C3xHe3:S3 - GroupNames

G = C₃×He₃⋊S₃ order 486 = 2·3⁵

Direct product of C₃ and He₃⋊S₃

direct product, non-abelian, supersoluble, monomial

Aliases: C₃×He₃⋊S₃, He₃⋊₃(C₃×S₃), (C₃×He₃)⋊₁₁S₃, (C₃²×C₉)⋊₁₆S₃, He₃⋊C₃⋊₁₂C₆, C₃³.₃₄(C₃⋊S₃), C₃².₆(He₃⋊C₂), (C₃×C₉)⋊₁₄(C₃×S₃), C₃².₅(C₃×C₃⋊S₃), (C₃×He₃⋊C₃)⋊₅C₂, C₃.₇(C₃×He₃⋊C₂), SmallGroup(486,171)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — He₃⋊C₃ — C₃×He₃⋊S₃

Chief series

C₁ — C₃ — C₃² — He₃ — He₃⋊C₃ — C₃×He₃⋊C₃ — C₃×He₃⋊S₃

Lower central

He₃⋊C₃ — C₃×He₃⋊S₃

Upper central

C₁ — C₃

Generators and relations for C₃×He₃⋊S₃
G = < a,b,c,d,e,f | a³=b³=c³=d³=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe^-1=bc=cb, dbd^-1=bc^-1, bf=fb, cd=dc, ce=ec, fcf=c^-1, ede^-1=b^-1c^-1d, fdf=bc^-1d^-1, fef=e^-1 >

Subgroups: 740 in 108 conjugacy classes, 20 normal (12 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, C₃×C₉, He₃, He₃, C₃³, C₃³, C₃×D₉, C₃²⋊C₆, S₃×C₃², C₃×C₃⋊S₃, He₃⋊C₃, He₃⋊C₃, C₃²×C₉, C₃×He₃, He₃⋊S₃, C₃²×D₉, C₃×C₃²⋊C₆, C₃×He₃⋊C₃, C₃×He₃⋊S₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃×S₃, C₃⋊S₃, He₃⋊C₂, C₃×C₃⋊S₃, He₃⋊S₃, C₃×He₃⋊C₂, C₃×He₃⋊S₃

Smallest permutation representation of C₃×He₃⋊S₃
►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)

(4 5 6)(7 9 8)(10 12 11)(16 17 18)(19 20 21)(22 24 23)(31 32 33)(34 36 35)(37 39 38)(43 44 45)(46 47 48)(49 51 50)

(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 30 29)(31 33 32)(34 36 35)(37 39 38)(40 42 41)(43 45 44)(46 48 47)(49 51 50)(52 54 53)

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

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

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

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

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

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

39 conjugacy classes

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

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	He₃⋊S₃	C₃×He₃⋊S₃
kernel	C₃×He₃⋊S₃	C₃×He₃⋊C₃	He₃⋊S₃	He₃⋊C₃	C₃²×C₉	C₃×He₃	C₃×C₉	He₃	C₃²	C₃	C₁
# reps	1	1	2	2	1	3	2	6	12	3	6

Matrix representation of C₃×He₃⋊S₃ ►in GL₆(𝔽₁₉)

7	0	0	0	0	0
0	7	0	0	0	0
0	0	7	0	0	0
0	0	0	7	0	0
0	0	0	0	7	0
0	0	0	0	0	7

1	0	0	0	0	0
0	7	0	0	0	0
0	0	11	0	0	0
0	0	0	1	0	0
0	0	0	0	7	0
0	0	0	0	0	11

7	0	0	0	0	0
0	7	0	0	0	0
0	0	7	0	0	0
0	0	0	11	0	0
0	0	0	0	11	0
0	0	0	0	0	11

0	4	0	0	0	0
0	0	9	0	0	0
9	0	0	0	0	0
0	0	0	0	0	5
0	0	0	17	0	0
0	0	0	0	17	0

0	0	1	0	0	0
1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0

G:=sub<GL(6,GF(19))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,9,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,5,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

C₃×He₃⋊S₃ in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3\rtimes S_3

% in TeX

G:=Group("C3xHe3:S3");

// GroupNames label

G:=SmallGroup(486,171);

// by ID

G=gap.SmallGroup(486,171);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,867,303,1096,652,11669]);

// Polycyclic

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

// generators/relations

G = C3×He3⋊S3 order 486 = 2·35

Direct product of C3 and He3⋊S3

G = C₃×He₃⋊S₃ order 486 = 2·3⁵

Direct product of C₃ and He₃⋊S₃