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G = C3.(C33⋊S3)  order 486 = 2·35

5th non-split extension by C3 of C33⋊S3 acting via C33⋊S3/C3≀C3=C2

non-abelian, supersoluble, monomial

Aliases: C32⋊C9.8S3, C33.5(C3⋊S3), C32.28He3⋊C2, C3.5(C33⋊S3), (C3×3- 1+2).2S3, C32.17(He3⋊C2), C3.2(3- 1+2.S3), SmallGroup(486,47)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C32.28He3 — C3.(C33⋊S3)
Chief series
C1 — C3 — C32 — C33 — C32⋊C9 — C32.28He3 — C3.(C33⋊S3)
Lower central
C32.28He3 — C3.(C33⋊S3)
Upper central
C1

Generators and relations for C3.(C33⋊S3)
 G = < a,b,c,d,e,f | a3=c3=d3=f2=1, b3=e3=a, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, cbc-1=a-1b, bd=db, ebe-1=a-1bc, fbf=a-1b2, cd=dc, ece-1=acd, fcf=acd-1, de=ed, fdf=d-1, fef=a-1e2 >

Subgroups: 682 in 70 conjugacy classes, 13 normal (5 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, 3- 1+2, C33, C9⋊C6, C9⋊S3, C3×C3⋊S3, C32⋊C9, C3×3- 1+2, C32⋊D9, C33.S3, C32.28He3, C3.(C33⋊S3)
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊S3, 3- 1+2.S3, C3.(C33⋊S3)

Character table of C3.(C33⋊S3)

 class 123A3B3C3D3E3F6A6B9A9B9C9D9E9F9G9H9I9J9K9L
 size 1812222998181181818181818181818181818
ρ11111111111111111111111    trivial
ρ21-1111111-1-1111111111111    linear of order 2
ρ32022222200-1-1-1222-1-1-1-1-1-1    orthogonal lifted from S3
ρ42022222200-1-1-1-1-1-1-1-1-1222    orthogonal lifted from S3
ρ52022222200222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62022222200-1-1-1-1-1-1222-1-1-1    orthogonal lifted from S3
ρ7313333-3+3√-3/2-3-3√-3/2ζ3ζ32000000000000    complex lifted from He3⋊C2
ρ83-13333-3-3√-3/2-3+3√-3/2ζ6ζ65000000000000    complex lifted from He3⋊C2
ρ9313333-3-3√-3/2-3+3√-3/2ζ32ζ3000000000000    complex lifted from He3⋊C2
ρ103-13333-3+3√-3/2-3-3√-3/2ζ65ζ6000000000000    complex lifted from He3⋊C2
ρ1160-3-36-30000000-330000000    orthogonal lifted from C33⋊S3
ρ1260-3-36-3000000030-3000000    orthogonal lifted from C33⋊S3
ρ1360-3-36-300000000-33000000    orthogonal lifted from C33⋊S3
ρ14606-3-3-30000000000-330000    orthogonal lifted from C33⋊S3
ρ15606-3-3-300000000000-33000    orthogonal lifted from C33⋊S3
ρ16606-3-3-3000000000030-3000    orthogonal lifted from C33⋊S3
ρ1760-36-3-300002ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000000000    orthogonal lifted from 3- 1+2.S3
ρ1860-3-3-360000000000000-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9    orthogonal lifted from 3- 1+2.S3
ρ1960-36-3-30000ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9000000000    orthogonal lifted from 3- 1+2.S3
ρ2060-3-3-360000000000000ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92    orthogonal lifted from 3- 1+2.S3
ρ2160-36-3-30000-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9000000000    orthogonal lifted from 3- 1+2.S3
ρ2260-3-3-3600000000000002ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9    orthogonal lifted from 3- 1+2.S3

Smallest permutation representation of C3.(C33⋊S3)
►On 81 points
Generators in S81
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)

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

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

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

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

Matrix representation of C3.(C33⋊S3) ►in GL12(𝔽19)

100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
0000000180000
0000001180000
0000001417181800
0000001771000
00000025001818
0000005120010
,
100000000000
010000000000
1610181800000000
001000000000
1612001818000000
000010000000
0000001610181700
0000000011800
0000001171751818
0000004517510
00000091651700
00000091551700
,
0180000000000
1180000000000
1771000000000
1770100000000
5120001000000
25001818000000
000000100000
000000010000
000000000100
0000001610181800
0000001612001818
000000000010
,
1810000000000
1800000000000
520100000000
212181800000000
17140001000000
147001818000000
0000000180000
0000001180000
0000001417181800
0000001771000
00000025001818
0000005120010
,
0000181000000
1612001718000000
1580072000000
1680072000000
813101214000000
813011214000000
0000001612001817
0000000000118
00000091400175
00000091300175
000000111818517
00000041010517
,
010000000000
100000000000
17140010000000
147001818000000
521000000000
212181800000000
0000001810000
000000010000
0000000000018
0000000000180
0000000001800
0000000018000

G:=sub<GL(12,GF(19))| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,14,17,2,5,0,0,0,0,0,0,18,18,17,7,5,12,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0],[1,0,16,0,16,0,0,0,0,0,0,0,0,1,10,0,12,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,1,4,9,9,0,0,0,0,0,0,10,0,17,5,16,15,0,0,0,0,0,0,18,1,17,17,5,5,0,0,0,0,0,0,17,18,5,5,17,17,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0],[0,1,17,17,5,2,0,0,0,0,0,0,18,18,7,7,12,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,16,16,0,0,0,0,0,0,0,0,1,0,10,12,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0],[18,18,5,2,17,14,0,0,0,0,0,0,1,0,2,12,14,7,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,1,14,17,2,5,0,0,0,0,0,0,18,18,17,7,5,12,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0],[0,16,15,16,8,8,0,0,0,0,0,0,0,12,8,8,13,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,17,7,7,12,12,0,0,0,0,0,0,1,18,2,2,14,14,0,0,0,0,0,0,0,0,0,0,0,0,16,0,9,9,1,4,0,0,0,0,0,0,12,0,14,13,1,10,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,1,17,17,5,5,0,0,0,0,0,0,17,18,5,5,17,17],[0,1,17,14,5,2,0,0,0,0,0,0,1,0,14,7,2,12,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0] >;
 

C3.(C33⋊S3) in GAP, Magma, Sage, TeX 

C_3.(C_3^3\rtimes S_3)
 
% in TeX
 
G:=Group("C3.(C3^3:S3)");
 
// GroupNames label
 
G:=SmallGroup(486,47);
 
// by ID
 
G=gap.SmallGroup(486,47);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,697,655,7022,224,824,6915,2817,735,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=c^3=d^3=f^2=1,b^3=e^3=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,c*b*c^-1=a^-1*b,b*d=d*b,e*b*e^-1=a^-1*b*c,f*b*f=a^-1*b^2,c*d=d*c,e*c*e^-1=a*c*d,f*c*f=a*c*d^-1,d*e=e*d,f*d*f=d^-1,f*e*f=a^-1*e^2>;
 
// generators/relations
 

Export

Character table of C3.(C33⋊S3) in TeX

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