C4xC3^2:4D6 - GroupNames

G = C₄xC₃²:₄D₆ order 432 = 2⁴·3³

Direct product of C₄ and C₃²:₄D₆

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

Aliases: C₄xC₃²:₄D₆, C₁₂:₇S₃², (C₃xC₁₂):₁₈D₆, C₃:Dic₃:₁₉D₆, C₃³:₇(C₂²xC₄), (C₃²xC₁₂):₁₂C₂², (C₃²xC₆).₆₈C₂³, C₃:₄(C₄xS₃²), C₃:S₃:₃(C₄xS₃), C₆.₉₇(C₂xS₃²), C₃²:₉(S₃xC₂xC₄), (C₄xC₃:S₃):₁₃S₃, (C₁₂xC₃:S₃):₁₅C₂, (C₂xC₃:S₃).₄₈D₆, C₃³:₉(C₂xC₄):₁₁C₂, (C₆xC₃:S₃).₅₉C₂², C₂.₁(C₂xC₃²:₄D₆), (C₃xC₆).₁₁₈(C₂²xS₃), (C₃xC₃:Dic₃):₂₀C₂², (C₂xC₃²:₄D₆).₄C₂, (C₃xC₃:S₃):₇(C₂xC₄), SmallGroup(432,690)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₄xC₃²:₄D₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²xC₆ — C₆xC₃:S₃ — C₂xC₃²:₄D₆ — C₄xC₃²:₄D₆

Lower central

C₃³ — C₄xC₃²:₄D₆

Upper central

C₁ — C₄

Generators and relations for C₄xC₃²:₄D₆
G = < a,b,c,d,e | a⁴=b³=c³=d⁶=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=b^-1, be=eb, dcd^-1=ece=c^-1, ede=d^-1 >

Subgroups: 1320 in 270 conjugacy classes, 63 normal (8 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂xC₄, C₂³, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂²xC₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₂²xS₃, C₃³, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, C₃xC₁₂, S₃², S₃xC₆, C₂xC₃:S₃, S₃xC₂xC₄, C₃xC₃:S₃, C₃²xC₆, S₃xDic₃, C₆.D₆, S₃xC₁₂, C₄xC₃:S₃, C₂xS₃², C₃xC₃:Dic₃, C₃²xC₁₂, C₃²:₄D₆, C₆xC₃:S₃, C₄xS₃², C₃³:₉(C₂xC₄), C₁₂xC₃:S₃, C₂xC₃²:₄D₆, C₄xC₃²:₄D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, C₂³, D₆, C₂²xC₄, C₄xS₃, C₂²xS₃, S₃², S₃xC₂xC₄, C₂xS₃², C₃²:₄D₆, C₄xS₃², C₂xC₃²:₄D₆, C₄xC₃²:₄D₆

Smallest permutation representation of C₄xC₃²:₄D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

60 conjugacy classes

class	1	2A	2B	···	2G	3A	3B	3C	3D	···	3H	4A	4B	4C	···	4H	6A	6B	6C	6D	···	6H	6I	···	6N	12A	···	12F	12G	···	12P	12Q	···	12V
order	1	2	2	···	2	3	3	3	3	···	3	4	4	4	···	4	6	6	6	6	···	6	6	···	6	12	···	12	12	···	12	12	···	12
size	1	1	9	···	9	2	2	2	4	···	4	1	1	9	···	9	2	2	2	4	···	4	18	···	18	2	···	2	4	···	4	18	···	18

60 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+		+	+	+	+		+	+
image	C₁	C₂	C₂	C₂	C₄	S₃	D₆	D₆	D₆	C₄xS₃	S₃²	C₂xS₃²	C₃²:₄D₆	C₄xS₃²	C₂xC₃²:₄D₆	C₄xC₃²:₄D₆
kernel	C₄xC₃²:₄D₆	C₃³:₉(C₂xC₄)	C₁₂xC₃:S₃	C₂xC₃²:₄D₆	C₃²:₄D₆	C₄xC₃:S₃	C₃:Dic₃	C₃xC₁₂	C₂xC₃:S₃	C₃:S₃	C₁₂	C₆	C₄	C₃	C₂	C₁
# reps	1	3	3	1	8	3	3	3	3	12	3	3	2	6	2	4

Matrix representation of C₄xC₃²:₄D₆ ►in GL₆(F₁₃)

5	0	0	0	0	0
0	5	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	5	0
0	0	0	0	0	5

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

0	1	0	0	0	0
12	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

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

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

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

C₄xC₃²:₄D₆ in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes_4D_6

% in TeX

G:=Group("C4xC3^2:4D6");

// GroupNames label

G:=SmallGroup(432,690);

// by ID

G=gap.SmallGroup(432,690);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,1124,571,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = C4xC32:4D6 order 432 = 24·33

Direct product of C4 and C32:4D6

G = C₄xC₃²:₄D₆ order 432 = 2⁴·3³

Direct product of C₄ and C₃²:₄D₆