C2^2xC3wrC3 - GroupNames

G = C₂²×C₃≀C₃ order 324 = 2²·3⁴

Direct product of C₂² and C₃≀C₃

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

Aliases: C₂²×C₃≀C₃, C₃².₁C₆², C₆².₁₉C₃², He₃⋊₃(C₂×C₆), (C₂×He₃)⋊₂C₆, (C₃×C₆²)⋊₂C₃, (C₃²×C₆)⋊₅C₆, C₃³⋊₉(C₂×C₆), C₆.₈(C₂×He₃), (C₂×C₆).₁₄He₃, (C₂²×He₃)⋊₃C₃, C₃.₂(C₂²×He₃), (C₂×3_-¹⁺²)⋊₁C₆, 3_-¹⁺²⋊₁(C₂×C₆), (C₂²×3_-¹⁺²)⋊₃C₃, (C₃×C₆).₆(C₃×C₆), SmallGroup(324,86)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂²×C₃≀C₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃≀C₃ — C₂×C₃≀C₃ — C₂²×C₃≀C₃

Lower central

C₁ — C₃ — C₃² — C₂²×C₃≀C₃

Upper central

C₁ — C₂×C₆ — C₆² — C₂²×C₃≀C₃

Generators and relations for C₂²×C₃≀C₃
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, C₂×C₆, C₂×C₆, C₁₈, C₃×C₆, C₃×C₆, He₃, 3_-¹⁺², C₃³, C₂×C₁₈, C₆², C₆², C₂×He₃, C₂×3_-¹⁺², C₃²×C₆, C₃≀C₃, C₂²×He₃, C₂²×3_-¹⁺², C₃×C₆², C₂×C₃≀C₃, C₂²×C₃≀C₃
Quotients: C₁, C₂, C₃, C₂², C₆, C₃², C₂×C₆, C₃×C₆, He₃, C₆², C₂×He₃, C₃≀C₃, C₂²×He₃, C₂×C₃≀C₃, C₂²×C₃≀C₃

Smallest permutation representation of C₂²×C₃≀C₃
►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	C₂×He₃	C₃≀C₃	C₂×C₃≀C₃
kernel	C₂²×C₃≀C₃	C₂×C₃≀C₃	C₂²×He₃	C₂²×3_-¹⁺²	C₃×C₆²	C₂×He₃	C₂×3_-¹⁺²	C₃²×C₆	C₂×C₆	C₆	C₂²	C₂
# reps	1	3	2	4	2	6	12	6	2	6	6	18

Matrix representation of C₂²×C₃≀C₃ ►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] >;

C₂²×C₃≀C₃ 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 = C22×C3≀C3 order 324 = 22·34

Direct product of C22 and C3≀C3

G = C₂²×C₃≀C₃ order 324 = 2²·3⁴

Direct product of C₂² and C₃≀C₃