C3xC4:(C3^2:C4) - GroupNames

G = C₃×C₄⋊(C₃²⋊C₄) order 432 = 2⁴·3³

Direct product of C₃ and C₄⋊(C₃²⋊C₄)

direct product, metabelian, soluble, monomial

Aliases: C₃×C₄⋊(C₃²⋊C₄), (C₃×C₁₂)⋊₅C₁₂, C₃³⋊₈(C₄⋊C₄), C₁₂⋊₃(C₃²⋊C₄), C₃⋊Dic₃⋊₇C₁₂, (C₃²×C₁₂)⋊₄C₄, C₄⋊(C₃×C₃²⋊C₄), C₃²⋊₅(C₃×C₄⋊C₄), C₃⋊S₃.₄(C₃×D₄), C₂.₅(C₆×C₃²⋊C₄), C₃⋊S₃.₂(C₃×Q₈), (C₃×C₃⋊S₃).₅Q₈, (C₄×C₃⋊S₃).₁₂C₆, (C₃×C₃⋊S₃).₁₄D₄, (C₆×C₃²⋊C₄).₃C₂, (C₂×C₃²⋊C₄).₃C₆, C₆.₂₂(C₂×C₃²⋊C₄), (C₁₂×C₃⋊S₃).₁₉C₂, (C₃×C₆).₁₃(C₂×C₁₂), (C₃×C₃⋊Dic₃)⋊₁₁C₄, (C₆×C₃⋊S₃).₃₉C₂², (C₃²×C₆).₁₁(C₂×C₄), (C₂×C₃⋊S₃).₁₄(C₂×C₆), SmallGroup(432,631)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₃×C₄⋊(C₃²⋊C₄)

Chief series

C₁ — C₃² — C₃×C₆ — C₂×C₃⋊S₃ — C₆×C₃⋊S₃ — C₆×C₃²⋊C₄ — C₃×C₄⋊(C₃²⋊C₄)

Lower central

C₃² — C₃×C₆ — C₃×C₄⋊(C₃²⋊C₄)

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×C₄⋊(C₃²⋊C₄)
G = < a,b,c,d,e | a³=b⁴=c³=d³=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=b^-1, ede^-1=cd=dc, ece^-1=c^-1d >

Subgroups: 444 in 96 conjugacy classes, 28 normal (24 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂×C₄, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₄⋊C₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₄×S₃, C₂×C₁₂, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, C₃²⋊C₄, S₃×C₆, C₂×C₃⋊S₃, C₃×C₄⋊C₄, C₃×C₃⋊S₃, C₃²×C₆, S₃×C₁₂, C₄×C₃⋊S₃, C₂×C₃²⋊C₄, C₃×C₃⋊Dic₃, C₃²×C₁₂, C₃×C₃²⋊C₄, C₆×C₃⋊S₃, C₄⋊(C₃²⋊C₄), C₁₂×C₃⋊S₃, C₆×C₃²⋊C₄, C₃×C₄⋊(C₃²⋊C₄)
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, Q₈, C₁₂, C₂×C₆, C₄⋊C₄, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₃²⋊C₄, C₃×C₄⋊C₄, C₂×C₃²⋊C₄, C₃×C₃²⋊C₄, C₄⋊(C₃²⋊C₄), C₆×C₃²⋊C₄, C₃×C₄⋊(C₃²⋊C₄)

Smallest permutation representation of C₃×C₄⋊(C₃²⋊C₄)
►On 48 points

Generators in S₄₈

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

(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)

(13 47 30)(14 48 31)(15 45 32)(16 46 29)(33 44 38)(34 41 39)(35 42 40)(36 43 37)

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

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4	4	4
type	+	+	+								+	-			+	+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	D₄	Q₈	C₃×D₄	C₃×Q₈	C₃²⋊C₄	C₂×C₃²⋊C₄	C₃×C₃²⋊C₄	C₄⋊(C₃²⋊C₄)	C₆×C₃²⋊C₄	C₃×C₄⋊(C₃²⋊C₄)
kernel	C₃×C₄⋊(C₃²⋊C₄)	C₁₂×C₃⋊S₃	C₆×C₃²⋊C₄	C₄⋊(C₃²⋊C₄)	C₃×C₃⋊Dic₃	C₃²×C₁₂	C₄×C₃⋊S₃	C₂×C₃²⋊C₄	C₃⋊Dic₃	C₃×C₁₂	C₃×C₃⋊S₃	C₃×C₃⋊S₃	C₃⋊S₃	C₃⋊S₃	C₁₂	C₆	C₄	C₃	C₂	C₁
# reps	1	1	2	2	2	2	2	4	4	4	1	1	2	2	2	2	4	4	4	8

Matrix representation of C₃×C₄⋊(C₃²⋊C₄) ►in GL₆(𝔽₁₃)

9	0	0	0	0	0
0	9	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	9	0
0	0	0	0	0	9

1	11	0	0	0	0
1	12	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	1	0	0	0
0	0	0	1	0	0
0	0	4	0	9	0
0	0	1	0	0	3

1	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	0	0
0	0	3	9	0	0
0	0	3	0	9	0
0	0	0	0	0	3

8	0	0	0	0	0
8	5	0	0	0	0
0	0	1	0	2	0
0	0	0	0	12	1
0	0	0	1	12	0
0	0	0	0	12	0

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

C₃×C₄⋊(C₃²⋊C₄) in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes (C_3^2\rtimes C_4)

% in TeX

G:=Group("C3xC4:(C3^2:C4)");

// GroupNames label

G:=SmallGroup(432,631);

// by ID

G=gap.SmallGroup(432,631);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,365,176,14117,362,18822,1203]);

// Polycyclic

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

// generators/relations

G = C3×C4⋊(C32⋊C4) order 432 = 24·33

Direct product of C3 and C4⋊(C32⋊C4)

G = C₃×C₄⋊(C₃²⋊C₄) order 432 = 2⁴·3³

Direct product of C₃ and C₄⋊(C₃²⋊C₄)