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## G = C3×3- 1+2.S3order 486 = 2·35

### Direct product of C3 and 3- 1+2.S3

Aliases: C3×3- 1+2.S3, C3.He37C6, (C32×C9).15S3, C33.35(C3⋊S3), C32.7(He3⋊C2), (C3×3- 1+2).7S3, 3- 1+2.1(C3×S3), (C3×C9).21(C3×S3), C32.7(C3×C3⋊S3), (C3×C3.He3)⋊1C2, C3.8(C3×He3⋊C2), SmallGroup(486,174)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3.He3 — C3×3- 1+2.S3
 Chief series C1 — C3 — C32 — C3×C9 — C3.He3 — C3×C3.He3 — C3×3- 1+2.S3
 Lower central C3.He3 — C3×3- 1+2.S3
 Upper central C1 — C3

Generators and relations for C3×3- 1+2.S3
G = < a,b,c,d,e | a3=b9=c3=e2=1, d3=b6, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b4, dbd-1=b7c, ebe=b5c-1, dcd-1=b3c, ce=ec, ede=b3d2 >

Subgroups: 416 in 90 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×D9, C9⋊C6, S3×C32, C3.He3, C3.He3, C32×C9, C3×3- 1+2, 3- 1+2.S3, C32×D9, C3×C9⋊C6, C3×C3.He3, C3×3- 1+2.S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, He3⋊C2, C3×C3⋊S3, 3- 1+2.S3, C3×He3⋊C2, C3×3- 1+2.S3

Smallest permutation representation of C3×3- 1+2.S3
On 54 points
Generators in S54
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(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)
(1 4 7)(3 9 6)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(28 31 34)(30 36 33)(37 43 40)(38 41 44)(46 49 52)(48 54 51)
(1 30 45 7 36 42 4 33 39)(2 34 40 8 31 37 5 28 43)(3 32 41 9 29 38 6 35 44)(10 47 26 16 53 23 13 50 20)(11 48 24 17 54 21 14 51 27)(12 52 19 18 49 25 15 46 22)
(1 10)(2 15)(3 17)(4 16)(5 12)(6 14)(7 13)(8 18)(9 11)(19 31)(20 30)(21 35)(22 28)(23 36)(24 32)(25 34)(26 33)(27 29)(37 52)(38 51)(39 47)(40 49)(41 48)(42 53)(43 46)(44 54)(45 50)

G:=sub<Sym(54)| (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (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), (1,4,7)(3,9,6)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(28,31,34)(30,36,33)(37,43,40)(38,41,44)(46,49,52)(48,54,51), (1,30,45,7,36,42,4,33,39)(2,34,40,8,31,37,5,28,43)(3,32,41,9,29,38,6,35,44)(10,47,26,16,53,23,13,50,20)(11,48,24,17,54,21,14,51,27)(12,52,19,18,49,25,15,46,22), (1,10)(2,15)(3,17)(4,16)(5,12)(6,14)(7,13)(8,18)(9,11)(19,31)(20,30)(21,35)(22,28)(23,36)(24,32)(25,34)(26,33)(27,29)(37,52)(38,51)(39,47)(40,49)(41,48)(42,53)(43,46)(44,54)(45,50)>;

G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (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), (1,4,7)(3,9,6)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(28,31,34)(30,36,33)(37,43,40)(38,41,44)(46,49,52)(48,54,51), (1,30,45,7,36,42,4,33,39)(2,34,40,8,31,37,5,28,43)(3,32,41,9,29,38,6,35,44)(10,47,26,16,53,23,13,50,20)(11,48,24,17,54,21,14,51,27)(12,52,19,18,49,25,15,46,22), (1,10)(2,15)(3,17)(4,16)(5,12)(6,14)(7,13)(8,18)(9,11)(19,31)(20,30)(21,35)(22,28)(23,36)(24,32)(25,34)(26,33)(27,29)(37,52)(38,51)(39,47)(40,49)(41,48)(42,53)(43,46)(44,54)(45,50) );

G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51)], [(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)], [(1,4,7),(3,9,6),(10,16,13),(11,14,17),(19,25,22),(20,23,26),(28,31,34),(30,36,33),(37,43,40),(38,41,44),(46,49,52),(48,54,51)], [(1,30,45,7,36,42,4,33,39),(2,34,40,8,31,37,5,28,43),(3,32,41,9,29,38,6,35,44),(10,47,26,16,53,23,13,50,20),(11,48,24,17,54,21,14,51,27),(12,52,19,18,49,25,15,46,22)], [(1,10),(2,15),(3,17),(4,16),(5,12),(6,14),(7,13),(8,18),(9,11),(19,31),(20,30),(21,35),(22,28),(23,36),(24,32),(25,34),(26,33),(27,29),(37,52),(38,51),(39,47),(40,49),(41,48),(42,53),(43,46),(44,54),(45,50)]])

39 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 6A ··· 6H 9A ··· 9I 9J ··· 9R order 1 2 3 3 3 3 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 size 1 27 1 1 2 2 2 3 ··· 3 27 ··· 27 6 ··· 6 18 ··· 18

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 6 6 type + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 C3×S3 He3⋊C2 3- 1+2.S3 C3×3- 1+2.S3 kernel C3×3- 1+2.S3 C3×C3.He3 3- 1+2.S3 C3.He3 C32×C9 C3×3- 1+2 C3×C9 3- 1+2 C32 C3 C1 # reps 1 1 2 2 1 3 2 6 12 3 6

Matrix representation of C3×3- 1+2.S3 in GL6(𝔽19)

 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 4 5 0 0 0 0 0 15 4 0 0 0 4 15 0 0 0 0 17 4 0 0 0 16 3 4 0 5 0 0 17 4 0 0 16 0
,
 11 0 0 0 0 0 12 7 0 0 0 0 11 0 1 0 0 0 0 0 0 11 0 0 7 0 0 0 7 0 8 0 0 0 0 1
,
 7 0 4 0 0 0 18 0 12 0 0 0 18 1 12 0 0 0 1 0 7 0 1 0 1 0 7 0 0 1 12 0 7 11 0 0
,
 18 0 0 6 0 0 0 0 0 18 1 0 0 0 0 18 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0

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

C3×3- 1+2.S3 in GAP, Magma, Sage, TeX

C_3\times 3_-^{1+2}.S_3
% in TeX

G:=Group("C3xES-(3,1).S3");
// GroupNames label

G:=SmallGroup(486,174);
// by ID

G=gap.SmallGroup(486,174);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,986,4755,303,453,11344,1096,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^9=c^3=e^2=1,d^3=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^4,d*b*d^-1=b^7*c,e*b*e=b^5*c^-1,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=b^3*d^2>;
// generators/relations

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