C3xD6.3D6 - GroupNames

G = C₃xD₆.₃D₆ order 432 = 2⁴·3³

Direct product of C₃ and D₆.₃D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xD₆.₃D₆, C₆².₈₇D₆, D₆.₃(S₃xC₆), C₃:D₁₂:₂C₆, (C₆xDic₃):₆C₆, (C₆xDic₃):₉S₃, (S₃xDic₃):₅C₆, (S₃xC₆).₂₃D₆, C₆.D₆:₂C₆, C₃²:₂Q₈:₄C₆, C₃²:₇D₄:₅C₆, C₃³:₂₁(C₄oD₄), C₆².₂₃(C₂xC₆), Dic₃.₃(S₃xC₆), (C₃xDic₃).₄₇D₆, C₃²:₂₄(C₄oD₁₂), (C₃xC₆²).₁₇C₂², (C₃²xC₆).₃₀C₂³, C₃²:₂₃(D₄:₂S₃), (C₃²xDic₃).₂₇C₂², C₂.₁₂(S₃²xC₆), (C₂xC₆).₁₃S₃², C₆.₁₁(S₃xC₂xC₆), C₆.₁₁₄(C₂xS₃²), C₃:₄(C₃xC₄oD₁₂), (C₃xC₃:D₄):₇S₃, C₃:D₄:₃(C₃xS₃), (C₃xC₃:D₄):₁C₆, C₂².₁(C₃xS₃²), (Dic₃xC₃xC₆):₉C₂, C₃:₃(C₃xD₄:₂S₃), (S₃xC₆).₃(C₂xC₆), (C₂xC₆).₁₄(S₃xC₆), (C₃xS₃xDic₃):₁₂C₂, C₃²:₈(C₃xC₄oD₄), (C₂xDic₃):₃(C₃xS₃), (C₃xC₆.D₆):₅C₂, (C₃²xC₃:D₄):₁C₂, (C₃xC₃²:₇D₄):₇C₂, (S₃xC₃xC₆).₁₂C₂², (C₃xC₃:D₁₂):₁₄C₂, (C₃xC₃²:₂Q₈):₁₀C₂, (C₆xC₃:S₃).₂₅C₂², C₃:Dic₃.₁₀(C₂xC₆), (C₃xC₆).₂₁(C₂²xC₆), (C₃xDic₃).₄(C₂xC₆), (C₃xC₆).₁₃₅(C₂²xS₃), (C₃xC₃:Dic₃).₃₆C₂², (C₂xC₃:S₃).₈(C₂xC₆), SmallGroup(432,652)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xD₆.₃D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — S₃xC₃xC₆ — C₃xS₃xDic₃ — C₃xD₆.₃D₆

Lower central

C₃² — C₃xC₆ — C₃xD₆.₃D₆

Upper central

C₁ — C₆ — C₂xC₆

Generators and relations for C₃xD₆.₃D₆
G = < a,b,c,d,e | a³=b⁶=c²=d⁶=1, e²=b³, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, dcd^-1=ece^-1=b³c, ede^-1=d^-1 >

Subgroups: 712 in 218 conjugacy classes, 64 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, D₄, Q₈, C₃², C₃², Dic₃, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₄oD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₂xDic₃, C₃:D₄, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₃³, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₆², C₆², C₄oD₁₂, D₄:₂S₃, C₃xC₄oD₄, S₃xC₃², C₃xC₃:S₃, C₃²xC₆, C₃²xC₆, S₃xDic₃, C₆.D₆, C₃:D₁₂, C₃²:₂Q₈, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₆xDic₃, C₆xDic₃, C₃xC₃:D₄, C₃xC₃:D₄, C₃²:₇D₄, C₆xC₁₂, D₄xC₃², C₃²xDic₃, C₃xC₃:Dic₃, S₃xC₃xC₆, C₆xC₃:S₃, C₃xC₆², D₆.₃D₆, C₃xC₄oD₁₂, C₃xD₄:₂S₃, C₃xS₃xDic₃, C₃xC₆.D₆, C₃xC₃:D₁₂, C₃xC₃²:₂Q₈, Dic₃xC₃xC₆, C₃²xC₃:D₄, C₃xC₃²:₇D₄, C₃xD₆.₃D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂xC₆, C₄oD₄, C₃xS₃, C₂²xS₃, C₂²xC₆, S₃², S₃xC₆, C₄oD₁₂, D₄:₂S₃, C₃xC₄oD₄, C₂xS₃², S₃xC₂xC₆, C₃xS₃², D₆.₃D₆, C₃xC₄oD₁₂, C₃xD₄:₂S₃, S₃²xC₆, C₃xD₆.₃D₆

