C3xDic3.D4 - GroupNames

G = C₃xDic₃.D₄ order 288 = 2⁵·3²

Direct product of C₃ and Dic₃.D₄

direct product, metabelian, supersoluble, monomial

Aliases: C₃xDic₃.D₄, C₆²:₅Q₈, C₆².₁₆₉C₂³, C₆.₄(C₆xQ₈), C₄:Dic₃:₂C₆, (C₂xC₆):₆Dic₆, C₆.₁₆(C₆xD₄), Dic₃:C₄:₄C₆, (C₂xDic₆):₂C₆, C₆.₁₇₄(S₃xD₄), C₂.₆(C₆xDic₆), (C₆xDic₆):₂₆C₂, (C₂xC₁₂).₂₃₀D₆, C₂³.₂₂(S₃xC₆), Dic₃.₆(C₃xD₄), C₆.₅₀(C₂xDic₆), C₂²:₃(C₃xDic₆), (C₃xDic₃).₄₃D₄, C₆.D₄.₂C₆, (C₂²xC₆).₁₀₃D₆, C₃²:₁₆(C₂²:Q₈), (C₆xC₁₂).₁₈₈C₂², (C₂xC₆²).₄₅C₂², C₆.₁₁₀(D₄:₂S₃), (C₂²xDic₃).₄C₆, (C₆xDic₃).₉₁C₂², C₂.₆(C₃xS₃xD₄), (C₂xC₆):₂(C₃xQ₈), (C₂xC₄).₅(S₃xC₆), C₃:₁(C₃xC₂²:Q₈), (C₂xC₁₂).₁(C₂xC₆), C₆.₂₀(C₃xC₄oD₄), C₂².₃₉(S₃xC₂xC₆), (C₃xC₆).₄₇(C₂xQ₈), (C₃xC₄:Dic₃):₂₆C₂, C₂.₆(C₃xD₄:₂S₃), (C₃xC₆).₂₀₃(C₂xD₄), (C₃xC₂²:C₄).₂C₆, C₂²:C₄.₁(C₃xS₃), (Dic₃xC₂xC₆).₁₁C₂, (C₃xDic₃:C₄):₂₃C₂, (C₃xC₂²:C₄).₁₅S₃, (C₂²xC₆).₁₉(C₂xC₆), (C₂xC₆).₂₄(C₂²xC₆), (C₂xDic₃).₅(C₂xC₆), (C₃xC₆).₁₂₆(C₄oD₄), (C₂xC₆).₃₀₂(C₂²xS₃), (C₃xC₆.D₄).₇C₂, (C₃²xC₂²:C₄).₄C₂, SmallGroup(288,649)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₃xDic₃.D₄

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₆² — C₆xDic₃ — Dic₃xC₂xC₆ — C₃xDic₃.D₄

Lower central

C₃ — C₂xC₆ — C₃xDic₃.D₄

Upper central

C₁ — C₂xC₆ — C₃xC₂²:C₄

Generators and relations for C₃xDic₃.D₄
G = < a,b,c,d,e | a³=b⁶=d⁴=e²=1, c²=b³, ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=b^-1, bd=db, be=eb, dcd^-1=b³c, ce=ec, ede=b³d^-1 >

