C3xS3xD8 - GroupNames

G = C₃xS₃xD₈ order 288 = 2⁵·3²

Direct product of C₃, S₃ and D₈

direct product, metabelian, supersoluble, monomial

Aliases: C₃xS₃xD₈, D₂₄:₄C₆, C₂₄:₁₈D₆, C₈:₄(S₃xC₆), C₃:₂(C₆xD₈), (S₃xC₈):₁C₆, C₂₄:₂(C₂xC₆), D₄:S₃:₁C₆, (C₃xD₈):₂C₆, (S₃xD₄):₁C₆, D₄:₁(S₃xC₆), (S₃xC₂₄):₅C₂, (C₃xD₄):₁₂D₆, D₁₂:₁(C₂xC₆), C₆.₂₇(C₆xD₄), C₃²:₁₁(C₂xD₈), (C₃xD₂₄):₁₂C₂, (C₃xC₂₄):₈C₂², D₆.₁₂(C₃xD₄), (S₃xC₆).₄₈D₄, C₆.₁₈₇(S₃xD₄), (C₃²xD₈):₃C₂, C₁₂.₁(C₂²xC₆), Dic₃.₃(C₃xD₄), (C₃xD₁₂):₁₀C₂², (C₃xC₁₂).₇₂C₂³, (C₃xDic₃).₃₀D₄, (D₄xC₃²):₅C₂², (S₃xC₁₂).₄₇C₂², C₁₂.₁₅₂(C₂²xS₃), C₃:C₈:₅(C₂xC₆), (C₃xS₃xD₄):₄C₂, C₄.₁(S₃xC₂xC₆), C₂.₁₅(C₃xS₃xD₄), (C₃xD₄:S₃):₉C₂, (C₃xD₄):₁(C₂xC₆), (C₃xC₃:C₈):₃₁C₂², (C₄xS₃).₇(C₂xC₆), (C₃xC₆).₂₁₅(C₂xD₄), SmallGroup(288,681)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₃xS₃xD₈

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₃xC₁₂ — S₃xC₁₂ — C₃xS₃xD₄ — C₃xS₃xD₈

Lower central

C₃ — C₆ — C₁₂ — C₃xS₃xD₈

Upper central

C₁ — C₆ — C₁₂ — C₃xD₈

Generators and relations for C₃xS₃xD₈
G = < a,b,c,d,e | a³=b³=c²=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 522 in 163 conjugacy classes, 58 normal (34 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, S₃, C₆, C₆, C₈, C₈, C₂xC₄, D₄, D₄, C₂³, C₃², Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂xC₈, D₈, D₈, C₂xD₄, C₃xS₃, C₃xS₃, C₃xC₆, C₃xC₆, C₃:C₈, C₂₄, C₂₄, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xD₄, C₂²xS₃, C₂²xC₆, C₂xD₈, C₃xDic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, C₆², S₃xC₈, D₂₄, D₄:S₃, C₂xC₂₄, C₃xD₈, C₃xD₈, S₃xD₄, C₆xD₄, C₃xC₃:C₈, C₃xC₂₄, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, D₄xC₃², S₃xC₂xC₆, S₃xD₈, C₆xD₈, S₃xC₂₄, C₃xD₂₄, C₃xD₄:S₃, C₃²xD₈, C₃xS₃xD₄, C₃xS₃xD₈
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, D₈, C₂xD₄, C₃xS₃, C₃xD₄, C₂²xS₃, C₂²xC₆, C₂xD₈, S₃xC₆, C₃xD₈, S₃xD₄, C₆xD₄, S₃xC₂xC₆, S₃xD₈, C₆xD₈, C₃xS₃xD₄, C₃xS₃xD₈

Smallest permutation representation of C₃xS₃xD₈
►On 48 points

Generators in S₄₈

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

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

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

(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)

(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 19)(20 24)(21 23)(25 29)(26 28)(30 32)(33 39)(34 38)(35 37)(41 43)(44 48)(45 47)

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

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

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

63 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	6M	6N	···	6S	6T	6U	6V	6W	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	12G	24A	24B	24C	24D	24E	···	24J	24K	24L	24M	24N
order	1	2	2	2	2	2	2	2	3	3	3	3	3	4	4	6	6	6	6	6	6	6	6	6	6	6	6	6	6	···	6	6	6	6	6	8	8	8	8	12	12	12	12	12	12	12	24	24	24	24	24	···	24	24	24	24	24
size	1	1	3	3	4	4	12	12	1	1	2	2	2	2	6	1	1	2	2	2	3	3	3	3	4	4	4	4	8	···	8	12	12	12	12	2	2	6	6	2	2	4	4	4	6	6	2	2	2	2	4	···	4	6	6	6	6

63 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	4	4	4	4
type	+	+	+	+	+	+							+	+	+	+	+	+							+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	D₄	D₄	D₆	D₆	D₈	C₃xS₃	C₃xD₄	C₃xD₄	S₃xC₆	S₃xC₆	C₃xD₈	S₃xD₄	S₃xD₈	C₃xS₃xD₄	C₃xS₃xD₈
kernel	C₃xS₃xD₈	S₃xC₂₄	C₃xD₂₄	C₃xD₄:S₃	C₃²xD₈	C₃xS₃xD₄	S₃xD₈	S₃xC₈	D₂₄	D₄:S₃	C₃xD₈	S₃xD₄	C₃xD₈	C₃xDic₃	S₃xC₆	C₂₄	C₃xD₄	C₃xS₃	D₈	Dic₃	D₆	C₈	D₄	S₃	C₆	C₃	C₂	C₁
# reps	1	1	1	2	1	2	2	2	2	4	2	4	1	1	1	1	2	4	2	2	2	2	4	8	1	2	2	4

Matrix representation of C₃xS₃xD₈ ►in GL₄(F₇) generated by

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

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

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

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

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

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

C₃xS₃xD₈ in GAP, Magma, Sage, TeX

C_3\times S_3\times D_8

% in TeX

G:=Group("C3xS3xD8");

// GroupNames label

G:=SmallGroup(288,681);

// by ID

G=gap.SmallGroup(288,681);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,303,1271,648,102,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=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,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C3xS3xD8 order 288 = 25·32

Direct product of C3, S3 and D8

G = C₃xS₃xD₈ order 288 = 2⁵·3²

Direct product of C₃, S₃ and D₈