C3xHe3.3S3 - GroupNames

G = C₃×He₃.₃S₃ order 486 = 2·3⁵

Direct product of C₃ and He₃.₃S₃

direct product, non-abelian, supersoluble, monomial

Aliases: C₃×He₃.₃S₃, (C₃²×C₉)⋊₁₅S₃, He₃.C₃⋊₇C₆, He₃.₆(C₃×S₃), (C₃×He₃).₁₅S₃, C₃³.₃₃(C₃⋊S₃), 3_-¹⁺²⋊₂(C₃×S₃), (C₃×3_-¹⁺²)⋊₆S₃, C₃².₅(He₃⋊C₂), (C₃×C₉)⋊₁₃(C₃×S₃), (C₃×He₃.C₃)⋊₁C₂, C₃².₃(C₃×C₃⋊S₃), C₃.₆(C₃×He₃⋊C₂), SmallGroup(486,168)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — He₃.C₃ — C₃×He₃.₃S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — He₃.C₃ — C₃×He₃.C₃ — C₃×He₃.₃S₃

Lower central

He₃.C₃ — C₃×He₃.₃S₃

Upper central

C₁ — C₃

Generators and relations for C₃×He₃.₃S₃
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=fbf=bc^-1, be=eb, cd=dc, ce=ec, ede^-1=b^-1cd, fdf=d^-1, fef=ce² >

Subgroups: 524 in 96 conjugacy classes, 20 normal (14 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, C₃×C₉, He₃, He₃, 3_-¹⁺², 3_-¹⁺², C₃³, C₃³, C₃×D₉, C₃²⋊C₆, C₉⋊C₆, S₃×C₃², C₃×C₃⋊S₃, He₃.C₃, He₃.C₃, C₃²×C₉, C₃×He₃, C₃×3_-¹⁺², He₃.₃S₃, C₃²×D₉, C₃×C₃²⋊C₆, C₃×C₉⋊C₆, C₃×He₃.C₃, C₃×He₃.₃S₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃×S₃, C₃⋊S₃, He₃⋊C₂, C₃×C₃⋊S₃, He₃.₃S₃, C₃×He₃⋊C₂, C₃×He₃.₃S₃

Smallest permutation representation of C₃×He₃.₃S₃
►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)

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

39 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	3	6	6
type	+	+			+	+	+					+
image	C₁	C₂	C₃	C₆	S₃	S₃	S₃	C₃×S₃	C₃×S₃	C₃×S₃	He₃⋊C₂	He₃.₃S₃	C₃×He₃.₃S₃
kernel	C₃×He₃.₃S₃	C₃×He₃.C₃	He₃.₃S₃	He₃.C₃	C₃²×C₉	C₃×He₃	C₃×3_-¹⁺²	C₃×C₉	He₃	3_-¹⁺²	C₃²	C₃	C₁
# reps	1	1	2	2	1	1	2	2	2	4	12	3	6

Matrix representation of C₃×He₃.₃S₃ ►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
6	7	0	0	0	0
3	0	11	0	0	0
0	0	0	11	0	0
0	0	0	9	1	0
5	0	0	11	0	7

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
16	0	0	0	0	11

6	6	0	0	0	0
0	13	1	0	0	0
18	2	0	0	0	0
5	0	0	12	0	6
14	0	0	4	0	13
4	14	0	15	1	7

16	0	0	0	0	0
0	16	0	0	0	0
10	0	5	0	0	0
0	0	0	6	0	0
0	0	0	0	6	0
9	0	0	15	0	4

0	0	0	1	0	0
0	0	0	0	1	0
4	0	0	15	0	1
1	0	0	0	0	0
0	1	0	0	0	0
4	0	1	15	0	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],[1,6,3,0,0,5,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,9,11,0,0,0,0,1,0,0,0,0,0,0,7],[7,0,0,0,0,16,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[6,0,18,5,14,4,6,13,2,0,0,14,0,1,0,0,0,0,0,0,0,12,4,15,0,0,0,0,0,1,0,0,0,6,13,7],[16,0,10,0,0,9,0,16,0,0,0,0,0,0,5,0,0,0,0,0,0,6,0,15,0,0,0,0,6,0,0,0,0,0,0,4],[0,0,4,1,0,4,0,0,0,0,1,0,0,0,0,0,0,1,1,0,15,0,0,15,0,1,0,0,0,0,0,0,1,0,0,0] >;

C₃×He₃.₃S₃ in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3._3S_3

% in TeX

G:=Group("C3xHe3.3S3");

// GroupNames label

G:=SmallGroup(486,168);

// by ID

G=gap.SmallGroup(486,168);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,8643,303,237,11344,1096,11669]);

// 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=f*b*f=b*c^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*c*d,f*d*f=d^-1,f*e*f=c*e^2>;

// generators/relations

G = C3×He3.3S3 order 486 = 2·35

Direct product of C3 and He3.3S3

G = C₃×He₃.₃S₃ order 486 = 2·3⁵

Direct product of C₃ and He₃.₃S₃