C3xD6:D6 - GroupNames

G = C₃xD₆:D₆ order 432 = 2⁴·3³

Direct product of C₃ and D₆:D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xD₆:D₆, C₁₂:₈S₃², C₁₂:₁(S₃xC₆), D₆:₂(S₃xC₆), (S₃xC₆):₁₂D₆, (C₃xD₁₂):₇C₆, (C₃xD₁₂):₅S₃, D₁₂:₄(C₃xS₃), (C₃xC₁₂):₁₀D₆, C₃²:₆(C₆xD₄), C₃³:₁₂(C₂xD₄), D₆:S₃:₃C₆, C₃²:₂₂(S₃xD₄), (C₃²xD₁₂):₉C₂, (C₃²xC₁₂):₂C₂², (C₃²xC₆).₂₈C₂³, C₄:₂(C₃xS₃²), C₃:₂(C₃xS₃xD₄), (S₃²xC₆):₆C₂, (C₂xS₃²):₂C₆, C₆.₉(S₃xC₂xC₆), (C₄xC₃:S₃):₇C₆, C₂.₁₁(S₃²xC₆), (C₃xC₃:S₃):₇D₄, C₃:S₃:₃(C₃xD₄), (C₁₂xC₃:S₃):₄C₂, (S₃xC₆):₂(C₂xC₆), C₆.₁₁₂(C₂xS₃²), (C₃xC₁₂):₆(C₂xC₆), (S₃xC₃xC₆):₈C₂², C₃:Dic₃:₇(C₂xC₆), (C₃xD₆:S₃):₁₀C₂, (C₆xC₃:S₃).₄₈C₂², (C₃xC₆).₁₉(C₂²xC₆), (C₃xC₆).₁₃₃(C₂²xS₃), (C₃xC₃:Dic₃):₂₂C₂², (C₂xC₃:S₃).₂₀(C₂xC₆), SmallGroup(432,650)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xD₆:D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — S₃xC₃xC₆ — S₃²xC₆ — C₃xD₆:D₆

Lower central

C₃² — C₃xC₆ — C₃xD₆:D₆

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃xD₆:D₆
G = < a,b,c,d,e | a³=b⁶=c²=d⁶=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd^-1=b^-1, be=eb, dcd^-1=bc, ece=b³c, ede=d^-1 >

Subgroups: 1056 in 270 conjugacy classes, 68 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₆, C₂xC₄, D₄, C₂³, C₃², C₃², C₃², Dic₃, C₁₂, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃xC₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, C₃³, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, C₃xC₁₂, C₃xC₁₂, S₃², S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₆², S₃xD₄, C₆xD₄, S₃xC₃², C₃xC₃:S₃, C₃²xC₆, D₆:S₃, S₃xC₁₂, C₃xD₁₂, C₃xD₁₂, C₃xC₃:D₄, C₄xC₃:S₃, D₄xC₃², C₂xS₃², S₃xC₂xC₆, C₃xC₃:Dic₃, C₃²xC₁₂, C₃xS₃², S₃xC₃xC₆, C₆xC₃:S₃, D₆:D₆, C₃xS₃xD₄, C₃xD₆:S₃, C₃²xD₁₂, C₁₂xC₃:S₃, S₃²xC₆, C₃xD₆:D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃xD₄, C₂²xS₃, C₂²xC₆, S₃², S₃xC₆, S₃xD₄, C₆xD₄, C₂xS₃², S₃xC₂xC₆, C₃xS₃², D₆:D₆, C₃xS₃xD₄, S₃²xC₆, C₃xD₆:D₆

Smallest permutation representation of C₃xD₆:D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	6A	6B	6C	···	6H	6I	6J	6K	6L	···	6S	6T	6U	6V	6W	6X	···	6AI	12A	12B	12C	···	12N	12O	12P
order	1	2	2	2	2	2	2	2	3	3	3	···	3	3	3	3	4	4	6	6	6	···	6	6	6	6	6	···	6	6	6	6	6	6	···	6	12	12	12	···	12	12	12
size	1	1	6	6	6	6	9	9	1	1	2	···	2	4	4	4	2	18	1	1	2	···	2	4	4	4	6	···	6	9	9	9	9	12	···	12	2	2	4	···	4	18	18

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+	+						+	+	+	+					+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	S₃	D₄	D₆	D₆	C₃xS₃	C₃xD₄	S₃xC₆	S₃xC₆	S₃²	S₃xD₄	C₂xS₃²	C₃xS₃²	D₆:D₆	C₃xS₃xD₄	S₃²xC₆	C₃xD₆:D₆
kernel	C₃xD₆:D₆	C₃xD₆:S₃	C₃²xD₁₂	C₁₂xC₃:S₃	S₃²xC₆	D₆:D₆	D₆:S₃	C₃xD₁₂	C₄xC₃:S₃	C₂xS₃²	C₃xD₁₂	C₃xC₃:S₃	C₃xC₁₂	S₃xC₆	D₁₂	C₃:S₃	C₁₂	D₆	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	2	1	2	2	4	4	2	4	2	2	2	4	4	4	4	8	1	2	1	2	2	4	2	4

Matrix representation of C₃xD₆:D₆ ►in GL₆(F₁₃)

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

12	0	0	0	0	0
0	12	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

7	2	0	0	0	0
2	6	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	12
0	0	0	0	0	1

11	7	0	0	0	0
7	2	0	0	0	0
0	0	12	12	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	12	12

11	7	0	0	0	0
7	2	0	0	0	0
0	0	12	0	0	0
0	0	1	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

C₃xD₆:D₆ in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes D_6

% in TeX

G:=Group("C3xD6:D6");

// GroupNames label

G:=SmallGroup(432,650);

// by ID

G=gap.SmallGroup(432,650);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,303,142,2028,14118]);

// Polycyclic

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

// generators/relations

G = C3xD6:D6 order 432 = 24·33

Direct product of C3 and D6:D6

G = C₃xD₆:D₆ order 432 = 2⁴·3³

Direct product of C₃ and D₆:D₆