C6xD6:S3 - GroupNames

G = C₆xD₆:S₃ order 432 = 2⁴·3³

Direct product of C₆ and D₆:S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₆xD₆:S₃, C₆².₁₁₀D₆, D₆:₄(S₃xC₆), (S₃xC₆):₁₅D₆, (C₃²xC₆):₄D₄, C₃²:₇(C₆xD₄), C₃³:₁₇(C₂xD₄), (S₃xC₆²):₁C₂, C₆².₂₆(C₂xC₆), (C₃²xC₆).₃₃C₂³, (C₃xC₆²).₂₀C₂², (S₃xC₂xC₆):₃C₆, (S₃xC₂xC₆):₃S₃, C₂.₁₄(S₃²xC₆), (C₂xC₆).₇₃S₃², (C₃xC₆):₄(C₃xD₄), C₆:₂(C₃xC₃:D₄), C₃:₃(C₆xC₃:D₄), C₆.₁₄(S₃xC₂xC₆), (S₃xC₆):₄(C₂xC₆), C₆.₁₁₇(C₂xS₃²), (C₂xC₆).₂₈(S₃xC₆), (S₃xC₃xC₆):₁₆C₂², C₃:Dic₃:₉(C₂xC₆), C₂².₁₀(C₃xS₃²), (C₃xC₆):₁₁(C₃:D₄), (C₂²xS₃):₃(C₃xS₃), (C₂xC₃:Dic₃):₁₂C₆, (C₆xC₃:Dic₃):₁₆C₂, C₃²:₂₀(C₂xC₃:D₄), (C₃xC₆).₂₄(C₂²xC₆), (C₃xC₆).₁₃₈(C₂²xS₃), (C₃xC₃:Dic₃):₂₃C₂², SmallGroup(432,655)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₆xD₆:S₃

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — S₃xC₃xC₆ — C₃xD₆:S₃ — C₆xD₆:S₃

Lower central

C₃² — C₃xC₆ — C₆xD₆:S₃

Upper central

C₁ — C₂xC₆

Generators and relations for C₆xD₆:S₃
G = < a,b,c,d,e | a⁶=b⁶=c²=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ece=b³c, ede=d^-1 >

Subgroups: 960 in 306 conjugacy classes, 80 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, D₄, C₂³, C₃², C₃², C₃², Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₃xC₆, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, C₃³, C₃xDic₃, C₃:Dic₃, S₃xC₆, S₃xC₆, C₆², C₆², C₆², C₂xC₃:D₄, C₆xD₄, S₃xC₃², C₃²xC₆, C₃²xC₆, D₆:S₃, C₆xDic₃, C₃xC₃:D₄, C₂xC₃:Dic₃, S₃xC₂xC₆, S₃xC₂xC₆, C₂xC₆², C₃xC₃:Dic₃, S₃xC₃xC₆, S₃xC₃xC₆, C₃xC₆², C₂xD₆:S₃, C₆xC₃:D₄, C₃xD₆:S₃, C₆xC₃:Dic₃, S₃xC₆², C₆xD₆:S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃:D₄, C₃xD₄, C₂²xS₃, C₂²xC₆, S₃², S₃xC₆, C₂xC₃:D₄, C₆xD₄, D₆:S₃, C₃xC₃:D₄, C₂xS₃², S₃xC₂xC₆, C₃xS₃², C₂xD₆:S₃, C₆xC₃:D₄, C₃xD₆:S₃, S₃²xC₆, C₆xD₆:S₃

Smallest permutation representation of C₆xD₆:S₃
►On 48 points

Generators in S₄₈

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

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

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

(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 40)(14 41)(15 42)(16 37)(17 38)(18 39)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)

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

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

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

90 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	6A	···	6F	6G	···	6X	6Y	···	6AG	6AH	···	6BM	12A	12B	12C	12D
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	12	12	12	12
size	1	1	1	1	6	6	6	6	1	1	2	···	2	4	4	4	18	18	1	···	1	2	···	2	4	···	4	6	···	6	18	18	18	18

90 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+					+	+	+	+							+	-	+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	D₄	D₆	D₆	C₃xS₃	C₃:D₄	C₃xD₄	S₃xC₆	S₃xC₆	C₃xC₃:D₄	S₃²	D₆:S₃	C₂xS₃²	C₃xS₃²	C₃xD₆:S₃	S₃²xC₆
kernel	C₆xD₆:S₃	C₃xD₆:S₃	C₆xC₃:Dic₃	S₃xC₆²	C₂xD₆:S₃	D₆:S₃	C₂xC₃:Dic₃	S₃xC₂xC₆	S₃xC₂xC₆	C₃²xC₆	S₃xC₆	C₆²	C₂²xS₃	C₃xC₆	C₃xC₆	D₆	C₂xC₆	C₆	C₂xC₆	C₆	C₆	C₂²	C₂	C₂
# reps	1	4	1	2	2	8	2	4	2	2	4	2	4	8	4	8	4	16	1	2	1	2	4	2

Matrix representation of C₆xD₆:S₃ ►in GL₆(F₁₃)

3	0	0	0	0	0
0	3	0	0	0	0
0	0	4	0	0	0
0	0	0	4	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	0	12
0	0	0	0	1	12

1	3	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	1	12
0	0	0	0	0	12

1	0	0	0	0	0
0	1	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	0	1

4	4	0	0	0	0
6	9	0	0	0	0
0	0	0	12	0	0
0	0	12	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

C₆xD₆:S₃ in GAP, Magma, Sage, TeX

C_6\times D_6\rtimes S_3

% in TeX

G:=Group("C6xD6:S3");

// GroupNames label

G:=SmallGroup(432,655);

// by ID

G=gap.SmallGroup(432,655);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,2028,14118]);

// Polycyclic

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

// generators/relations

G = C6xD6:S3 order 432 = 24·33

Direct product of C6 and D6:S3

G = C₆xD₆:S₃ order 432 = 2⁴·3³

Direct product of C₆ and D₆:S₃