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## G = S3×C3.He3order 486 = 2·35

### Direct product of S3 and C3.He3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S3×C3.He3
 Chief series C1 — C3 — C32 — C33 — C3×3- 1+2 — C3×C3.He3 — S3×C3.He3
 Lower central C3 — C32 — C33 — S3×C3.He3
 Upper central C1 — C3 — C32 — C3.He3

Generators and relations for S3×C3.He3
G = < a,b,c,d,e,f | a3=b2=c3=e3=1, d3=c-1, f3=c, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=cde-1, fef-1=c-1e >

Subgroups: 218 in 72 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, S3×C9, C3×C18, C2×3- 1+2, S3×C32, C3.He3, C3.He3, C32×C9, C3×3- 1+2, C2×C3.He3, S3×C3×C9, S3×3- 1+2, C3×C3.He3, S3×C3.He3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, He3, C2×He3, S3×C32, C3.He3, C2×C3.He3, S3×He3, S3×C3.He3

Smallest permutation representation of S3×C3.He3
On 54 points
Generators in S54
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 19)(8 20)(9 21)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)(37 50)(38 51)(39 52)(40 53)(41 54)(42 46)(43 47)(44 48)(45 49)
(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)
(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)
(10 13 16)(11 14 17)(12 15 18)(28 31 34)(29 32 35)(30 33 36)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(1 33 45 7 30 42 4 36 39)(2 28 37 8 34 43 5 31 40)(3 32 38 9 29 44 6 35 41)(10 51 21 16 48 27 13 54 24)(11 49 19 17 46 25 14 52 22)(12 47 26 18 53 23 15 50 20)

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

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

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

51 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 6C 6D 9A ··· 9F 9G ··· 9L 9M ··· 9R 9S ··· 9X 18A ··· 18F 18G ··· 18L order 1 2 3 3 3 3 3 3 3 3 3 6 6 6 6 9 ··· 9 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 3 1 1 2 2 2 3 3 6 6 3 3 9 9 3 ··· 3 6 ··· 6 9 ··· 9 18 ··· 18 9 ··· 9 27 ··· 27

51 irreducible representations

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

Matrix representation of S3×C3.He3 in GL5(𝔽19)

 0 1 0 0 0 18 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 18 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 17
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 11
,
 7 0 0 0 0 0 7 0 0 0 0 0 0 16 0 0 0 0 0 17 0 0 17 0 0

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

S3×C3.He3 in GAP, Magma, Sage, TeX

S_3\times C_3.{\rm He}_3
% in TeX

G:=Group("S3xC3.He3");
// GroupNames label

G:=SmallGroup(486,124);
// by ID

G=gap.SmallGroup(486,124);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,176,303,11669]);
// Polycyclic

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

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