C6xD4:2S3 - GroupNames

G = C₆xD₄:₂S₃ order 288 = 2⁵·3²

Direct product of C₆ and D₄:₂S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₆xD₄:₂S₃, C₆².₂₆₉C₂³, D₄:₅(S₃xC₆), (C₆xD₄):₆C₆, (C₆xD₄):₁₇S₃, (C₃xD₄):₂₃D₆, Dic₆:₇(C₂xC₆), C₆.₆(C₂³xC₆), (C₆xDic₆):₂₂C₂, (C₂xDic₆):₁₂C₆, (C₂xC₁₂).₃₃₄D₆, C₂³.₂₉(S₃xC₆), C₆.₇₄(S₃xC₂³), (C₃xC₆).₄₃C₂⁴, D₆.₂(C₂²xC₆), (S₃xC₁₂):₂₀C₂², (S₃xC₆).₂₉C₂³, C₁₂.₂₀(C₂²xC₆), (C₂²xC₆).₁₀₉D₆, C₁₂.₁₇₁(C₂²xS₃), (C₆xC₁₂).₁₆₃C₂², (C₃xC₁₂).₁₂₁C₂³, (C₃xDic₆):₃₂C₂², (C₆xDic₃):₃₄C₂², (C₂²xDic₃):₁₁C₆, (D₄xC₃²):₁₉C₂², (C₂xC₆²).₈₃C₂², Dic₃.₃(C₂²xC₆), (C₃xDic₃).₃₀C₂³, (S₃xC₂xC₄):₄C₆, (D₄xC₃xC₆):₁₀C₂, C₃:₂(C₆xC₄oD₄), C₆:₂(C₃xC₄oD₄), C₄.₂₀(S₃xC₂xC₆), (S₃xC₂xC₁₂):₁₂C₂, (C₄xS₃):₄(C₂xC₆), (C₂xD₄):₈(C₃xS₃), (C₃xD₄):₆(C₂xC₆), C₃:D₄:₂(C₂xC₆), C₂.₇(S₃xC₂²xC₆), C₂².₁(S₃xC₂xC₆), (C₃xC₆):₉(C₄oD₄), (C₆xC₃:D₄):₂₄C₂, (C₂xC₃:D₄):₁₀C₆, (C₂xC₄).₆₀(S₃xC₆), (Dic₃xC₂xC₆):₁₉C₂, C₃²:₁₅(C₂xC₄oD₄), (C₂xC₁₂).₄₅(C₂xC₆), (C₂xDic₃):₉(C₂xC₆), (C₂xC₆).₁(C₂²xC₆), (C₃xC₃:D₄):₁₆C₂², (S₃xC₂xC₆).₁₁₀C₂², (C₂²xC₆).₃₄(C₂xC₆), (C₂xC₆).₂₀(C₂²xS₃), (C₂²xS₃).₃₀(C₂xC₆), SmallGroup(288,993)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₆xD₄:₂S₃

Chief series

C₁ — C₃ — C₆ — C₃xC₆ — S₃xC₆ — S₃xC₂xC₆ — S₃xC₂xC₁₂ — C₆xD₄:₂S₃

Lower central

C₃ — C₆ — C₆xD₄:₂S₃

Upper central

C₁ — C₂xC₆ — C₆xD₄

Generators and relations for C₆xD₄:₂S₃
G = < a,b,c,d,e | a⁶=b⁴=c²=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ece=b²c, ede=d^-1 >

