C3:S3xES-(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, C₃⁴.₇C₆, (C₃²×C₉)⋊₂₆C₆, C₃⋊(S₃×3_-¹⁺²), C₃³.₆₈(C₃×S₃), C₃³.₅₅(C₃×C₆), C₃².₄₉(S₃×C₃²), (C₃×3_-¹⁺²)⋊₂₃S₃, C₃²⋊₆(C₂×3_-¹⁺²), (C₃²×3_-¹⁺²)⋊₅C₂, C₉⋊₂(C₃×C₃⋊S₃), (C₉×C₃⋊S₃)⋊₃C₃, (C₃×C₉)⋊₂₃(C₃×S₃), (C₃×C₃⋊S₃).₈C₃², (C₃²×C₃⋊S₃).₃C₃, C₃².₂₁(C₃×C₃⋊S₃), C₃.₁₀(C₃²×C₃⋊S₃), SmallGroup(486,233)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃⋊S₃×3_-¹⁺²

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₉ — C₃²×3_-¹⁺² — C₃⋊S₃×3_-¹⁺²

Lower central

C₃² — C₃³ — C₃⋊S₃×3_-¹⁺²

Upper central

C₁ — 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, cac=a^-1, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d⁴ >

Subgroups: 616 in 216 conjugacy classes, 49 normal (12 characteristic)
C₁, C₂, C₃, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², C₃², C₁₈, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, C₃×C₉, 3_-¹⁺², 3_-¹⁺², C₃³, C₃³, C₃³, S₃×C₉, C₂×3_-¹⁺², S₃×C₃², C₃×C₃⋊S₃, C₃×C₃⋊S₃, C₃²×C₉, C₃×3_-¹⁺², C₃×3_-¹⁺², C₃⁴, S₃×3_-¹⁺², C₉×C₃⋊S₃, C₃²×C₃⋊S₃, C₃²×3_-¹⁺², C₃⋊S₃×3_-¹⁺²
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃⋊S₃, C₃×C₆, 3_-¹⁺², C₂×3_-¹⁺², S₃×C₃², C₃×C₃⋊S₃, S₃×3_-¹⁺², C₃²×C₃⋊S₃, C₃⋊S₃×3_-¹⁺²

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

Generators in S₅₄

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

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

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

(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)

(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(37 40 43)(39 45 42)(46 52 49)(47 50 53)

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

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

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

66 conjugacy classes

class	1	2	3A	3B	3C	···	3N	3O	3P	3Q	···	3X	6A	6B	6C	6D	9A	···	9F	9G	···	9AD	18A	···	18F
order	1	2	3	3	3	···	3	3	3	3	···	3	6	6	6	6	9	···	9	9	···	9	18	···	18
size	1	9	1	1	2	···	2	3	3	6	···	6	9	9	27	27	3	···	3	6	···	6	27	···	27

66 irreducible representations

dim	1	1	1	1	1	1	2	2	2	3	3	6
type	+	+					+
image	C₁	C₂	C₃	C₃	C₆	C₆	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	C₉×C₃⋊S₃	C₃²×C₃⋊S₃	C₃²×C₉	C₃⁴	C₃×3_-¹⁺²	C₃×C₉	C₃³	C₃⋊S₃	C₃²	C₃
# reps	1	1	6	2	6	2	4	24	8	2	2	8

Matrix representation of C₃⋊S₃×3_-¹⁺² ►in GL₇(𝔽₁₉)

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

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

11	0	0	0	0	0	0
0	11	0	0	0	0	0
0	0	11	0	0	0	0
0	0	0	11	0	0	0
0	0	0	0	18	9	12
0	0	0	0	7	1	0
0	0	0	0	1	0	0

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

G:=sub<GL(7,GF(19))| [0,1,0,0,0,0,0,18,18,0,0,0,0,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[18,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,18,7,1,0,0,0,0,9,1,0,0,0,0,0,12,0,0],[11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,11,18,0,0,0,0,0,0,1,0,0,0,0,0,0,11,7] >;

C₃⋊S₃×3_-¹⁺² in GAP, Magma, Sage, TeX

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

% in TeX

G:=Group("C3:S3xES-(3,1)");

// GroupNames label

G:=SmallGroup(486,233);

// by ID

G=gap.SmallGroup(486,233);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,68,3244,11669]);

// Polycyclic

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