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G = (C3×He3).S3  order 486 = 2·35

4th non-split extension by C3×He3 of S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C32⋊C9⋊5S3, (C3×He3).4S3, C33.2(C3⋊S3), C3.2(C33⋊S3), C33.C32⋊1C2, (C3×3- 1+2)⋊1S3, C3.1(He3.3S3), C32.14(He3⋊C2), SmallGroup(486,44)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for (C3×He3).S3
 G = < a,b,c,d,e,f | a3=b3=c3=d3=f2=1, e3=fcf=c-1, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, bc=cb, dbd-1=fbf=bc-1, ebe-1=a-1bc-1, cd=dc, ce=ec, ede-1=a-1b-1d, fdf=d-1, fef=ce2 >

Subgroups: 1096 in 92 conjugacy classes, 13 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, He3, 3- 1+2, C33, C33, C32⋊C6, C9⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C3×He3, C3×3- 1+2, C32⋊D9, He3⋊4S3, C33.S3, C33.C32, (C3×He3).S3
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊S3, He3.3S3, (C3×He3).S3

Character table of (C3×He3).S3

 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
ρ11606-3-3-30003-300000000000    orthogonal lifted from C33⋊S3
ρ1260-3-36-300000000-30000003    orthogonal lifted from C33⋊S3
ρ13606-3-3-3003-3000000000000    orthogonal lifted from C33⋊S3
ρ1460-36-3-300000003000-30000    orthogonal lifted from C33⋊S3
ρ1560-3-36-3000000000300000-3    orthogonal lifted from C33⋊S3
ρ1660-36-3-3000000000003-3000    orthogonal lifted from C33⋊S3
ρ1760-3-36-3000000003-3000000    orthogonal lifted from C33⋊S3
ρ1860-36-3-30000000-300003000    orthogonal lifted from C33⋊S3
ρ19606-3-3-300-30300000000000    orthogonal lifted from C33⋊S3
ρ2060-3-3-3600000000002ζ95+ζ94+ζ92-ζ900ζ98+ζ94-ζ92+2ζ9ζ98+ζ97-ζ94+2ζ920    orthogonal lifted from He3.3S3
ρ2160-3-3-360000000000ζ98+ζ97-ζ94+2ζ92002ζ95+ζ94+ζ92-ζ9ζ98+ζ94-ζ92+2ζ90    orthogonal lifted from He3.3S3
ρ2260-3-3-360000000000ζ98+ζ94-ζ92+2ζ900ζ98+ζ97-ζ94+2ζ922ζ95+ζ94+ζ92-ζ90    orthogonal lifted from He3.3S3

Smallest permutation representation of (C3×He3).S3
►On 81 points
Generators in S81
(1 30 44)(2 31 45)(3 32 37)(4 33 38)(5 34 39)(6 35 40)(7 36 41)(8 28 42)(9 29 43)(10 74 23)(11 75 24)(12 76 25)(13 77 26)(14 78 27)(15 79 19)(16 80 20)(17 81 21)(18 73 22)(46 69 57)(47 70 58)(48 71 59)(49 72 60)(50 64 61)(51 65 62)(52 66 63)(53 67 55)(54 68 56)

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

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

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

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

Matrix representation of (C3×He3).S3 ►in GL12(ℤ)

100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
000000010000
000000-1-10000
000000000100
00000000-1-100
000000000001
0000000000-1-1
,
-1-10000000000
100000000000
000100000000
00-1-100000000
000010000000
000001000000
000000100000
000000010000
00000000-1-100
000000001000
000000000001
0000000000-1-1
,
010000000000
-1-10000000000
000100000000
00-1-100000000
000001000000
0000-1-1000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
000001000000
0000-1-1000000
-1-10000000000
100000000000
001000000000
000100000000
000000010000
000000-1-10000
000000001000
000000000100
000000000010
000000000001
,
0000-1-1000000
000010000000
100000000000
010000000000
001000000000
000100000000
000000000010
000000000001
000000100000
000000010000
000000001000
000000000100
,
-100000000000
110000000000
000011000000
00000-1000000
001100000000
000-100000000
000000100000
000000-1-10000
000000000010
0000000000-1-1
000000001000
00000000-1-100

G:=sub<GL(12,Integers())| [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,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-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,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-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],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,-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,-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,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0] >;
 

(C3×He3).S3 in GAP, Magma, Sage, TeX 

(C_3\times {\rm He}_3).S_3
 
% in TeX
 
G:=Group("(C3xHe3).S3");
 
// GroupNames label
 
G:=SmallGroup(486,44);
 
// by ID
 
G=gap.SmallGroup(486,44);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,2162,224,176,6915,873,1383,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=f^2=1,e^3=f*c*f=c^-1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,b*c=c*b,d*b*d^-1=f*b*f=b*c^-1,e*b*e^-1=a^-1*b*c^-1,c*d=d*c,c*e=e*c,e*d*e^-1=a^-1*b^-1*d,f*d*f=d^-1,f*e*f=c*e^2>;
 
// generators/relations
 

Export

Character table of (C3×He3).S3 in TeX

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