S3xD4.S3 - GroupNames

G = S₃×D₄.S₃ order 288 = 2⁵·3²

Direct product of S₃ and D₄.S₃

direct product, metabelian, supersoluble, monomial

Aliases: S₃×D₄.S₃, Dic₆⋊₉D₆, D₁₂.₆D₆, C₃⋊C₈⋊₁₄D₆, D₄.₁S₃², (S₃×D₄).S₃, C₃⋊₆(S₃×SD₁₆), (S₃×Dic₆)⋊₄C₂, (C₃×S₃)⋊₂SD₁₆, (C₄×S₃).₁₉D₆, (C₃×D₄).₂₀D₆, (S₃×C₆).₃₂D₄, C₆.₁₄₈(S₃×D₄), C₃²⋊₉(C₂×SD₁₆), (C₃×C₁₂).₅C₂³, C₁₂.₅(C₂²×S₃), Dic₆⋊S₃⋊₈C₂, C₃²⋊₉SD₁₆⋊₁C₂, D₆.₂₀(C₃⋊D₄), C₃²⋊₄C₈⋊₅C₂², (C₃×Dic₃).₁₂D₄, (C₃×Dic₆)⋊₆C₂², D₁₂.S₃⋊₁₁C₂, C₃²⋊₄Q₈⋊₄C₂², (S₃×C₁₂).₁₂C₂², Dic₃.₄(C₃⋊D₄), (C₃×D₁₂).₁₂C₂², (D₄×C₃²).₁C₂², (S₃×C₃⋊C₈)⋊₄C₂, C₄.₅(C₂×S₃²), (C₃×S₃×D₄).₁C₂, C₃⋊₂(C₂×D₄.S₃), (C₃×C₃⋊C₈)⋊₁₄C₂², (C₃×D₄.S₃)⋊₅C₂, C₆.₄₄(C₂×C₃⋊D₄), C₂.₂₂(S₃×C₃⋊D₄), (C₃×C₆).₁₂₀(C₂×D₄), SmallGroup(288,576)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — S₃×D₄.S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — S₃×C₁₂ — S₃×Dic₆ — S₃×D₄.S₃

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — S₃×D₄.S₃

Upper central

C₁ — C₂ — C₄ — D₄

Generators and relations for S₃×D₄.S₃
G = < a,b,c,d,e,f | a³=b²=c⁴=d²=e³=1, f²=c², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf^-1=c^-1, ce=ec, de=ed, fdf^-1=cd, fef^-1=e^-1 >

Subgroups: 554 in 146 conjugacy classes, 44 normal (40 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, S₃, C₆, C₆, C₈, C₂×C₄, D₄, D₄, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂×C₆, C₂×C₈, SD₁₆, C₂×D₄, C₂×Q₈, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₃⋊C₈, C₂₄, Dic₆, Dic₆, C₄×S₃, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₂²×S₃, C₂²×C₆, C₂×SD₁₆, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, S₃×C₆, S₃×C₆, C₆², S₃×C₈, C₂₄⋊C₂, C₂×C₃⋊C₈, D₄.S₃, D₄.S₃, Q₈⋊₂S₃, C₃×SD₁₆, C₂×Dic₆, S₃×D₄, S₃×Q₈, C₆×D₄, C₃×C₃⋊C₈, C₃²⋊₄C₈, S₃×Dic₃, C₃²⋊₂Q₈, C₃×Dic₆, S₃×C₁₂, C₃×D₁₂, C₃×C₃⋊D₄, C₃²⋊₄Q₈, D₄×C₃², S₃×C₂×C₆, S₃×SD₁₆, C₂×D₄.S₃, S₃×C₃⋊C₈, Dic₆⋊S₃, D₁₂.S₃, C₃×D₄.S₃, C₃²⋊₉SD₁₆, S₃×Dic₆, C₃×S₃×D₄, S₃×D₄.S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, SD₁₆, C₂×D₄, C₃⋊D₄, C₂²×S₃, C₂×SD₁₆, S₃², D₄.S₃, S₃×D₄, C₂×C₃⋊D₄, C₂×S₃², S₃×SD₁₆, C₂×D₄.S₃, S₃×C₃⋊D₄, S₃×D₄.S₃

Smallest permutation representation of S₃×D₄.S₃
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	8A	8B	8C	8D	12A	12B	12C	12D	12E	24A	24B
order	1	2	2	2	2	2	3	3	3	4	4	4	4	6	6	6	6	6	6	6	6	6	6	6	6	8	8	8	8	12	12	12	12	12	24	24
size	1	1	3	3	4	12	2	2	4	2	6	12	36	2	2	4	4	4	6	6	8	8	8	12	12	6	6	18	18	4	4	8	12	24	12	12

36 irreducible representations

dim	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	8
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+				+	-	+	+			-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₄	D₄	D₆	D₆	D₆	D₆	D₆	SD₁₆	C₃⋊D₄	C₃⋊D₄	S₃²	D₄.S₃	S₃×D₄	C₂×S₃²	S₃×SD₁₆	S₃×C₃⋊D₄	S₃×D₄.S₃
kernel	S₃×D₄.S₃	S₃×C₃⋊C₈	Dic₆⋊S₃	D₁₂.S₃	C₃×D₄.S₃	C₃²⋊₉SD₁₆	S₃×Dic₆	C₃×S₃×D₄	D₄.S₃	S₃×D₄	C₃×Dic₃	S₃×C₆	C₃⋊C₈	Dic₆	C₄×S₃	D₁₂	C₃×D₄	C₃×S₃	Dic₃	D₆	D₄	S₃	C₆	C₄	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	4	2	2	1	2	1	1	2	2	1

Matrix representation of S₃×D₄.S₃ ►in GL₆(𝔽₇₃)

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

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

72	29	0	0	0	0
10	1	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	29	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

1	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	72	72
0	0	0	0	1	0

61	28	0	0	0	0
60	12	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	72	72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,10,0,0,0,0,29,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,29,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[61,60,0,0,0,0,28,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

S₃×D₄.S₃ in GAP, Magma, Sage, TeX

S_3\times D_4.S_3

% in TeX

G:=Group("S3xD4.S3");

// GroupNames label

G:=SmallGroup(288,576);

// by ID

G=gap.SmallGroup(288,576);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,135,346,185,80,1356,9414]);

// Polycyclic

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

// generators/relations

G = S3×D4.S3 order 288 = 25·32

Direct product of S3 and D4.S3

G = S₃×D₄.S₃ order 288 = 2⁵·3²

Direct product of S₃ and D₄.S₃