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## G = (C3×He3).S3order 486 = 2·35

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

Aliases: C32⋊C95S3, (C3×He3).4S3, C33.2(C3⋊S3), C3.2(C33⋊S3), C33.C321C2, (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, He34S3, 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 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I size 1 81 2 2 2 2 9 9 18 18 18 81 81 18 18 18 18 18 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 2 2 2 2 2 2 -1 -1 -1 0 0 -1 -1 -1 2 -1 -1 2 2 -1 orthogonal lifted from S3 ρ4 2 0 2 2 2 2 2 2 -1 -1 -1 0 0 2 -1 -1 -1 2 2 -1 -1 -1 orthogonal lifted from S3 ρ5 2 0 2 2 2 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 0 2 2 2 2 2 2 -1 -1 -1 0 0 -1 2 2 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ7 3 -1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 ζ6 ζ65 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ8 3 1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 ζ3 ζ32 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ9 3 -1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 ζ65 ζ6 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ10 3 1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 ζ32 ζ3 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ11 6 0 6 -3 -3 -3 0 0 0 3 -3 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ12 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 -3 0 0 0 0 0 0 3 orthogonal lifted from C33⋊S3 ρ13 6 0 6 -3 -3 -3 0 0 3 -3 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ14 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 3 0 0 0 -3 0 0 0 0 orthogonal lifted from C33⋊S3 ρ15 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 -3 orthogonal lifted from C33⋊S3 ρ16 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 3 -3 0 0 0 orthogonal lifted from C33⋊S3 ρ17 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 3 -3 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ18 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 -3 0 0 0 0 3 0 0 0 orthogonal lifted from C33⋊S3 ρ19 6 0 6 -3 -3 -3 0 0 -3 0 3 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ20 6 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 0 0 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 0 orthogonal lifted from He3.3S3 ρ21 6 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 0 orthogonal lifted from He3.3S3 ρ22 6 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 0 0 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 0 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)

 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 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -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 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 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 -1 -1 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 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 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 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 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 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 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

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

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