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G = C33.(C3⋊S3)  order 486 = 2·35

3rd non-split extension by C33 of C3⋊S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C32⋊C9.7S3, C33.3(C3⋊S3), C33.3C32⋊C2, C3.3(C33⋊S3), C3.2(He3.3S3), (C3×3- 1+2).1S3, C32.15(He3⋊C2), C3.1(3- 1+2.S3), SmallGroup(486,45)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C33.3C32 — C33.(C3⋊S3)
Chief series
C1 — C3 — C32 — C33 — C32⋊C9 — C33.3C32 — C33.(C3⋊S3)
Lower central
C33.3C32 — C33.(C3⋊S3)
Upper central
C1

Generators and relations for C33.(C3⋊S3)
 G = < a,b,c,d,e,f | a3=b3=c3=f2=1, d3=bc-1, e3=c, eae-1=ab=ba, ac=ca, dad-1=ac-1, faf=ab-1c-1, bc=cb, bd=db, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, ede-1=ac-1d, fdf=b-1cd2, fef=c-1e2 >

Subgroups: 646 in 60 conjugacy classes, 13 normal (all 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, C33.3C32, C33.(C3⋊S3)
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊S3, He3.3S3, 3- 1+2.S3, C33.(C3⋊S3)

Character table of C33.(C3⋊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
ρ11606-3-3-30000000000-330000    orthogonal lifted from C33⋊S3
ρ12606-3-3-300000000000-33000    orthogonal lifted from C33⋊S3
ρ13606-3-3-3000000000030-3000    orthogonal lifted from C33⋊S3
ρ1460-36-3-300002ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ92000000000    orthogonal lifted from He3.3S3
ρ1560-3-36-300000002ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000000    orthogonal lifted from 3- 1+2.S3
ρ1660-36-3-30000ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ9000000000    orthogonal lifted from He3.3S3
ρ1760-3-3-3600000000000002ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ9    orthogonal lifted from He3.3S3
ρ1860-3-36-30000000ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9000000    orthogonal lifted from 3- 1+2.S3
ρ1960-3-36-30000000-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9000000    orthogonal lifted from 3- 1+2.S3
ρ2060-3-3-360000000000000ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92    orthogonal lifted from He3.3S3
ρ2160-3-3-360000000000000ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9    orthogonal lifted from He3.3S3
ρ2260-36-3-30000ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ9000000000    orthogonal lifted from He3.3S3

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

(1 39 29)(2 40 30)(3 41 31)(4 42 32)(5 43 33)(6 44 34)(7 45 35)(8 37 36)(9 38 28)(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 67 57)(47 68 58)(48 69 59)(49 70 60)(50 71 61)(51 72 62)(52 64 63)(53 65 55)(54 66 56)

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

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

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

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

18180000000000
100000000000
001000000000
000100000000
000001000000
00001818000000
0000001160000
0000001170000
00000017141000
0000006180100
0000000100181
00000018200180
,
100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
0000001730000
0000001810000
0000004118100
0000009618000
0000005700181
000000171600180
,
010000000000
18180000000000
000100000000
00181800000000
000001000000
00001818000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
000100000000
00181800000000
000001000000
00001818000000
100000000000
010000000000
0000001060000
00000017160000
0000001122500
00000021014700
0000001840025
000000181400147
,
750000000000
1420000000000
00171200000000
007500000000
00001712000000
000075000000
00000011100030
000000890011
00000012600917
00000017110086
00000018510113
00000014101137
,
1800000000000
110000000000
0000018000000
0000180000000
0001800000000
0018000000000
0000001830000
000000010000
000000720001
000000350010
000000350100
0000007151000

G:=sub<GL(12,GF(19))| [18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,1,1,17,6,0,18,0,0,0,0,0,0,16,17,14,18,1,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,17,18,4,9,5,17,0,0,0,0,0,0,3,1,1,6,7,16,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,0,1,0],[0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,17,11,2,18,18,0,0,0,0,0,0,6,16,2,10,4,14,0,0,0,0,0,0,0,0,2,14,0,0,0,0,0,0,0,0,0,0,5,7,0,0,0,0,0,0,0,0,0,0,0,0,2,14,0,0,0,0,0,0,0,0,0,0,5,7],[7,14,0,0,0,0,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,0,0,11,8,12,17,18,14,0,0,0,0,0,0,10,9,6,11,5,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,9,8,11,13,0,0,0,0,0,0,0,1,17,6,3,7],[18,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,7,3,3,7,0,0,0,0,0,0,3,1,2,5,5,15,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C33.(C3⋊S3) in GAP, Magma, Sage, TeX 

C_3^3.(C_3\rtimes S_3)
 
% in TeX
 
G:=Group("C3^3.(C3:S3)");
 
// GroupNames label
 
G:=SmallGroup(486,45);
 
// by ID
 
G=gap.SmallGroup(486,45);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1993,1951,2162,224,176,6915,2817,735,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=f^2=1,d^3=b*c^-1,e^3=c,e*a*e^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a*c^-1,f*a*f=a*b^-1*c^-1,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,f*c*f=c^-1,e*d*e^-1=a*c^-1*d,f*d*f=b^-1*c*d^2,f*e*f=c^-1*e^2>;
 
// generators/relations
 

Export

Character table of C33.(C3⋊S3) in TeX

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