Subgroups: 682 in 355 conjugacy classes, 178 normal (30 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, C₂³, C₃², Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₃xC₆, Dic₆, C₄xS₃, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂xC₄oD₄, C₃xDic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, C₆², C₆², C₆², C₂xDic₆, S₃xC₂xC₄, D₄:₂S₃, C₂²xDic₃, C₂xC₃:D₄, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₆xQ₈, C₃xC₄oD₄, C₃xDic₆, S₃xC₁₂, C₆xDic₃, C₆xDic₃, C₃xC₃:D₄, C₆xC₁₂, D₄xC₃², S₃xC₂xC₆, C₂xC₆², C₂xD₄:₂S₃, C₆xC₄oD₄, C₆xDic₆, S₃xC₂xC₁₂, C₃xD₄:₂S₃, Dic₃xC₂xC₆, C₆xC₃:D₄, D₄xC₃xC₆, C₆xD₄:₂S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂xC₆, C₄oD₄, C₂⁴, C₃xS₃, C₂²xS₃, C₂²xC₆, C₂xC₄oD₄, S₃xC₆, D₄:₂S₃, C₃xC₄oD₄, S₃xC₂³, C₂³xC₆, S₃xC₂xC₆, C₂xD₄:₂S₃, C₆xC₄oD₄, C₃xD₄:₂S₃, S₃xC₂²xC₆, C₆xD₄:₂S₃

Smallest permutation representation of C₆xD₄:₂S₃
►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 12 23 28)(2 7 24 29)(3 8 19 30)(4 9 20 25)(5 10 21 26)(6 11 22 27)(13 33 37 46)(14 34 38 47)(15 35 39 48)(16 36 40 43)(17 31 41 44)(18 32 42 45)

(7 29)(8 30)(9 25)(10 26)(11 27)(12 28)(31 44)(32 45)(33 46)(34 47)(35 48)(36 43)

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

90 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	6A	···	6F	6G	···	6W	6X	···	6AI	6AJ	6AK	6AL	6AM	12A	12B	12C	12D	12E	···	12L	12M	···	12R	12S	···	12Z
order	1	2	2	2	2	2	2	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	4	4	4	6	···	6	6	···	6	6	···	6	6	6	6	6	12	12	12	12	12	···	12	12	···	12	12	···	12
size	1	1	1	1	2	2	2	2	6	6	1	1	2	2	2	2	2	3	3	3	3	6	6	6	6	1	···	1	2	···	2	4	···	4	6	6	6	6	2	2	2	2	3	···	3	4	···	4	6	···	6

90 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	4	4
type	+	+	+	+	+	+	+								+	+	+	+							-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	S₃	D₆	D₆	D₆	C₄oD₄	C₃xS₃	S₃xC₆	S₃xC₆	S₃xC₆	C₃xC₄oD₄	D₄:₂S₃	C₃xD₄:₂S₃
kernel	C₆xD₄:₂S₃	C₆xDic₆	S₃xC₂xC₁₂	C₃xD₄:₂S₃	Dic₃xC₂xC₆	C₆xC₃:D₄	D₄xC₃xC₆	C₂xD₄:₂S₃	C₂xDic₆	S₃xC₂xC₄	D₄:₂S₃	C₂²xDic₃	C₂xC₃:D₄	C₆xD₄	C₆xD₄	C₂xC₁₂	C₃xD₄	C₂²xC₆	C₃xC₆	C₂xD₄	C₂xC₄	D₄	C₂³	C₆	C₆	C₂
# reps	1	1	1	8	2	2	1	2	2	2	16	4	4	2	1	1	4	2	4	2	2	8	4	8	2	4

Matrix representation of C₆xD₄:₂S₃ ►in GL₄(F₁₃) generated by

10	0	0	0
0	10	0	0
0	0	1	0
0	0	0	1

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

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

3	0	0	0
0	9	0	0
0	0	1	0
0	0	0	1

0	12	0	0
12	0	0	0
0	0	0	5
0	0	8	0

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

C₆xD₄:₂S₃ in GAP, Magma, Sage, TeX

C_6\times D_4\rtimes_2S_3

% in TeX

G:=Group("C6xD4:2S3");

// GroupNames label

G:=SmallGroup(288,993);

// by ID

G=gap.SmallGroup(288,993);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,1571,409,9414]);

// Polycyclic

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

// generators/relations

G = C6xD4:2S3 order 288 = 25·32

Direct product of C6 and D4:2S3

G = C₆xD₄:₂S₃ order 288 = 2⁵·3²

Direct product of C₆ and D₄:₂S₃