Subgroups: 354 in 165 conjugacy classes, 70 normal (58 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₂xC₄, C₂xC₄, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xQ₈, C₃xC₆, C₃xC₆, Dic₆, C₂xDic₃, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xQ₈, C₂²xC₆, C₂²xC₆, C₂²:Q₈, C₃xDic₃, C₃xDic₃, C₃xC₁₂, C₆², C₆², C₆², Dic₃:C₄, C₄:Dic₃, C₆.D₄, C₃xC₂²:C₄, C₃xC₂²:C₄, C₃xC₄:C₄, C₂xDic₆, C₂²xDic₃, C₂²xC₁₂, C₆xQ₈, C₃xDic₆, C₆xDic₃, C₆xDic₃, C₆xC₁₂, C₂xC₆², Dic₃.D₄, C₃xC₂²:Q₈, C₃xDic₃:C₄, C₃xC₄:Dic₃, C₃xC₆.D₄, C₃²xC₂²:C₄, C₆xDic₆, Dic₃xC₂xC₆, C₃xDic₃.D₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, Q₈, C₂³, D₆, C₂xC₆, C₂xD₄, C₂xQ₈, C₄oD₄, C₃xS₃, Dic₆, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₂²:Q₈, S₃xC₆, C₂xDic₆, S₃xD₄, D₄:₂S₃, C₆xD₄, C₆xQ₈, C₃xC₄oD₄, C₃xDic₆, S₃xC₂xC₆, Dic₃.D₄, C₃xC₂²:Q₈, C₆xDic₆, C₃xS₃xD₄, C₃xD₄:₂S₃, C₃xDic₃.D₄

Smallest permutation representation of C₃xDic₃.D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6F	6G	···	6S	6T	···	6Y	12A	···	12P	12Q	···	12X	12Y	12Z	12AA	12AB
order	1	2	2	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	4	6	···	6	6	···	6	6	···	6	12	···	12	12	···	12	12	12	12	12
size	1	1	1	1	2	2	1	1	2	2	2	4	4	6	6	6	6	12	12	1	···	1	2	···	2	4	···	4	4	···	4	6	···	6	12	12	12	12

72 irreducible representations

dim	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
type	+	+	+	+	+	+	+								+	+	-	+	+				-						+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	S₃	D₄	Q₈	D₆	D₆	C₄oD₄	C₃xS₃	C₃xD₄	Dic₆	C₃xQ₈	S₃xC₆	S₃xC₆	C₃xC₄oD₄	C₃xDic₆	S₃xD₄	D₄:₂S₃	C₃xS₃xD₄	C₃xD₄:₂S₃
kernel	C₃xDic₃.D₄	C₃xDic₃:C₄	C₃xC₄:Dic₃	C₃xC₆.D₄	C₃²xC₂²:C₄	C₆xDic₆	Dic₃xC₂xC₆	Dic₃.D₄	Dic₃:C₄	C₄:Dic₃	C₆.D₄	C₃xC₂²:C₄	C₂xDic₆	C₂²xDic₃	C₃xC₂²:C₄	C₃xDic₃	C₆²	C₂xC₁₂	C₂²xC₆	C₃xC₆	C₂²:C₄	Dic₃	C₂xC₆	C₂xC₆	C₂xC₄	C₂³	C₆	C₂²	C₆	C₆	C₂	C₂
# reps	1	2	1	1	1	1	1	2	4	2	2	2	2	2	1	2	2	2	1	2	2	4	4	4	4	2	4	8	1	1	2	2

Matrix representation of C₃xDic₃.D₄ ►in GL₄(F₁₃) generated by

1	0	0	0
0	1	0	0
0	0	3	0
0	0	0	3

1	0	0	0
0	1	0	0
0	0	10	0
0	0	9	4

12	0	0	0
0	12	0	0
0	0	1	8
0	0	3	12

0	12	0	0
1	0	0	0
0	0	5	0
0	0	2	8

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

G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,10,9,0,0,0,4],[12,0,0,0,0,12,0,0,0,0,1,3,0,0,8,12],[0,1,0,0,12,0,0,0,0,0,5,2,0,0,0,8],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12] >;

C₃xDic₃.D₄ in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3.D_4

% in TeX

G:=Group("C3xDic3.D4");

// GroupNames label

G:=SmallGroup(288,649);

// by ID

G=gap.SmallGroup(288,649);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,590,555,142,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=b^3*d^-1>;

// generators/relations

G = C3xDic3.D4 order 288 = 25·32

Direct product of C3 and Dic3.D4

G = C₃xDic₃.D₄ order 288 = 2⁵·3²

Direct product of C₃ and Dic₃.D₄