C3^2xC3^3:C2 - GroupNames

G = C₃²×C₃³⋊C₂ order 486 = 2·3⁵

Direct product of C₃² and C₃³⋊C₂

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₃²×C₃³⋊C₂, C₃⁵⋊₃C₂, C₃⁴⋊₁₅S₃, C₃⁴⋊₁₆C₆, C₃³⋊₁₅(C₃×C₆), C₃³⋊₁₉(C₃×S₃), C₃³⋊₁₁(C₃⋊S₃), C₃²⋊₆(S₃×C₃²), C₃⋊(C₃²×C₃⋊S₃), C₃²⋊₅(C₃×C₃⋊S₃), SmallGroup(486,258)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃²×C₃³⋊C₂

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃⁴ — C₃⁵ — C₃²×C₃³⋊C₂

Lower central

C₃³ — C₃²×C₃³⋊C₂

Upper central

C₁ — C₃²

Generators and relations for C₃²×C₃³⋊C₂
G = < a,b,c,d,e,f | a³=b³=c³=d³=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=c^-1, de=ed, fdf=d^-1, fef=e^-1 >

Subgroups: 3768 in 1584 conjugacy classes, 174 normal (6 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₃², C₃², C₃², C₃×S₃, C₃⋊S₃, C₃×C₆, C₃³, C₃³, C₃³, S₃×C₃², C₃×C₃⋊S₃, C₃³⋊C₂, C₃⁴, C₃⁴, C₃²×C₃⋊S₃, C₃×C₃³⋊C₂, C₃⁵, C₃²×C₃³⋊C₂
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃⋊S₃, C₃×C₆, S₃×C₃², C₃×C₃⋊S₃, C₃³⋊C₂, C₃²×C₃⋊S₃, C₃×C₃³⋊C₂, C₃²×C₃³⋊C₂

Smallest permutation representation of C₃²×C₃³⋊C₂
►On 54 points

Generators in S₅₄

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

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

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

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

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

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

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

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

135 conjugacy classes

class	1	2	3A	···	3H	3I	···	3DU	6A	···	6H
order	1	2	3	···	3	3	···	3	6	···	6
size	1	27	1	···	1	2	···	2	27	···	27

135 irreducible representations

dim	1	1	1	1	2	2
type	+	+			+
image	C₁	C₂	C₃	C₆	S₃	C₃×S₃
kernel	C₃²×C₃³⋊C₂	C₃⁵	C₃×C₃³⋊C₂	C₃⁴	C₃⁴	C₃³
# reps	1	1	8	8	13	104

Matrix representation of C₃²×C₃³⋊C₂ ►in GL₆(𝔽₇)

2	0	0	0	0	0
0	2	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	4	0
0	0	0	0	0	4

2	0	0	0	0	0
0	2	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	2	4	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	2

2	0	0	0	0	0
0	4	0	0	0	0
0	0	2	4	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

4	0	0	0	0	0
0	2	0	0	0	0
0	0	2	4	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	2

0	1	0	0	0	0
1	0	0	0	0	0
0	0	6	0	0	0
0	0	3	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(7))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,4,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,4,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,4,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,3,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₃²×C₃³⋊C₂ in GAP, Magma, Sage, TeX

C_3^2\times C_3^3\rtimes C_2

% in TeX

G:=Group("C3^2xC3^3:C2");

// GroupNames label

G:=SmallGroup(486,258);

// by ID

G=gap.SmallGroup(486,258);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,3244,11669]);

// Polycyclic

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

// generators/relations

G = C32×C33⋊C2 order 486 = 2·35

Direct product of C32 and C33⋊C2

G = C₃²×C₃³⋊C₂ order 486 = 2·3⁵

Direct product of C₃² and C₃³⋊C₂