C3xHe3.S3 - GroupNames

G = C₃×He₃.S₃ order 486 = 2·3⁵

Direct product of C₃ and He₃.S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₃×He₃.S₃, C₉⋊S₃⋊₆C₃², (C₃²×C₉)⋊₈C₆, He₃.C₃⋊₈C₆, (C₃×He₃).₈S₃, He₃.₁(C₃×S₃), C₃³.₅₈(C₃×S₃), C₃².₁₄(S₃×C₃²), C₃².₄₃(C₃²⋊C₆), (C₃×C₉⋊S₃)⋊₂C₃, (C₃×C₉)⋊₁₀(C₃×C₆), (C₃×He₃.C₃)⋊₃C₂, C₃.₅(C₃×C₃²⋊C₆), SmallGroup(486,119)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₉ — C₃×He₃.S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃²×C₉ — C₃×He₃.C₃ — C₃×He₃.S₃

Lower central

C₃×C₉ — 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=f²=1, e³=fcf=c^-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd^-1=bc^-1, be=eb, fbf=b^-1, cd=dc, ce=ec, ede^-1=b^-1cd, df=fd, fef=ce² >

Subgroups: 492 in 78 conjugacy classes, 22 normal (13 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, C₃×C₉, He₃, He₃, 3_-¹⁺², C₃³, C₃³, C₃×D₉, C₃²⋊C₆, C₉⋊S₃, S₃×C₃², C₃×C₃⋊S₃, He₃.C₃, He₃.C₃, C₃²×C₉, C₃×He₃, C₃×3_-¹⁺², He₃.S₃, C₃×C₃²⋊C₆, C₃×C₉⋊S₃, C₃×He₃.C₃, C₃×He₃.S₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, C₃²⋊C₆, S₃×C₃², He₃.S₃, C₃×C₃²⋊C₆, C₃×He₃.S₃

Smallest permutation representation of C₃×He₃.S₃
►On 54 points

Generators in S₅₄

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

(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)

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

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

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

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

39 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	3I	···	3N	6A	···	6H	9A	···	9I	9J	···	9O
order	1	2	3	3	3	3	3	3	3	3	3	···	3	6	···	6	9	···	9	9	···	9
size	1	27	1	1	2	2	2	6	6	6	9	···	9	27	···	27	6	···	6	18	···	18

39 irreducible representations

dim	1	1	1	1	1	1	2	2	2	6	6	6	6
type	+	+					+			+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	S₃	C₃×S₃	C₃×S₃	C₃²⋊C₆	He₃.S₃	C₃×C₃²⋊C₆	C₃×He₃.S₃
kernel	C₃×He₃.S₃	C₃×He₃.C₃	He₃.S₃	C₃×C₉⋊S₃	He₃.C₃	C₃²×C₉	C₃×He₃	He₃	C₃³	C₃²	C₃	C₃	C₁
# reps	1	1	6	2	6	2	1	6	2	1	3	2	6

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

11	0	0	0	0	0
0	11	0	0	0	0
0	0	11	0	0	0
0	0	0	11	0	0
0	0	0	0	11	0
0	0	0	0	0	11

1	0	0	0	0	0
0	7	0	0	0	0
1	18	11	0	0	0
0	0	0	1	0	0
8	12	0	8	11	0
8	0	0	1	0	7

11	0	0	0	0	0
0	11	0	0	0	0
0	0	11	0	0	0
0	0	0	7	0	0
12	7	0	0	7	0
1	18	0	0	0	7

12	8	6	0	0	0
1	0	0	0	0	0
0	0	7	0	0	0
8	11	0	1	0	6
0	0	18	0	0	18
0	0	11	0	1	18

6	0	0	0	0	0
0	4	0	0	0	0
0	9	6	0	0	0
0	0	0	16	0	0
3	3	0	14	5	0
7	3	0	0	0	16

0	0	0	1	0	0
1	18	0	1	6	0
0	0	0	0	11	1
1	0	0	0	0	0
0	0	0	0	1	0
0	0	1	0	8	0

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[1,0,1,0,8,8,0,7,18,0,12,0,0,0,11,0,0,0,0,0,0,1,8,1,0,0,0,0,11,0,0,0,0,0,0,7],[11,0,0,0,12,1,0,11,0,0,7,18,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[12,1,0,8,0,0,8,0,0,11,0,0,6,0,7,0,18,11,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,6,18,18],[6,0,0,0,3,7,0,4,9,0,3,3,0,0,6,0,0,0,0,0,0,16,14,0,0,0,0,0,5,0,0,0,0,0,0,16],[0,1,0,1,0,0,0,18,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,6,11,0,1,8,0,0,1,0,0,0] >;

C₃×He₃.S₃ in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3.S_3

% in TeX

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

// GroupNames label

G:=SmallGroup(486,119);

// by ID

G=gap.SmallGroup(486,119);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,8643,873,237,3244,3250,11669]);

// Polycyclic

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

// 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