C2xS3xES-(3,1) - GroupNames

G = C₂×S₃×3_-¹⁺² order 324 = 2²·3⁴

Direct product of C₂, S₃ and 3_-¹⁺²

direct product, metabelian, supersoluble, monomial

Aliases: C₂×S₃×3_-¹⁺², C₃².₉C₆², (S₃×C₁₈)⋊C₃, C₉⋊₃(S₃×C₆), (S₃×C₉)⋊₂C₆, C₁₈⋊₂(C₃×S₃), (C₃×C₁₈)⋊₇C₆, (S₃×C₃²).C₆, (C₃²×C₆).₈C₆, C₃³.₂(C₂×C₆), C₆⋊(C₂×3_-¹⁺²), (S₃×C₆).₃C₃², C₃².₁₁(S₃×C₆), C₆.₁₃(S₃×C₃²), C₃⋊(C₂²×3_-¹⁺²), (C₆×3_-¹⁺²)⋊₃C₂, (C₃×3_-¹⁺²)⋊₄C₂², (S₃×C₃×C₆).C₃, C₃.₆(S₃×C₃×C₆), (C₃×C₉)⋊₈(C₂×C₆), (C₃×S₃).₃(C₃×C₆), (C₃×C₆).₂₀(C₃×C₆), (C₃×C₆).₂₄(C₃×S₃), SmallGroup(324,141)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×S₃×3_-¹⁺²

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃×3_-¹⁺² — S₃×3_-¹⁺² — C₂×S₃×3_-¹⁺²

Lower central

C₃ — C₃² — C₂×S₃×3_-¹⁺²

Upper central

C₁ — C₆ — C₂×3_-¹⁺²

Generators and relations for C₂×S₃×3_-¹⁺²
G = < a,b,c,d,e | a²=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: 220 in 102 conjugacy classes, 49 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², S₃, C₆, C₆, C₉, C₉, C₃², C₃², D₆, C₂×C₆, C₁₈, C₁₈, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₆, C₃×C₉, 3_-¹⁺², 3_-¹⁺², C₃³, C₂×C₁₈, S₃×C₆, S₃×C₆, C₆², S₃×C₉, C₃×C₁₈, C₂×3_-¹⁺², C₂×3_-¹⁺², S₃×C₃², C₃²×C₆, C₃×3_-¹⁺², S₃×C₁₈, C₂²×3_-¹⁺², S₃×C₃×C₆, S₃×3_-¹⁺², C₆×3_-¹⁺², C₂×S₃×3_-¹⁺²
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₃², D₆, C₂×C₆, C₃×S₃, C₃×C₆, 3_-¹⁺², S₃×C₆, C₆², C₂×3_-¹⁺², S₃×C₃², C₂²×3_-¹⁺², S₃×C₃×C₆, S₃×3_-¹⁺², C₂×S₃×3_-¹⁺²

Smallest permutation representation of C₂×S₃×3_-¹⁺²
►On 36 points

Generators in S₃₆

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)

(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)

(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)

(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)(28 31 34)(30 36 33)

G:=sub<Sym(36)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)(28,31,34)(30,36,33)>;

G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)(28,31,34)(30,36,33) );

G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,10),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,19),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,22,25),(21,27,24),(28,31,34),(30,36,33)]])

66 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	3G	3H	3I	6A	6B	6C	6D	6E	6F	···	6K	6L	6M	6N	6O	6P	6Q	9A	···	9F	9G	···	9L	18A	···	18F	18G	···	18L	18M	···	18X
order	1	2	2	2	3	3	3	3	3	3	3	3	3	6	6	6	6	6	6	···	6	6	6	6	6	6	6	9	···	9	9	···	9	18	···	18	18	···	18	18	···	18
size	1	1	3	3	1	1	2	2	2	3	3	6	6	1	1	2	2	2	3	···	3	6	6	9	9	9	9	3	···	3	6	···	6	3	···	3	6	···	6	9	···	9

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	6	6
type	+	+	+							+	+
image	C₁	C₂	C₂	C₃	C₃	C₆	C₆	C₆	C₆	S₃	D₆	C₃×S₃	C₃×S₃	S₃×C₆	S₃×C₆	3_-¹⁺²	C₂×3_-¹⁺²	C₂×3_-¹⁺²	S₃×3_-¹⁺²	C₂×S₃×3_-¹⁺²
kernel	C₂×S₃×3_-¹⁺²	S₃×3_-¹⁺²	C₆×3_-¹⁺²	S₃×C₁₈	S₃×C₃×C₆	S₃×C₉	C₃×C₁₈	S₃×C₃²	C₃²×C₆	C₂×3_-¹⁺²	3_-¹⁺²	C₁₈	C₃×C₆	C₉	C₃²	D₆	S₃	C₆	C₂	C₁
# reps	1	2	1	6	2	12	6	4	2	1	1	6	2	6	2	2	4	2	2	2

Matrix representation of C₂×S₃×3_-¹⁺² ►in GL₅(𝔽₁₉)

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

18	1	0	0	0
18	0	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

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

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

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

C₂×S₃×3_-¹⁺² in GAP, Magma, Sage, TeX

C_2\times S_3\times 3_-^{1+2}

% in TeX

G:=Group("C2xS3xES-(3,1)");

// GroupNames label

G:=SmallGroup(324,141);

// by ID

G=gap.SmallGroup(324,141);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,303,93,7781]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=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 = C2×S3×3- 1+2 order 324 = 22·34

Direct product of C2, S3 and 3- 1+2

G = C₂×S₃×3_-¹⁺² order 324 = 2²·3⁴

Direct product of C₂, S₃ and 3_-¹⁺²