρ420222222-1-1-1002-1-1-122-1-1-1    orthogonal lifted from S3
ρ52022222222200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ620222222-1-1-100-122-1-1-1-1-12    orthogonal lifted from S3
ρ73-13333-3-3√-3/2-3+3√-3/2000ζ6ζ65000000000    complex lifted from He3⋊C2
ρ8313333-3+3√-3/2-3-3√-3/2000ζ3ζ32000000000    complex lifted from He3⋊C2
ρ93-13333-3+3√-3/2-3-3√-3/2000ζ65ζ6000000000    complex lifted from He3⋊C2
ρ10313333-3-3√-3/2-3+3√-3/2000ζ32ζ3000000000    complex lifted from He3⋊C2
ρ1160-36-3-300-33000000000000    orthogonal lifted from C33⋊S3
ρ1260-36-3-30030-300000000000    orthogonal lifted from C33⋊S3
ρ1360-36-3-3000-3300000000000    orthogonal lifted from C33⋊S3
ρ1460-3-36-3000000002ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ900000ζ98+ζ97-ζ94+2ζ92    orthogonal lifted from He3.3S3
ρ15606-3-3-30000000ζ95+2ζ94-ζ92+ζ9000-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9000    orthogonal lifted from He3⋊S3
ρ16606-3-3-300000002ζ98-ζ94+ζ92+ζ9000ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000    orthogonal lifted from He3⋊S3
ρ1760-3-3-3600000000002ζ95+ζ94+ζ92-ζ900ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ920    orthogonal lifted from He3.3S3
ρ1860-3-3-360000000000ζ98+ζ97-ζ94+2ζ92002ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ90    orthogonal lifted from He3.3S3
ρ1960-3-36-300000000ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ900000ζ98+ζ94-ζ92+2ζ9    orthogonal lifted from He3.3S3
ρ2060-3-36-300000000ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ92000002ζ95+ζ94+ζ92-ζ9    orthogonal lifted from He3.3S3
ρ21606-3-3-30000000-ζ98+2ζ97+ζ94+ζ920002ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9000    orthogonal lifted from He3⋊S3
ρ2260-3-3-360000000000ζ98+ζ94-ζ92+2ζ900ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ90    orthogonal lifted from He3.3S3

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

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

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

Matrix representation of C32⋊C9⋊6S3 ►in GL12(𝔽19)

100000000000
010000000000
000100000000
00181800000000
00001818000000
000010000000
0000000180000
0000001180000
0000000181000
0000000180100
0000000180001
00000010001818
,
010000000000
18180000000000
000100000000
00181800000000
000001000000
00001818000000
0000001810000
0000001800000
0000001800100
00000001181800
0000001800001
00000001001818
,
000100000000
00181800000000
000001000000
00001818000000
100000000000
010000000000
00000011181700
0000000011800
000000110181818
0000000001810
0000000001800
0000001001800
,
100000000000
010000000000
00181800000000
001000000000
00001818000000
000010000000
000000141400312
0000002200103
00000014000175
00000021400175
000000141457175
000000221217175
,
0180000000000
1800000000000
000011000000
0000018000000
001100000000
0001800000000
0000002231000
0000007710700
00000070171200
00000002171200
000000221712517
0000007717121214

G:=sub<GL(12,GF(19))| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,18,18,18,18,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18],[0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,18,18,18,0,18,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,17,18,18,18,18,18,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,14,2,14,2,14,2,0,0,0,0,0,0,14,2,0,14,14,2,0,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,0,0,7,17,0,0,0,0,0,0,3,10,17,17,17,17,0,0,0,0,0,0,12,3,5,5,5,5],[0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,7,7,0,2,7,0,0,0,0,0,0,2,7,0,2,2,7,0,0,0,0,0,0,3,10,17,17,17,17,0,0,0,0,0,0,10,7,12,12,12,12,0,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,0,0,17,14] >;
 

C32⋊C9⋊6S3 in GAP, Magma, Sage, TeX 

C_3^2\rtimes C_9\rtimes_6S_3
 
% in TeX
 
G:=Group("C3^2:C9:6S3");
 
// GroupNames label
 
G:=SmallGroup(486,46);
 
// by ID
 
G=gap.SmallGroup(486,46);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,2162,224,176,6915,2817,735,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^9=d^3=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=e*a*e=a*b*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=d^-1>;
 
// generators/relations
 

Export

Character table of C32⋊C9⋊6S3 in TeX

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