S3xHe3.C3 - GroupNames

G = S₃×He₃.C₃ order 486 = 2·3⁵

Direct product of S₃ and He₃.C₃

direct product, metabelian, supersoluble, monomial

Aliases: S₃×He₃.C₃, (S₃×He₃).C₃, (C₃²×C₉)⋊₃C₆, C₃.₆(S₃×He₃), (C₃×He₃).₆C₆, (C₃×S₃).₃He₃, He₃.₁₁(C₃×S₃), C₃³.₂₆(C₃×C₆), C₃².₁₅(C₂×He₃), (S₃×C₃²).₂C₃², C₃².₂₃(S₃×C₃²), (C₃×3_-¹⁺²)⋊₇C₆, 3_-¹⁺²⋊₆(C₃×S₃), (S₃×3_-¹⁺²)⋊₂C₃, (S₃×C₃×C₉)⋊₂C₃, C₃⋊(C₂×He₃.C₃), (C₃×C₉)⋊₂₅(C₃×S₃), (C₃×He₃.C₃)⋊₆C₂, SmallGroup(486,120)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — S₃×He₃.C₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₉ — C₃×He₃.C₃ — S₃×He₃.C₃

Lower central

C₃ — C₃² — C₃³ — S₃×He₃.C₃

Upper central

C₁ — C₃ — C₃² — He₃.C₃

Generators and relations for S₃×He₃.C₃
G = < a,b,c,d,e,f | a³=b²=c³=d³=e³=1, f³=d, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece^-1=cd^-1, cf=fc, de=ed, df=fd, fef^-1=cd^-1e >

Subgroups: 308 in 78 conjugacy classes, 24 normal (18 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₁₈, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, He₃, He₃, 3_-¹⁺², 3_-¹⁺², C₃³, C₃³, S₃×C₉, C₃×C₁₈, C₂×He₃, C₂×3_-¹⁺², S₃×C₃², S₃×C₃², He₃.C₃, He₃.C₃, C₃²×C₉, C₃×He₃, C₃×3_-¹⁺², C₂×He₃.C₃, S₃×C₃×C₉, S₃×He₃, S₃×3_-¹⁺², C₃×He₃.C₃, S₃×He₃.C₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, He₃, C₂×He₃, S₃×C₃², He₃.C₃, C₂×He₃.C₃, S₃×He₃, S₃×He₃.C₃

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

Generators in S₅₄

(1 4 7)(2 5 8)(3 6 9)(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 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(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)

(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 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(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)

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

(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)

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

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

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

51 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	3I	3J	3K	3L	3M	6A	6B	6C	6D	6E	6F	9A	···	9F	9G	···	9L	9M	9N	9O	9P	9Q	9R	9S	9T	18A	···	18F	18G	18H	18I	18J
order	1	2	3	3	3	3	3	3	3	3	3	3	3	3	3	6	6	6	6	6	6	9	···	9	9	···	9	9	9	9	9	9	9	9	9	18	···	18	18	18	18	18
size	1	3	1	1	2	2	2	3	3	6	6	9	9	18	18	3	3	9	9	27	27	3	···	3	6	···	6	9	9	9	9	18	18	18	18	9	···	9	27	27	27	27

51 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	3	3	3	3	6	6
type	+	+							+
image	C₁	C₂	C₃	C₃	C₃	C₆	C₆	C₆	S₃	C₃×S₃	C₃×S₃	C₃×S₃	He₃	C₂×He₃	He₃.C₃	C₂×He₃.C₃	S₃×He₃	S₃×He₃.C₃
kernel	S₃×He₃.C₃	C₃×He₃.C₃	S₃×C₃×C₉	S₃×He₃	S₃×3_-¹⁺²	C₃²×C₉	C₃×He₃	C₃×3_-¹⁺²	He₃.C₃	C₃×C₉	He₃	3_-¹⁺²	C₃×S₃	C₃²	S₃	C₃	C₃	C₁
# reps	1	1	2	2	4	2	2	4	1	2	2	4	2	2	6	6	2	6

Matrix representation of S₃×He₃.C₃ ►in GL₅(𝔽₁₉)

18	1	0	0	0
18	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

1	0	0	0	0
0	1	0	0	0
0	0	7	0	0
0	0	0	7	0
0	0	0	0	7

7	0	0	0	0
0	7	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

11	0	0	0	0
0	11	0	0	0
0	0	6	0	0
0	0	0	6	0
0	0	0	0	4

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

S₃×He₃.C₃ in GAP, Magma, Sage, TeX

S_3\times {\rm He}_3.C_3

% in TeX

G:=Group("S3xHe3.C3");

// GroupNames label

G:=SmallGroup(486,120);

// by ID

G=gap.SmallGroup(486,120);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,176,873,11669]);

// Polycyclic

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

// generators/relations

G = S3×He3.C3 order 486 = 2·35

Direct product of S3 and He3.C3

G = S₃×He₃.C₃ order 486 = 2·3⁵

Direct product of S₃ and He₃.C₃