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## G = S3×C9○He3order 486 = 2·35

### Direct product of S3 and C9○He3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S3×C9○He3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×C9○He3 — S3×C9○He3
 Lower central C3 — C32 — S3×C9○He3
 Upper central C1 — C9 — C9○He3

Generators and relations for S3×C9○He3
G = < a,b,c,d,e,f | a3=b2=c9=d3=f3=1, e1=c6, 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=c3d, ef=fe >

Subgroups: 452 in 219 conjugacy classes, 87 normal (15 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, C18, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, S3×C9, S3×C9, C3×C18, C2×He3, C2×3- 1+2, S3×C32, C32×C9, C3×He3, C3×3- 1+2, C9○He3, C9○He3, S3×C3×C9, S3×He3, S3×3- 1+2, C2×C9○He3, C3×C9○He3, S3×C9○He3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C33, S3×C32, C32×C6, C9○He3, C2×C9○He3, S3×C33, S3×C9○He3

Smallest permutation representation of S3×C9○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 50)(2 51)(3 52)(4 53)(5 54)(6 46)(7 47)(8 48)(9 49)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)(19 44)(20 45)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)
(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 35 44)(2 36 45)(3 28 37)(4 29 38)(5 30 39)(6 31 40)(7 32 41)(8 33 42)(9 34 43)(10 25 47)(11 26 48)(12 27 49)(13 19 50)(14 20 51)(15 21 52)(16 22 53)(17 23 54)(18 24 46)
(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 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(28 31 34)(29 32 35)(30 33 36)(46 52 49)(47 53 50)(48 54 51)

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

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

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

99 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3M 3N ··· 3U 6A 6B 6C ··· 6J 9A ··· 9F 9G ··· 9L 9M ··· 9AB 9AC ··· 9AR 18A ··· 18F 18G ··· 18V 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 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 3 ··· 3 9 ··· 9

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 6 type + + + image C1 C2 C3 C3 C3 C6 C6 C6 S3 C3×S3 C3×S3 C3×S3 C9○He3 C2×C9○He3 S3×C9○He3 kernel S3×C9○He3 C3×C9○He3 S3×C3×C9 S3×He3 S3×3- 1+2 C32×C9 C3×He3 C3×3- 1+2 C9○He3 C3×C9 He3 3- 1+2 S3 C3 C1 # reps 1 1 8 2 16 8 2 16 1 8 2 16 6 6 6

Matrix representation of S3×C9○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
,
 18 0 0 0 0 1 1 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 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 7 0 0 0 0 0 7 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 7

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],[18,1,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[7,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,7] >;

S3×C9○He3 in GAP, Magma, Sage, TeX

S_3\times C_9\circ {\rm He}_3
% in TeX

G:=Group("S3xC9oHe3");
// GroupNames label

G:=SmallGroup(486,226);
// by ID

G=gap.SmallGroup(486,226);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,176,873,11669]);
// Polycyclic

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

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