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## G = (C3×He3)⋊S3order 486 = 2·35

### 2nd semidirect product of C3×He3 and S3 acting faithfully

Aliases: C32⋊C94S3, (C3×He3)⋊2S3, C33.1(C3⋊S3), C3.1(He3⋊S3), C3.1(C33⋊S3), C32.24He33C2, C32.13(He3⋊C2), SmallGroup(486,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C32.24He3 — (C3×He3)⋊S3
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C32.24He3 — (C3×He3)⋊S3
 Lower central C32.24He3 — (C3×He3)⋊S3
 Upper central C1

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

Subgroups: 1438 in 94 conjugacy classes, 13 normal (5 characteristic)
C1, C2, 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, C3×He3, C32⋊D9, He34S3, C32.24He3, (C3×He3)⋊S3
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊S3, He3⋊S3, (C3×He3)⋊S3

Character table of (C3×He3)⋊S3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 6A 6B 9A 9B 9C 9D 9E 9F size 1 81 2 2 2 2 9 9 18 18 18 18 18 18 81 81 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 -1 -1 -1 0 0 -1 -1 2 2 2 -1 orthogonal lifted from S3 ρ4 2 0 2 2 2 2 2 2 -1 2 -1 2 -1 2 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ5 2 0 2 2 2 2 2 2 2 -1 2 -1 2 -1 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 0 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 0 0 2 2 -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 0 0 0 ζ65 ζ6 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 0 0 0 ζ3 ζ32 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 0 0 0 ζ6 ζ65 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 0 0 0 ζ32 ζ3 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ11 6 0 -3 -3 -3 6 0 0 0 -3 0 0 0 3 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ12 6 0 -3 -3 -3 6 0 0 0 0 0 3 0 -3 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ13 6 0 -3 6 -3 -3 0 0 -3 0 3 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 -3 0 3 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ15 6 0 -3 -3 -3 6 0 0 0 3 0 -3 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ16 6 0 -3 6 -3 -3 0 0 3 0 0 0 -3 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ17 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 2ζ98-ζ94+ζ92+ζ9 orthogonal lifted from He3⋊S3 ρ18 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 orthogonal lifted from He3⋊S3 ρ19 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 0 orthogonal lifted from He3⋊S3 ρ20 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 0 orthogonal lifted from He3⋊S3 ρ21 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 0 0 0 -ζ98+2ζ97+ζ94+ζ92 orthogonal lifted from He3⋊S3 ρ22 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 0 0 0 ζ95+2ζ94-ζ92+ζ9 orthogonal lifted from He3⋊S3

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

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

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

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

Matrix representation of (C3×He3)⋊S3 in GL12(𝔽19)

 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 7 15 1 0 0 1 0 0 0 0 0 0 15 16 0 18 18 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 18 1 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 1 0 18 0 1 0 0 0 0 0 0 1 0 18 1 18 18
,
 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 18 0 0 0 0 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 7 15 0 1 1 0 0 0 0 0 0 0 7 15 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 18 2 18 18 0 0 0 0 0 0 18 1 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 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 1 18 0 1 18 18 0 0 0 0 0 0 18 0 1 18 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 0 18 0 0 0 0 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 8 1 18 0 18 18 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 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 2 18 18 2 17 18 0 0 0 0 0 0 1 18 0 1 18 18 0 0 0 0 0 0 2 18 0 1 18 18
,
 0 0 0 0 18 1 0 0 0 0 0 0 8 1 18 18 17 18 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 15 0 12 12 3 15 0 0 0 0 0 0 15 0 13 12 3 15 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 18 0 0 0 1 0 0 0 0 0 0 0 18 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 8 1 18 18 17 18 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 4 15 0 0 1 0 0 0 0 0 0 0 4 15 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 0 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 17 1 1 17 1 2 0 0 0 0 0 0 18 1 0 18 1 1 0 0 0 0 0 0 18 1 1 18 1 1

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

(C3×He3)⋊S3 in GAP, Magma, Sage, TeX

(C_3\times {\rm He}_3)\rtimes S_3
% in TeX

G:=Group("(C3xHe3):S3");
// GroupNames label

G:=SmallGroup(486,43);
// by ID

G=gap.SmallGroup(486,43);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,224,3027,2817,1383,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,f*b*f=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=b*c^-1,e*b*e^-1=a*b*c,c*d=d*c,c*e=e*c,f*c*f=c^-1,e*d*e^-1=a*b*d,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

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