C3xS3xDic6 - GroupNames

G = C₃xS₃xDic₆ order 432 = 2⁴·3³

Direct product of C₃, S₃ and Dic₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xS₃xDic₆, C₁₂.₇₈S₃², C₃³:₇(C₂xQ₈), D₆.₇(S₃xC₆), C₃:₁(C₆xDic₆), (S₃xDic₃).C₆, C₃²:₄(C₆xQ₈), (S₃xC₁₂).₈S₃, (S₃xC₁₂).₆C₆, C₁₂.₄₂(S₃xC₆), (C₃xDic₆):₄C₆, (S₃xC₆).₄₆D₆, C₃²:₂Q₈:₂C₆, (S₃xC₃²):₃Q₈, C₃²:₁₃(S₃xQ₈), (C₃xC₁₂).₁₇₆D₆, Dic₃.₅(S₃xC₆), C₃²:₄Q₈:₁₀C₆, (C₃xDic₃).₄₅D₆, (C₃²xDic₆):₆C₂, C₃²:₁₂(C₂xDic₆), (C₃²xC₆).₂₀C₂³, (C₃²xC₁₂).₃₅C₂², (C₃²xDic₃).₈C₂², C₃:₂(C₃xS₃xQ₈), C₂.₄(S₃²xC₆), C₄.₅(C₃xS₃²), (C₃xS₃):(C₃xQ₈), C₆.₁(S₃xC₂xC₆), C₆.₁₀₄(C₂xS₃²), (S₃xC₃xC₁₂).₃C₂, (C₄xS₃).₁(C₃xS₃), (C₃xS₃xDic₃).₂C₂, (S₃xC₆).₁₁(C₂xC₆), (C₃xC₁₂).₅₀(C₂xC₆), (C₃xC₃²:₂Q₈):₈C₂, (S₃xC₃xC₆).₁₈C₂², C₃:Dic₃.₈(C₂xC₆), (C₃xC₃²:₄Q₈):₆C₂, (C₃xC₆).₁₁(C₂²xC₆), (C₃xDic₃).₁(C₂xC₆), (C₃xC₆).₁₂₅(C₂²xS₃), (C₃xC₃:Dic₃).₃₄C₂², SmallGroup(432,642)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xS₃xDic₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — S₃xC₃xC₆ — C₃xS₃xDic₃ — C₃xS₃xDic₆

Lower central

C₃² — C₃xC₆ — C₃xS₃xDic₆

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃xS₃xDic₆
G = < a,b,c,d,e | a³=b³=c²=d¹²=1, e²=d⁶, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 568 in 198 conjugacy classes, 72 normal (44 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂xC₄, Q₈, C₃², C₃², Dic₃, Dic₃, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xQ₈, C₃xS₃, C₃xS₃, C₃xC₆, C₃xC₆, Dic₆, Dic₆, C₄xS₃, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₃xQ₈, C₃³, C₃xDic₃, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, C₃xC₁₂, S₃xC₆, S₃xC₆, C₆², C₂xDic₆, S₃xQ₈, C₆xQ₈, S₃xC₃², C₃²xC₆, S₃xDic₃, C₃²:₂Q₈, C₃xDic₆, C₃xDic₆, S₃xC₁₂, S₃xC₁₂, C₆xDic₃, C₃²:₄Q₈, C₆xC₁₂, Q₈xC₃², C₃²xDic₃, C₃²xDic₃, C₃xC₃:Dic₃, C₃²xC₁₂, S₃xC₃xC₆, S₃xDic₆, C₆xDic₆, C₃xS₃xQ₈, C₃xS₃xDic₃, C₃xC₃²:₂Q₈, C₃²xDic₆, S₃xC₃xC₁₂, C₃xC₃²:₄Q₈, C₃xS₃xDic₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, Q₈, C₂³, D₆, C₂xC₆, C₂xQ₈, C₃xS₃, Dic₆, C₃xQ₈, C₂²xS₃, C₂²xC₆, S₃², S₃xC₆, C₂xDic₆, S₃xQ₈, C₆xQ₈, C₃xDic₆, C₂xS₃², S₃xC₂xC₆, C₃xS₃², S₃xDic₆, C₆xDic₆, C₃xS₃xQ₈, S₃²xC₆, C₃xS₃xDic₆

Smallest permutation representation of C₃xS₃xDic₆
►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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)

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

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

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

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

81 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	4C	4D	4E	4F	6A	6B	6C	···	6H	6I	6J	6K	6L	6M	6N	6O	6P	···	6U	12A	···	12H	12I	···	12Q	12R	···	12AC	12AD	···	12AI	12AJ	12AK	12AL	12AM
order	1	2	2	2	3	3	3	···	3	3	3	3	4	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	3	3	1	1	2	···	2	4	4	4	2	6	6	6	18	18	1	1	2	···	2	3	3	3	3	4	4	4	6	···	6	2	···	2	4	···	4	6	···	6	12	···	12	18	18	18	18

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	Q₈	D₆	D₆	D₆	C₃xS₃	C₃xS₃	Dic₆	C₃xQ₈	S₃xC₆	S₃xC₆	S₃xC₆	C₃xDic₆	S₃²	S₃xQ₈	C₂xS₃²	C₃xS₃²	S₃xDic₆	C₃xS₃xQ₈	S₃²xC₆	C₃xS₃xDic₆
kernel	C₃xS₃xDic₆	C₃xS₃xDic₃	C₃xC₃²:₂Q₈	C₃²xDic₆	S₃xC₃xC₁₂	C₃xC₃²:₄Q₈	S₃xDic₆	S₃xDic₃	C₃²:₂Q₈	C₃xDic₆	S₃xC₁₂	C₃²:₄Q₈	C₃xDic₆	S₃xC₁₂	S₃xC₃²	C₃xDic₃	C₃xC₁₂	S₃xC₆	Dic₆	C₄xS₃	C₃xS₃	C₃xS₃	Dic₃	C₁₂	D₆	S₃	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	2	1	1	1	2	4	4	2	2	2	1	1	2	3	2	1	2	2	4	4	6	4	2	8	1	1	1	2	2	2	2	4

Matrix representation of C₃xS₃xDic₆ ►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

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

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

0	1	0	0	0	0
12	0	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

10	4	0	0	0	0
4	3	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

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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[10,4,0,0,0,0,4,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

C₃xS₃xDic₆ in GAP, Magma, Sage, TeX

C_3\times S_3\times {\rm Dic}_6

% in TeX

G:=Group("C3xS3xDic6");

// GroupNames label

G:=SmallGroup(432,642);

// by ID

G=gap.SmallGroup(432,642);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=1,e^2=d^6,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,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C3xS3xDic6 order 432 = 24·33

Direct product of C3, S3 and Dic6

G = C₃xS₃xDic₆ order 432 = 2⁴·3³

Direct product of C₃, S₃ and Dic₆