C3xS3xES-(3,1) - GroupNames

G = C₃×S₃×3_-¹⁺² order 486 = 2·3⁵

Direct product of C₃, S₃ and 3_-¹⁺²

direct product, metabelian, supersoluble, monomial

Aliases: C₃×S₃×3_-¹⁺², C₃⁴.₆C₆, (S₃×C₉)⋊C₃², C₉⋊₂(S₃×C₃²), (C₃²×C₉)⋊₂₄C₆, C₃.₆(S₃×C₃³), C₃⋊(C₆×3_-¹⁺²), (C₃×S₃).₃C₃³, (S₃×C₃³).₂C₃, C₃³.₂₉(C₃×C₆), C₃³.₇₂(C₃×S₃), (S₃×C₃²).₅C₃², C₃².₁₅(C₃²×C₆), C₃².₅₄(S₃×C₃²), (C₃×3_-¹⁺²)⋊₂₁C₆, (C₃²×3_-¹⁺²)⋊₄C₂, C₃²⋊₅(C₂×3_-¹⁺²), (S₃×C₃×C₉)⋊₄C₃, (C₃×C₉)⋊₁₅(C₃×C₆), (C₃×C₉)⋊₂₇(C₃×S₃), SmallGroup(486,225)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃×S₃×3_-¹⁺²

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃×3_-¹⁺² — C₃²×3_-¹⁺² — C₃×S₃×3_-¹⁺²

Lower central

C₃ — C₃² — C₃×S₃×3_-¹⁺²

Upper central

C₁ — C₃² — C₃×3_-¹⁺²

Generators and relations for C₃×S₃×3_-¹⁺²
G = < a,b,c,d,e | a³=b³=c²=d⁹=e³=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d⁴ >

Subgroups: 596 in 258 conjugacy classes, 96 normal (15 characteristic)
C₁, C₂, C₃, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², C₃², C₁₈, C₃×S₃, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, 3_-¹⁺², 3_-¹⁺², C₃³, C₃³, C₃³, S₃×C₉, C₃×C₁₈, C₂×3_-¹⁺², S₃×C₃², S₃×C₃², S₃×C₃², C₃²×C₆, C₃²×C₉, C₃×3_-¹⁺², C₃×3_-¹⁺², C₃×3_-¹⁺², C₃⁴, S₃×C₃×C₉, S₃×3_-¹⁺², C₆×3_-¹⁺², S₃×C₃³, C₃²×3_-¹⁺², C₃×S₃×3_-¹⁺²
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, 3_-¹⁺², C₃³, C₂×3_-¹⁺², S₃×C₃², C₃²×C₆, C₃×3_-¹⁺², S₃×3_-¹⁺², C₆×3_-¹⁺², S₃×C₃³, C₃×S₃×3_-¹⁺²

Smallest permutation representation of C₃×S₃×3_-¹⁺²
►On 54 points

Generators in S₅₄

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

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

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

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

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

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

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

99 conjugacy classes

class	1	2	3A	···	3H	3I	···	3Q	3R	···	3W	3X	···	3AC	6A	···	6H	6I	···	6N	9A	···	9R	9S	···	9AJ	18A	···	18R
order	1	2	3	···	3	3	···	3	3	···	3	3	···	3	6	···	6	6	···	6	9	···	9	9	···	9	18	···	18
size	1	3	1	···	1	2	···	2	3	···	3	6	···	6	3	···	3	9	···	9	3	···	3	6	···	6	9	···	9

99 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	3	3	6
type	+	+							+
image	C₁	C₂	C₃	C₃	C₃	C₆	C₆	C₆	S₃	C₃×S₃	C₃×S₃	C₃×S₃	3_-¹⁺²	C₂×3_-¹⁺²	S₃×3_-¹⁺²
kernel	C₃×S₃×3_-¹⁺²	C₃²×3_-¹⁺²	S₃×C₃×C₉	S₃×3_-¹⁺²	S₃×C₃³	C₃²×C₉	C₃×3_-¹⁺²	C₃⁴	C₃×3_-¹⁺²	C₃×C₉	3_-¹⁺²	C₃³	C₃×S₃	C₃²	C₃
# reps	1	1	6	18	2	6	18	2	1	6	18	2	6	6	6

Matrix representation of C₃×S₃×3_-¹⁺² ►in GL₅(𝔽₁₉)

11	0	0	0	0
0	11	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

0	1	0	0	0
1	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

11	0	0	0	0
0	11	0	0	0
0	0	11	11	11
0	0	0	0	7
0	0	15	8	8

1	0	0	0	0
0	1	0	0	0
0	0	7	8	7
0	0	0	11	0
0	0	0	0	1

G:=sub<GL(5,GF(19))| [11,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[7,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,11,0,0,0,0,0,11,0,15,0,0,11,0,8,0,0,11,7,8],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,8,11,0,0,0,7,0,1] >;

C₃×S₃×3_-¹⁺² in GAP, Magma, Sage, TeX

C_3\times S_3\times 3_-^{1+2}

% in TeX

G:=Group("C3xS3xES-(3,1)");

// GroupNames label

G:=SmallGroup(486,225);

// by ID

G=gap.SmallGroup(486,225);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,303,93,11669]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^9=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;

// generators/relations

G = C3×S3×3- 1+2 order 486 = 2·35

Direct product of C3, S3 and 3- 1+2

G = C₃×S₃×3_-¹⁺² order 486 = 2·3⁵

Direct product of C₃, S₃ and 3_-¹⁺²