C2^2xC3^3:C4 - GroupNames

G = C₂²×C₃³⋊C₄ order 432 = 2⁴·3³

Direct product of C₂² and C₃³⋊C₄

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

Aliases: C₂²×C₃³⋊C₄, C₆²⋊₁₂Dic₃, (C₃×C₆²)⋊₄C₄, C₃³⋊₉(C₂²×C₄), C₃²⋊₅(C₂²×Dic₃), C₆⋊₂(C₂×C₃²⋊C₄), (C₆×C₃⋊S₃)⋊₁₁C₄, (C₂×C₆)⋊₃(C₃²⋊C₄), (C₂×C₃⋊S₃)⋊₇Dic₃, C₃⋊S₃⋊₄(C₂×Dic₃), (C₃²×C₆)⋊₃(C₂×C₄), (C₂×C₃⋊S₃).₅₀D₆, (C₃×C₆)⋊₄(C₂×Dic₃), C₃⋊₂(C₂²×C₃²⋊C₄), C₃⋊S₃.₇(C₂²×S₃), (C₂²×C₃⋊S₃).₈S₃, (C₆×C₃⋊S₃).₆₅C₂², (C₃×C₃⋊S₃).₁₀C₂³, (C₂×C₆×C₃⋊S₃).₁₁C₂, (C₃×C₃⋊S₃)⋊₁₂(C₂×C₄), SmallGroup(432,766)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂²×C₃³⋊C₄

Chief series

C₁ — C₃ — C₃³ — C₃×C₃⋊S₃ — C₃³⋊C₄ — C₂×C₃³⋊C₄ — C₂²×C₃³⋊C₄

Lower central

C₃³ — C₂²×C₃³⋊C₄

Upper central

C₁ — C₂²

Generators and relations for C₂²×C₃³⋊C₄
G = < a,b,c,d,e,f | a²=b²=c³=d³=e³=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf^-1=cd^-1, de=ed, fdf^-1=c^-1d^-1, fef^-1=e^-1 >

Subgroups: 1064 in 192 conjugacy classes, 53 normal (13 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂³, C₃², C₃², Dic₃, D₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₂×Dic₃, C₂²×S₃, C₂²×C₆, C₃³, C₃²⋊C₄, S₃×C₆, C₂×C₃⋊S₃, C₆², C₆², C₂²×Dic₃, C₃×C₃⋊S₃, C₃×C₃⋊S₃, C₃²×C₆, C₂×C₃²⋊C₄, S₃×C₂×C₆, C₂²×C₃⋊S₃, C₃³⋊C₄, C₆×C₃⋊S₃, C₃×C₆², C₂²×C₃²⋊C₄, C₂×C₃³⋊C₄, C₂×C₆×C₃⋊S₃, C₂²×C₃³⋊C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, Dic₃, D₆, C₂²×C₄, C₂×Dic₃, C₂²×S₃, C₃²⋊C₄, C₂²×Dic₃, C₂×C₃²⋊C₄, C₃³⋊C₄, C₂²×C₃²⋊C₄, C₂×C₃³⋊C₄, C₂²×C₃³⋊C₄

Smallest permutation representation of C₂²×C₃³⋊C₄
►On 48 points

Generators in S₄₈

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

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

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

(1 16 17)(3 19 14)(6 35 30)(8 32 33)(10 26 43)(12 41 28)(22 38 47)(24 45 40)

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

(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)

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	···	3G	4A	···	4H	6A	6B	6C	6D	···	6U	6V	6W	6X	6Y
order	1	2	2	2	2	2	2	2	3	3	···	3	4	···	4	6	6	6	6	···	6	6	6	6	6
size	1	1	1	1	9	9	9	9	2	4	···	4	27	···	27	2	2	2	4	···	4	18	18	18	18

48 irreducible representations

dim	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+	+			+	-	+	-	+	+
image	C₁	C₂	C₂	C₄	C₄	S₃	Dic₃	D₆	Dic₃	C₃²⋊C₄	C₂×C₃²⋊C₄	C₃³⋊C₄	C₂×C₃³⋊C₄
kernel	C₂²×C₃³⋊C₄	C₂×C₃³⋊C₄	C₂×C₆×C₃⋊S₃	C₆×C₃⋊S₃	C₃×C₆²	C₂²×C₃⋊S₃	C₂×C₃⋊S₃	C₂×C₃⋊S₃	C₆²	C₂×C₆	C₆	C₂²	C₂
# reps	1	6	1	6	2	1	3	3	1	2	6	4	12

Matrix representation of C₂²×C₃³⋊C₄ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	0	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	12	0
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	9	0	0	10
0	0	0	3	6	0
0	0	0	0	9	0
0	0	0	0	0	3

1	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	2	12
0	0	0	9	5	12
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
12	12	0	0	0	0
0	0	3	0	7	3
0	0	0	3	6	4
0	0	0	0	9	0
0	0	0	0	0	9

8	0	0	0	0	0
5	5	0	0	0	0
0	0	12	2	3	10
0	0	3	11	5	8
0	0	0	11	2	10
0	0	11	0	11	1

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,6,9,0,0,0,10,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,2,5,1,0,0,0,12,12,0,1],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,7,6,9,0,0,0,3,4,0,9],[8,5,0,0,0,0,0,5,0,0,0,0,0,0,12,3,0,11,0,0,2,11,11,0,0,0,3,5,2,11,0,0,10,8,10,1] >;

C₂²×C₃³⋊C₄ in GAP, Magma, Sage, TeX

C_2^2\times C_3^3\rtimes C_4

% in TeX

G:=Group("C2^2xC3^3:C4");

// GroupNames label

G:=SmallGroup(432,766);

// by ID

G=gap.SmallGroup(432,766);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,2804,165,2693,530,14118]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^3=f^4=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^-1=c*d^-1,d*e=e*d,f*d*f^-1=c^-1*d^-1,f*e*f^-1=e^-1>;

// generators/relations

G = C22×C33⋊C4 order 432 = 24·33

Direct product of C22 and C33⋊C4

G = C₂²×C₃³⋊C₄ order 432 = 2⁴·3³

Direct product of C₂² and C₃³⋊C₄