Permutation representations of C₃xD₆.₃D₆
►On 24 points - transitive group 24T1281

Generators in S₂₄

(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)

(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

(1 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)

(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 14 15 16 17 18)(19 24 23 22 21 20)

(1 12 4 9)(2 7 5 10)(3 8 6 11)(13 23 16 20)(14 24 17 21)(15 19 18 22)

G:=sub<Sym(24)| (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,22)(8,21)(9,20)(10,19)(11,24)(12,23), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,12,4,9)(2,7,5,10)(3,8,6,11)(13,23,16,20)(14,24,17,21)(15,19,18,22)>;

G:=Group( (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,22)(8,21)(9,20)(10,19)(11,24)(12,23), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,12,4,9)(2,7,5,10)(3,8,6,11)(13,23,16,20)(14,24,17,21)(15,19,18,22) );

G=PermutationGroup([[(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,22),(8,21),(9,20),(10,19),(11,24),(12,23)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,14,15,16,17,18),(19,24,23,22,21,20)], [(1,12,4,9),(2,7,5,10),(3,8,6,11),(13,23,16,20),(14,24,17,21),(15,19,18,22)]])

G:=TransitiveGroup(24,1281);

81 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	4C	4D	4E	6A	6B	6C	···	6P	6Q	···	6AB	6AC	6AD	6AE	6AF	6AG	6AH	6AI	12A	12B	12C	12D	12E	···	12T	12U	12V	12W	12X	12Y
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
size	1	1	2	6	18	1	1	2	···	2	4	4	4	3	3	6	6	18	1	1	2	···	2	4	···	4	6	6	12	12	12	18	18	3	3	3	3	6	···	6	12	12	12	18	18

81 irreducible representations

dim	1	1	1	1	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	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	C₆	S₃	S₃	D₆	D₆	D₆	C₄oD₄	C₃xS₃	C₃xS₃	S₃xC₆	S₃xC₆	S₃xC₆	C₄oD₁₂	C₃xC₄oD₄	C₃xC₄oD₁₂	S₃²	D₄:₂S₃	C₂xS₃²	C₃xS₃²	D₆.₃D₆	C₃xD₄:₂S₃	S₃²xC₆	C₃xD₆.₃D₆
kernel	C₃xD₆.₃D₆	C₃xS₃xDic₃	C₃xC₆.D₆	C₃xC₃:D₁₂	C₃xC₃²:₂Q₈	Dic₃xC₃xC₆	C₃²xC₃:D₄	C₃xC₃²:₇D₄	D₆.₃D₆	S₃xDic₃	C₆.D₆	C₃:D₁₂	C₃²:₂Q₈	C₆xDic₃	C₃xC₃:D₄	C₃²:₇D₄	C₆xDic₃	C₃xC₃:D₄	C₃xDic₃	S₃xC₆	C₆²	C₃³	C₂xDic₃	C₃:D₄	Dic₃	D₆	C₂xC₆	C₃²	C₃²	C₃	C₂xC₆	C₃²	C₆	C₂²	C₃	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	1	1	3	1	2	2	2	2	6	2	4	4	4	8	1	1	1	2	2	2	2	4

Matrix representation of C₃xD₆.₃D₆ ►in GL₄(F₇) generated by

4	0	0	0
0	4	0	0
0	0	4	0
0	0	0	4

0	0	3	1
5	4	6	0
1	6	5	5
3	3	2	0

3	5	0	1
6	4	4	5
2	1	0	4
4	0	1	0

4	1	4	4
5	6	5	3
3	1	0	2
5	2	4	4

4	5	4	6
6	0	6	0
3	3	3	1
3	4	2	0

G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,5,1,3,0,4,6,3,3,6,5,2,1,0,5,0],[3,6,2,4,5,4,1,0,0,4,0,1,1,5,4,0],[4,5,3,5,1,6,1,2,4,5,0,4,4,3,2,4],[4,6,3,3,5,0,3,4,4,6,3,2,6,0,1,0] >;

C₃xD₆.₃D₆ in GAP, Magma, Sage, TeX

C_3\times D_6._3D_6

% in TeX

G:=Group("C3xD6.3D6");

// GroupNames label

G:=SmallGroup(432,652);

// by ID

G=gap.SmallGroup(432,652);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,2028,14118]);

// Polycyclic

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

// generators/relations

G = C3xD6.3D6 order 432 = 24·33

Direct product of C3 and D6.3D6

G = C₃xD₆.₃D₆ order 432 = 2⁴·3³

Direct product of C₃ and D₆.₃D₆