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## G = C3×S3×He3order 486 = 2·35

### Direct product of C3, S3 and He3

Aliases: C3×S3×He3, C347C6, C3⋊(C6×He3), C339(C3×C6), (S3×C33)⋊2C3, (C3×He3)⋊21C6, C3317(C3×S3), (S3×C32)⋊C32, C3.5(S3×C33), (C32×He3)⋊2C2, C323(C2×He3), (C3×S3).2C33, C324(S3×C32), C32.14(C32×C6), SmallGroup(486,223)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×S3×He3
G = < a,b,c,d,e,f | a3=b3=c2=d3=e3=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=de-1, ef=fe >

Subgroups: 1298 in 393 conjugacy classes, 96 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3×S3, C3×S3, C3×C6, He3, He3, C33, C33, C33, C2×He3, S3×C32, S3×C32, S3×C32, C32×C6, C3×He3, C3×He3, C3×He3, C34, S3×He3, C6×He3, S3×C33, C32×He3, C3×S3×He3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, He3, C33, C2×He3, S3×C32, C32×C6, C3×He3, S3×He3, C6×He3, S3×C33, C3×S3×He3

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

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

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

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

99 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 3R ··· 3AO 3AP ··· 3BM 6A ··· 6H 6I ··· 6AF order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 size 1 3 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 3 ··· 3 9 ··· 9

99 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 6 type + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 He3 C2×He3 S3×He3 kernel C3×S3×He3 C32×He3 S3×He3 S3×C33 C3×He3 C34 C3×He3 He3 C33 C3×S3 C32 C3 # reps 1 1 18 8 18 8 1 18 8 6 6 6

Matrix representation of C3×S3×He3 in GL5(𝔽7)

 4 0 0 0 0 0 4 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2
,
 4 3 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 6 0 0 0 0 3 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 4 3 0 0 0 2 0 0 0 0 0 4
,
 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2
,
 4 0 0 0 0 0 4 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 2 6 5

G:=sub<GL(5,GF(7))| [4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[4,0,0,0,0,3,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,3,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,4,2,0,0,0,3,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[4,0,0,0,0,0,4,0,0,0,0,0,2,0,2,0,0,0,0,6,0,0,0,4,5] >;

C3×S3×He3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xS3xHe3");
// GroupNames label

G:=SmallGroup(486,223);
// by ID

G=gap.SmallGroup(486,223);
# by ID

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

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

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