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, He₃⋊₇(C₃×S₃), (S₃×He₃)⋊₂C₃, (C₃×He₃)⋊₆C₆, (C₃²×C₉)⋊₄C₆, C₃.₇(S₃×He₃), (C₃×S₃).₄He₃, C₃³.₂₇(C₃×C₆), C₃².₁₆(C₂×He₃), (S₃×C₃²).₃C₃², C₃².₂₄(S₃×C₃²), (S₃×C₃×C₉)⋊₃C₃, (C₃×C₉)⋊₂₆(C₃×S₃), C₃⋊(C₂×He₃⋊C₃), (C₃×He₃⋊C₃)⋊₂C₂, SmallGroup(486,123)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — S₃×He₃⋊C₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×He₃ — 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=f³=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece^-1=fcf^-1=cd^-1, de=ed, df=fd, fef^-1=cde >

Subgroups: 488 in 90 conjugacy classes, 24 normal (15 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₁₈, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, He₃, He₃, C₃³, C₃³, S₃×C₉, C₃×C₁₈, C₂×He₃, S₃×C₃², S₃×C₃², He₃⋊C₃, He₃⋊C₃, C₃²×C₉, C₃×He₃, C₂×He₃⋊C₃, S₃×C₃×C₉, S₃×He₃, 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 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 29)(2 28)(3 30)(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 3 2)(7 8 9)(13 14 15)(16 18 17)(19 20 21)(22 24 23)(28 29 30)(34 36 35)(40 42 41)(43 44 45)(46 48 47)(49 50 51)

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

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

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

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

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

51 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	3I	3J	···	3O	3P	···	3U	6A	6B	6C	6D	6E	···	6J	9A	···	9F	9G	···	9L	18A	···	18F
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	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

51 irreducible representations

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

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

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

18	0	0	0	0
1	1	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	7	0	0
0	0	0	1	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	17
0	0	17	0	0
0	0	0	5	0

11	0	0	0	0
0	11	0	0	0
0	0	0	0	9
0	0	9	0	0
0	0	0	4	0

G:=sub<GL(5,GF(19))| [0,18,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,1,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,11],[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],[7,0,0,0,0,0,7,0,0,0,0,0,0,17,0,0,0,0,0,5,0,0,17,0,0],[11,0,0,0,0,0,11,0,0,0,0,0,0,9,0,0,0,0,0,4,0,0,9,0,0] >;

S₃×He₃⋊C₃ in GAP, Magma, Sage, TeX

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

% in TeX

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

// GroupNames label

G:=SmallGroup(486,123);

// by ID

G=gap.SmallGroup(486,123);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,873,303,11669]);

// Polycyclic

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