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## G = C3⋊S3×He3order 486 = 2·35

### Direct product of C3⋊S3 and He3

Aliases: C3⋊S3×He3, C349C6, C3⋊(S3×He3), (C3×He3)⋊21S3, C3314(C3×S3), (C32×He3)⋊5C2, C324(C2×He3), C33.54(C3×C6), C32.47(S3×C32), C324(C3×C3⋊S3), (C32×C3⋊S3)⋊3C3, C3.9(C32×C3⋊S3), (C3×C3⋊S3).7C32, SmallGroup(486,231)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3⋊S3×He3
 Chief series C1 — C3 — C32 — C33 — C34 — C32×He3 — C3⋊S3×He3
 Lower central C32 — C33 — C3⋊S3×He3
 Upper central C1 — 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, cac=a-1, 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: 1300 in 324 conjugacy classes, 49 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, C3×C6, He3, He3, C33, C33, C33, C2×He3, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C3×He3, C3×He3, C34, S3×He3, C32×C3⋊S3, C32×He3, C3⋊S3×He3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, He3, C2×He3, S3×C32, C3×C3⋊S3, S3×He3, C32×C3⋊S3, 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 5 8)(2 6 9)(3 4 7)(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 52 49)(47 53 50)(48 54 51)
(1 28)(2 30)(3 29)(4 32)(5 31)(6 33)(7 35)(8 34)(9 36)(10 38)(11 37)(12 39)(13 41)(14 40)(15 42)(16 44)(17 43)(18 45)(19 47)(20 46)(21 48)(22 50)(23 49)(24 51)(25 53)(26 52)(27 54)
(1 20 11)(2 21 12)(3 19 10)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)
(1 6 7)(2 4 8)(3 5 9)(10 14 18)(11 15 16)(12 13 17)(19 23 27)(20 24 25)(21 22 26)(28 33 35)(29 31 36)(30 32 34)(37 42 44)(38 40 45)(39 41 43)(46 51 53)(47 49 54)(48 50 52)
(1 21 14)(2 19 15)(3 20 13)(4 23 16)(5 24 17)(6 22 18)(7 26 10)(8 27 11)(9 25 12)(28 48 40)(29 46 41)(30 47 42)(31 51 43)(32 49 44)(33 50 45)(34 54 37)(35 52 38)(36 53 39)

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

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

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

66 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 3O ··· 3V 3W ··· 3BB 6A 6B 6C ··· 6J order 1 2 3 3 3 ··· 3 3 ··· 3 3 ··· 3 6 6 6 ··· 6 size 1 9 1 1 2 ··· 2 3 ··· 3 6 ··· 6 9 9 27 ··· 27

66 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 type + + + image C1 C2 C3 C6 S3 C3×S3 He3 C2×He3 S3×He3 kernel C3⋊S3×He3 C32×He3 C32×C3⋊S3 C34 C3×He3 C33 C3⋊S3 C32 C3 # reps 1 1 8 8 4 32 2 2 8

Matrix representation of C3⋊S3×He3 in GL7(𝔽7)

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

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

C3⋊S3×He3 in GAP, Magma, Sage, TeX

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

G:=Group("C3:S3xHe3");
// GroupNames label

G:=SmallGroup(486,231);
// by ID

G=gap.SmallGroup(486,231);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,3244,11669]);
// 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,c*a*c=a^-1,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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