C2^2xC3wrC3 - GroupNames

G = C₂²xC₃wrC₃ order 324 = 2²·3⁴

Direct product of C₂² and C₃wrC₃

direct product, metabelian, nilpotent (class 3), monomial

Aliases: C₂²xC₃wrC₃, C₃².₁C₆², C₆².₁₉C₃², He₃:₃(C₂xC₆), (C₂xHe₃):₂C₆, (C₃xC₆²):₂C₃, (C₃²xC₆):₅C₆, C₃³:₉(C₂xC₆), C₆.₈(C₂xHe₃), (C₂xC₆).₁₄He₃, (C₂²xHe₃):₃C₃, C₃.₂(C₂²xHe₃), (C₂x3_-¹⁺²):₁C₆, 3_-¹⁺²:₁(C₂xC₆), (C₂²x3_-¹⁺²):₃C₃, (C₃xC₆).₆(C₃xC₆), SmallGroup(324,86)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂²xC₃wrC₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃wrC₃ — C₂xC₃wrC₃ — C₂²xC₃wrC₃

Lower central

C₁ — C₃ — C₃² — C₂²xC₃wrC₃

Upper central

C₁ — C₂xC₆ — C₆² — C₂²xC₃wrC₃

Generators and relations for C₂²xC₃wrC₃
G = < a,b,c,d,e,f | a²=b²=c³=d³=e³=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece^-1=cd^-1, cf=fc, de=ed, df=fd, fef^-1=cd^-1e >

Subgroups: 250 in 100 conjugacy classes, 40 normal (12 characteristic)
C₁, C₂, C₃, C₃, C₂², C₆, C₆, C₉, C₃², C₃², C₂xC₆, C₂xC₆, C₁₈, C₃xC₆, C₃xC₆, He₃, 3_-¹⁺², C₃³, C₂xC₁₈, C₆², C₆², C₂xHe₃, C₂x3_-¹⁺², C₃²xC₆, C₃wrC₃, C₂²xHe₃, C₂²x3_-¹⁺², C₃xC₆², C₂xC₃wrC₃, C₂²xC₃wrC₃
Quotients: C₁, C₂, C₃, C₂², C₆, C₃², C₂xC₆, C₃xC₆, He₃, C₆², C₂xHe₃, C₃wrC₃, C₂²xHe₃, C₂xC₃wrC₃, C₂²xC₃wrC₃

Smallest permutation representation of C₂²xC₃wrC₃
►On 36 points

Generators in S₃₆

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

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

(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)

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

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

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

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

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

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

68 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3J	3K	3L	6A	···	6F	6G	···	6AD	6AE	···	6AJ	9A	9B	9C	9D	18A	···	18L
order	1	2	2	2	3	3	3	···	3	3	3	6	···	6	6	···	6	6	···	6	9	9	9	9	18	···	18
size	1	1	1	1	1	1	3	···	3	9	9	1	···	1	3	···	3	9	···	9	9	9	9	9	9	···	9

68 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₃	C₆	C₆	C₆	He₃	C₂xHe₃	C₃wrC₃	C₂xC₃wrC₃
kernel	C₂²xC₃wrC₃	C₂xC₃wrC₃	C₂²xHe₃	C₂²x3_-¹⁺²	C₃xC₆²	C₂xHe₃	C₂x3_-¹⁺²	C₃²xC₆	C₂xC₆	C₆	C₂²	C₂
# reps	1	3	2	4	2	6	12	6	2	6	6	18

Matrix representation of C₂²xC₃wrC₃ ►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

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

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

1	0	0	0	0
0	1	0	0	0
0	0	7	0	0
0	0	0	7	0
0	0	0	0	7

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

7	0	0	0	0
0	7	0	0	0
0	0	7	0	0
0	0	0	1	0
0	0	0	0	1

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],[1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[7,0,0,0,0,0,7,0,0,0,0,0,0,18,0,0,0,0,0,18,0,0,1,0,0],[7,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1] >;

C₂²xC₃wrC₃ in GAP, Magma, Sage, TeX

C_2^2\times C_3\wr C_3

% in TeX

G:=Group("C2^2xC3wrC3");

// GroupNames label

G:=SmallGroup(324,86);

// by ID

G=gap.SmallGroup(324,86);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,303,1096]);

// Polycyclic

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

// generators/relations

G = C22xC3wrC3 order 324 = 22·34

Direct product of C22 and C3wrC3

G = C₂²xC₃wrC₃ order 324 = 2²·3⁴

Direct product of C₂² and C₃wrC₃