C3^3:6C4^2 - GroupNames

G = C₃³:₆C₄² order 432 = 2⁴·3³

3^rd semidirect product of C₃³ and C₄² acting via C₄²/C₂²=C₂²

metabelian, supersoluble, monomial, A-group

Aliases: C₃³:₆C₄², C₆².₈₃D₆, C₃:₂Dic₃², C₃:Dic₃:₄Dic₃, C₆.₁₉(S₃xDic₃), C₃²:₇(C₄xDic₃), C₆.₂₀(C₆.D₆), (C₃xC₆²).₁₃C₂², C₂².₂(C₃²:₄D₆), (C₂xC₆).₅₉S₃², (C₃xC₆).₅₅(C₄xS₃), (C₃xC₃:Dic₃):₇C₄, (C₂xC₃:Dic₃).₈S₃, C₂.₂(C₃³:₉(C₂xC₄)), (C₆xC₃:Dic₃).₁₃C₂, (C₃²xC₆).₄₆(C₂xC₄), (C₃xC₆).₄₂(C₂xDic₃), SmallGroup(432,460)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃³:₆C₄²

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²xC₆ — C₃xC₆² — C₆xC₃:Dic₃ — C₃³:₆C₄²

Lower central

C₃³ — C₃³:₆C₄²

Upper central

C₁ — C₂²

Generators and relations for C₃³:₆C₄²
G = < a,b,c,d,e | a³=b³=c³=d⁴=e⁴=1, ab=ba, ac=ca, dad^-1=eae^-1=a^-1, bc=cb, bd=db, ebe^-1=b^-1, dcd^-1=c^-1, ce=ec, de=ed >

Subgroups: 600 in 178 conjugacy classes, 59 normal (5 characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₂xC₄, C₃², C₃², Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₄², C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₃³, C₃xDic₃, C₃:Dic₃, C₆², C₆², C₄xDic₃, C₃²xC₆, C₆xDic₃, C₂xC₃:Dic₃, C₃xC₃:Dic₃, C₃xC₆², Dic₃², C₆xC₃:Dic₃, C₃³:₆C₄²
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, Dic₃, D₆, C₄², C₄xS₃, C₂xDic₃, S₃², C₄xDic₃, S₃xDic₃, C₆.D₆, C₃²:₄D₆, Dic₃², C₃³:₉(C₂xC₄), C₃³:₆C₄²

Smallest permutation representation of C₃³:₆C₄²
►On 48 points

Generators in S₄₈

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

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

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

(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)

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

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	···	3H	4A	···	4L	6A	···	6I	6J	···	6X	12A	···	12L
order	1	2	2	2	3	3	3	3	···	3	4	···	4	6	···	6	6	···	6	12	···	12
size	1	1	1	1	2	2	2	4	···	4	9	···	9	2	···	2	4	···	4	18	···	18

60 irreducible representations

dim	1	1	1	2	2	2	2	4	4	4	4	4
type	+	+		+	-	+		+	-	+
image	C₁	C₂	C₄	S₃	Dic₃	D₆	C₄xS₃	S₃²	S₃xDic₃	C₆.D₆	C₃²:₄D₆	C₃³:₉(C₂xC₄)
kernel	C₃³:₆C₄²	C₆xC₃:Dic₃	C₃xC₃:Dic₃	C₂xC₃:Dic₃	C₃:Dic₃	C₆²	C₃xC₆	C₂xC₆	C₆	C₆	C₂²	C₂
# reps	1	3	12	3	6	3	12	3	6	3	2	6

Matrix representation of C₃³:₆C₄² ►in GL₆(F₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	12
0	0	0	0	1	0

12	12	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	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	0	12	0	0
0	0	1	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

8	0	0	0	0	0
0	8	0	0	0	0
0	0	0	8	0	0
0	0	8	0	0	0
0	0	0	0	8	0
0	0	0	0	5	5

12	0	0	0	0	0
1	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	5	0
0	0	0	0	8	8

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,8,5,0,0,0,0,0,5],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,8,0,0,0,0,0,8] >;

C₃³:₆C₄² in GAP, Magma, Sage, TeX

C_3^3\rtimes_6C_4^2

% in TeX

G:=Group("C3^3:6C4^2");

// GroupNames label

G:=SmallGroup(432,460);

// by ID

G=gap.SmallGroup(432,460);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,36,1124,571,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,d*e=e*d>;

// generators/relations

G = C33:6C42 order 432 = 24·33

3rd semidirect product of C33 and C42 acting via C42/C22=C22

G = C₃³:₆C₄² order 432 = 2⁴·3³

3^rd semidirect product of C₃³ and C₄² acting via C₄²/C₂²=C₂²