C3:S3xHe3 - GroupNames

G = C₃⋊S₃×He₃ order 486 = 2·3⁵

Direct product of C₃⋊S₃ and He₃

direct product, metabelian, supersoluble, monomial

Aliases: C₃⋊S₃×He₃, C₃⁴⋊₉C₆, C₃⋊(S₃×He₃), (C₃×He₃)⋊₂₁S₃, C₃³⋊₁₄(C₃×S₃), (C₃²×He₃)⋊₅C₂, C₃²⋊₄(C₂×He₃), C₃³.₅₄(C₃×C₆), C₃².₄₇(S₃×C₃²), C₃²⋊₄(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

G = C3⋊S3×He3 order 486 = 2·35

Direct product of C3⋊S3 and He3

G = C₃⋊S₃×He₃ order 486 = 2·3⁵

Direct product of C₃⋊S₃ and He₃