C3xD6.6D6 - GroupNames

G = C₃xD₆.₆D₆ order 432 = 2⁴·3³

Direct product of C₃ and D₆.₆D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xD₆.₆D₆, C₁₂.₈₀S₃², (S₃xC₁₂):₃S₃, (S₃xC₁₂):₃C₆, D₆.₆(S₃xC₆), C₃:D₁₂:₄C₆, C₁₂.₄₃(S₃xC₆), Dic₆:₅(C₃xS₃), (C₃xDic₆):₇C₆, (S₃xC₆).₃₈D₆, C₆.D₆:₁C₆, C₁₂:S₃:₁₀C₆, (C₃xDic₆):₁₁S₃, (C₃xC₁₂).₁₇₇D₆, C₃³:₁₁(C₄oD₄), Dic₃.₉(S₃xC₆), (C₃xDic₃).₂₆D₆, (C₃²xDic₆):₉C₂, C₃²:₁₉(C₄oD₁₂), (C₃²xC₆).₂₅C₂³, C₃²:₁₁(Q₈:₃S₃), (C₃²xC₁₂).₃₉C₂², (C₃²xDic₃).₁₁C₂², C₂.₉(S₃²xC₆), C₄.₇(C₃xS₃²), C₆.₆(S₃xC₂xC₆), (S₃xC₃xC₁₂):₄C₂, (C₄xS₃):₂(C₃xS₃), C₆.₁₀₉(C₂xS₃²), C₃:₁(C₃xC₄oD₁₂), C₃:₁(C₃xQ₈:₃S₃), C₃²:₇(C₃xC₄oD₄), (C₃xC₁₂:S₃):₆C₂, (S₃xC₆).₁₃(C₂xC₆), (C₃xC₁₂).₅₄(C₂xC₆), (C₃xC₆.D₆):₄C₂, (S₃xC₃xC₆).₂₀C₂², (C₃xC₃:D₁₂):₁₀C₂, (C₆xC₃:S₃).₂₄C₂², (C₃xC₆).₁₆(C₂²xC₆), (C₃xDic₃).₉(C₂xC₆), (C₃xC₆).₁₃₀(C₂²xS₃), (C₂xC₃:S₃).₇(C₂xC₆), SmallGroup(432,647)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xD₆.₆D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — S₃xC₃xC₆ — C₃xC₃:D₁₂ — C₃xD₆.₆D₆

Lower central

C₃² — C₃xC₆ — C₃xD₆.₆D₆

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃xD₆.₆D₆
G = < a,b,c,d,e | a³=b⁶=c²=1, d⁶=e²=b³, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ece^-1=b³c, ede^-1=b³d⁵ >

Subgroups: 744 in 210 conjugacy classes, 64 normal (44 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂xC₄, D₄, Q₈, C₃², C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₄oD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, Dic₆, C₄xS₃, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₃³, C₃xDic₃, C₃xDic₃, C₃xDic₃, C₃xC₁₂, C₃xC₁₂, S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₆², C₄oD₁₂, Q₈:₃S₃, C₃xC₄oD₄, S₃xC₃², C₃xC₃:S₃, C₃²xC₆, C₆.D₆, C₃:D₁₂, C₃xDic₆, C₃xDic₆, S₃xC₁₂, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₁₂:S₃, C₆xC₁₂, Q₈xC₃², C₃²xDic₃, C₃²xDic₃, C₃²xC₁₂, S₃xC₃xC₆, C₆xC₃:S₃, D₆.₆D₆, C₃xC₄oD₁₂, C₃xQ₈:₃S₃, C₃xC₆.D₆, C₃xC₃:D₁₂, C₃²xDic₆, S₃xC₃xC₁₂, C₃xC₁₂:S₃, C₃xD₆.₆D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂xC₆, C₄oD₄, C₃xS₃, C₂²xS₃, C₂²xC₆, S₃², S₃xC₆, C₄oD₁₂, Q₈:₃S₃, C₃xC₄oD₄, C₂xS₃², S₃xC₂xC₆, C₃xS₃², D₆.₆D₆, C₃xC₄oD₁₂, C₃xQ₈:₃S₃, S₃²xC₆, C₃xD₆.₆D₆

Smallest permutation representation of C₃xD₆.₆D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

81 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	4C	4D	4E	6A	6B	6C	···	6H	6I	6J	6K	6L	···	6S	6T	6U	6V	6W	12A	···	12H	12I	12J	12K	12L	12M	···	12U	12V	···	12AE	12AF	···	12AK
order	1	2	2	2	2	3	3	3	···	3	3	3	3	4	4	4	4	4	6	6	6	···	6	6	6	6	6	···	6	6	6	6	6	12	···	12	12	12	12	12	12	···	12	12	···	12	12	···	12
size	1	1	6	18	18	1	1	2	···	2	4	4	4	2	3	3	6	6	1	1	2	···	2	4	4	4	6	···	6	18	18	18	18	2	···	2	3	3	3	3	4	···	4	6	···	6	12	···	12

81 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+							+	+	+	+	+										+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	S₃	D₆	D₆	D₆	C₄oD₄	C₃xS₃	C₃xS₃	S₃xC₆	S₃xC₆	S₃xC₆	C₄oD₁₂	C₃xC₄oD₄	C₃xC₄oD₁₂	S₃²	Q₈:₃S₃	C₂xS₃²	C₃xS₃²	D₆.₆D₆	C₃xQ₈:₃S₃	S₃²xC₆	C₃xD₆.₆D₆
kernel	C₃xD₆.₆D₆	C₃xC₆.D₆	C₃xC₃:D₁₂	C₃²xDic₆	S₃xC₃xC₁₂	C₃xC₁₂:S₃	D₆.₆D₆	C₆.D₆	C₃:D₁₂	C₃xDic₆	S₃xC₁₂	C₁₂:S₃	C₃xDic₆	S₃xC₁₂	C₃xDic₃	C₃xC₁₂	S₃xC₆	C₃³	Dic₆	C₄xS₃	Dic₃	C₁₂	D₆	C₃²	C₃²	C₃	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	2	1	1	1	2	4	4	2	2	2	1	1	3	2	1	2	2	2	6	4	2	4	4	8	1	1	1	2	2	2	2	4

Matrix representation of C₃xD₆.₆D₆ ►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	0	1	0	0
0	0	12	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

8	11	0	0	0	0
0	5	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	0	12
0	0	0	0	1	12

10	8	0	0	0	0
2	3	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	12

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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,11,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[10,2,0,0,0,0,8,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;

C₃xD₆.₆D₆ in GAP, Magma, Sage, TeX

C_3\times D_6._6D_6

% in TeX

G:=Group("C3xD6.6D6");

// GroupNames label

G:=SmallGroup(432,647);

// by ID

G=gap.SmallGroup(432,647);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,176,303,142,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=c^2=1,d^6=e^2=b^3,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^-1=b^3*c,e*d*e^-1=b^3*d^5>;

// generators/relations

G = C3xD6.6D6 order 432 = 24·33

Direct product of C3 and D6.6D6

G = C₃xD₆.₆D₆ order 432 = 2⁴·3³

Direct product of C₃ and D₆.₆D₆