S3xC4.Dic3 - GroupNames

G = S₃×C₄.Dic₃ order 288 = 2⁵·3²

Direct product of S₃ and C₄.Dic₃

direct product, metabelian, supersoluble, monomial

Aliases: S₃×C₄.Dic₃, C₃⋊C₈⋊₁₉D₆, (S₃×C₁₂).₃C₄, C₁₂.₉₂(C₄×S₃), (C₄×S₃).₄₁D₆, C₃⋊₅(S₃×M₄(2)), (C₂×C₁₂).₂₈₉D₆, (C₃×S₃)⋊₂M₄(2), C₆².₄₃(C₂×C₄), (C₄×S₃).₁Dic₃, D₆.₇(C₂×Dic₃), (C₆×Dic₃).₉C₄, C₄.₁₄(S₃×Dic₃), C₃²⋊₅(C₂×M₄(2)), C₁₂.₅₈D₆⋊₇C₂, D₆.Dic₃⋊₁₃C₂, (C₆×C₁₂).₆₅C₂², C₁₂.₁₆(C₂×Dic₃), C₆.₂(C₂²×Dic₃), C₂².₆(S₃×Dic₃), (S₃×C₁₂).₅₇C₂², (C₃×C₁₂).₁₄₂C₂³, C₁₂.₁₄₁(C₂²×S₃), Dic₃.₅(C₂×Dic₃), C₃²⋊₄C₈⋊₁₅C₂², (C₂×Dic₃).₅Dic₃, (C₂²×S₃).₃Dic₃, (S₃×C₃⋊C₈)⋊₁₂C₂, C₄.₈₈(C₂×S₃²), (C₂×C₄).₆₁S₃², (S₃×C₂×C₆).₆C₄, (S₃×C₂×C₄).₁S₃, C₆.₈₂(S₃×C₂×C₄), (S₃×C₂×C₁₂).₄C₂, C₂.₄(C₂×S₃×Dic₃), (C₃×C₃⋊C₈)⋊₂₄C₂², (C₂×C₆).₇₂(C₄×S₃), C₃⋊₂(C₂×C₄.Dic₃), (S₃×C₆).₁₆(C₂×C₄), (C₃×C₁₂).₅₅(C₂×C₄), (C₃×C₄.Dic₃)⋊₉C₂, (C₂×C₆).₇(C₂×Dic₃), (C₃×C₆).₃₈(C₂²×C₄), (C₃×Dic₃).₂₁(C₂×C₄), SmallGroup(288,461)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — S₃×C₄.Dic₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — S₃×C₁₂ — S₃×C₃⋊C₈ — S₃×C₄.Dic₃

Lower central

C₃² — C₃×C₆ — S₃×C₄.Dic₃

Upper central

C₁ — C₄ — C₂×C₄

Generators and relations for S₃×C₄.Dic₃
G = < a,b,c,d,e | a³=b²=c⁴=1, d⁶=c², e²=c²d³, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=c^-1, ede^-1=d⁵ >

Subgroups: 370 in 146 conjugacy classes, 64 normal (50 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, C₂³, C₃², Dic₃, C₁₂, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₈, M₄(2), C₂²×C₄, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₃⋊C₈, C₂₄, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×S₃, C₂²×C₆, C₂×M₄(2), C₃×Dic₃, C₃×C₁₂, S₃×C₆, S₃×C₆, C₆², S₃×C₈, C₈⋊S₃, C₂×C₃⋊C₈, C₄.Dic₃, C₄.Dic₃, C₃×M₄(2), S₃×C₂×C₄, C₂²×C₁₂, C₃×C₃⋊C₈, C₃²⋊₄C₈, S₃×C₁₂, C₆×Dic₃, C₆×C₁₂, S₃×C₂×C₆, S₃×M₄(2), C₂×C₄.Dic₃, S₃×C₃⋊C₈, D₆.Dic₃, C₃×C₄.Dic₃, C₁₂.₅₈D₆, S₃×C₂×C₁₂, S₃×C₄.Dic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, Dic₃, D₆, M₄(2), C₂²×C₄, C₄×S₃, C₂×Dic₃, C₂²×S₃, C₂×M₄(2), S₃², C₄.Dic₃, S₃×C₂×C₄, C₂²×Dic₃, S₃×Dic₃, C₂×S₃², S₃×M₄(2), C₂×C₄.Dic₃, C₂×S₃×Dic₃, S₃×C₄.Dic₃

Smallest permutation representation of S₃×C₄.Dic₃
►On 48 points

Generators in S₄₈

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	8A	8B	8C	8D	8E	8F	8G	8H	12A	···	12F	12G	···	12K	12L	12M	12N	12O	24A	24B	24C	24D
order	1	2	2	2	2	2	3	3	3	4	4	4	4	4	4	6	6	6	6	6	6	6	6	6	6	6	6	8	8	8	8	8	8	8	8	12	···	12	12	···	12	12	12	12	12	24	24	24	24
size	1	1	2	3	3	6	2	2	4	1	1	2	3	3	6	2	2	2	2	4	4	4	4	6	6	6	6	6	6	6	6	18	18	18	18	2	···	2	4	···	4	6	6	6	6	12	12	12	12

54 irreducible representations

dim	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	4	4
type	+	+	+	+	+	+				+	+	+	-	+	-	+	-					+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	S₃	D₆	Dic₃	D₆	Dic₃	D₆	Dic₃	M₄(2)	C₄×S₃	C₄×S₃	C₄.Dic₃	S₃²	S₃×Dic₃	C₂×S₃²	S₃×Dic₃	S₃×M₄(2)	S₃×C₄.Dic₃
kernel	S₃×C₄.Dic₃	S₃×C₃⋊C₈	D₆.Dic₃	C₃×C₄.Dic₃	C₁₂.₅₈D₆	S₃×C₂×C₁₂	S₃×C₁₂	C₆×Dic₃	S₃×C₂×C₆	C₄.Dic₃	S₃×C₂×C₄	C₃⋊C₈	C₄×S₃	C₄×S₃	C₂×Dic₃	C₂×C₁₂	C₂²×S₃	C₃×S₃	C₁₂	C₂×C₆	S₃	C₂×C₄	C₄	C₄	C₂²	C₃	C₁
# reps	1	2	2	1	1	1	4	2	2	1	1	2	2	2	1	2	1	4	2	2	8	1	1	1	1	2	4

Matrix representation of S₃×C₄.Dic₃ ►in GL₆(𝔽₇₃)

0	1	0	0	0	0
72	72	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

72	0	0	0	0	0
1	1	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	27	0	0	0
0	0	72	46	0	0
0	0	0	0	1	0
0	0	0	0	0	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	27	0	0	0
0	0	0	27	0	0
0	0	0	0	1	72
0	0	0	0	1	0

27	0	0	0	0	0
0	27	0	0	0	0
0	0	27	71	0	0
0	0	59	46	0	0
0	0	0	0	0	27
0	0	0	0	27	0

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,72,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,59,0,0,0,0,71,46,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;

S₃×C₄.Dic₃ in GAP, Magma, Sage, TeX

S_3\times C_4.{\rm Dic}_3

% in TeX

G:=Group("S3xC4.Dic3");

// GroupNames label

G:=SmallGroup(288,461);

// by ID

G=gap.SmallGroup(288,461);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,219,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=1,d^6=c^2,e^2=c^2*d^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;

// generators/relations

G = S3×C4.Dic3 order 288 = 25·32

Direct product of S3 and C4.Dic3

G = S₃×C₄.Dic₃ order 288 = 2⁵·3²

Direct product of S₃ and C₄.Dic₃