S3xQ8:2S3 - GroupNames

G = S₃×Q₈⋊₂S₃ order 288 = 2⁵·3²

Direct product of S₃ and Q₈⋊₂S₃

direct product, metabelian, supersoluble, monomial

Aliases: S₃×Q₈⋊₂S₃, Dic₆⋊₆D₆, D₁₂.₂₃D₆, Q₈⋊₂S₃², C₃⋊C₈⋊₁₆D₆, (C₃×Q₈)⋊₉D₆, (S₃×Q₈)⋊₄S₃, C₃⋊₇(S₃×SD₁₆), (C₃×S₃)⋊₃SD₁₆, (C₄×S₃).₂₂D₆, (S₃×D₁₂).₂C₂, (S₃×C₆).₃₅D₄, C₆.₁₅₄(S₃×D₄), C₃²⋊₁₁(C₂×SD₁₆), C₃²⋊₁₁SD₁₆⋊₁C₂, D₆.₂₁(C₃⋊D₄), C₁₂.₁₅(C₂²×S₃), (C₃×C₁₂).₁₅C₂³, C₃²⋊₄C₈⋊₈C₂², (C₃×Dic₃).₁₅D₄, C₃²⋊₅SD₁₆⋊₁₃C₂, Dic₆⋊S₃⋊₁₁C₂, (Q₈×C₃²)⋊₁C₂², (S₃×C₁₂).₁₈C₂², Dic₃.₅(C₃⋊D₄), (C₃×Dic₆)⋊₁₀C₂², (C₃×D₁₂).₁₆C₂², C₁₂⋊S₃.₁₀C₂², (S₃×C₃⋊C₈)⋊₆C₂, (C₃×S₃×Q₈)⋊₁C₂, C₄.₁₅(C₂×S₃²), (C₃×C₃⋊C₈)⋊₁₆C₂², C₃⋊₂(C₂×Q₈⋊₂S₃), C₆.₅₀(C₂×C₃⋊D₄), C₂.₂₈(S₃×C₃⋊D₄), (C₃×Q₈⋊₂S₃)⋊₅C₂, (C₃×C₆).₁₃₀(C₂×D₄), SmallGroup(288,586)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — S₃×Q₈⋊₂S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — S₃×C₁₂ — S₃×D₁₂ — S₃×Q₈⋊₂S₃

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — S₃×Q₈⋊₂S₃

Upper central

C₁ — C₂ — C₄ — Q₈

Generators and relations for S₃×Q₈⋊₂S₃
G = < a,b,c,d,e,f | a³=b²=c⁴=e³=f²=1, d²=c², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd^-1=fcf=c^-1, ce=ec, de=ed, fdf=c^-1d, fef=e^-1 >

Subgroups: 682 in 146 conjugacy classes, 44 normal (40 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, S₃, C₆, C₆, C₈, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂×C₆, C₂×C₈, SD₁₆, C₂×D₄, C₂×Q₈, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃⋊C₈, C₃⋊C₈, C₂₄, Dic₆, Dic₆, C₄×S₃, C₄×S₃, D₁₂, D₁₂, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₃×Q₈, C₂²×S₃, C₂×SD₁₆, C₃×Dic₃, C₃×Dic₃, C₃×C₁₂, C₃×C₁₂, S₃², S₃×C₆, S₃×C₆, C₂×C₃⋊S₃, S₃×C₈, C₂₄⋊C₂, C₂×C₃⋊C₈, D₄.S₃, Q₈⋊₂S₃, Q₈⋊₂S₃, C₃×SD₁₆, C₂×D₁₂, S₃×D₄, S₃×Q₈, C₆×Q₈, C₃×C₃⋊C₈, C₃²⋊₄C₈, C₃⋊D₁₂, C₃×Dic₆, C₃×Dic₆, S₃×C₁₂, S₃×C₁₂, C₃×D₁₂, C₁₂⋊S₃, Q₈×C₃², C₂×S₃², S₃×SD₁₆, C₂×Q₈⋊₂S₃, S₃×C₃⋊C₈, Dic₆⋊S₃, C₃²⋊₅SD₁₆, C₃×Q₈⋊₂S₃, C₃²⋊₁₁SD₁₆, S₃×D₁₂, C₃×S₃×Q₈, S₃×Q₈⋊₂S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, SD₁₆, C₂×D₄, C₃⋊D₄, C₂²×S₃, C₂×SD₁₆, S₃², Q₈⋊₂S₃, S₃×D₄, C₂×C₃⋊D₄, C₂×S₃², S₃×SD₁₆, C₂×Q₈⋊₂S₃, S₃×C₃⋊D₄, S₃×Q₈⋊₂S₃

Smallest permutation representation of S₃×Q₈⋊₂S₃
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	12G	12H	12I	12J	12K	24A	24B
order	1	2	2	2	2	2	3	3	3	4	4	4	4	6	6	6	6	6	6	8	8	8	8	12	12	12	12	12	12	12	12	12	12	12	24	24
size	1	1	3	3	12	36	2	2	4	2	4	6	12	2	2	4	6	6	24	6	6	18	18	4	4	4	4	8	8	8	8	12	12	12	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	Q₈⋊₂S₃	S₃×D₄	C₂×S₃²	S₃×SD₁₆	S₃×C₃⋊D₄	S₃×Q₈⋊₂S₃
kernel	S₃×Q₈⋊₂S₃	S₃×C₃⋊C₈	Dic₆⋊S₃	C₃²⋊₅SD₁₆	C₃×Q₈⋊₂S₃	C₃²⋊₁₁SD₁₆	S₃×D₁₂	C₃×S₃×Q₈	Q₈⋊₂S₃	S₃×Q₈	C₃×Dic₃	S₃×C₆	C₃⋊C₈	Dic₆	C₄×S₃	D₁₂	C₃×Q₈	C₃×S₃	Dic₃	D₆	Q₈	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×Q₈⋊₂S₃ ►in GL₆(𝔽₇₃)

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

0	72	0	0	0	0
1	0	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

6	67	0	0	0	0
67	67	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	72	0
0	0	0	0	0	72

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	0	72
0	0	0	0	1	72

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

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,67,0,0,0,0,67,67,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

S₃×Q₈⋊₂S₃ in GAP, Magma, Sage, TeX

S_3\times Q_8\rtimes_2S_3

% in TeX

G:=Group("S3xQ8:2S3");

// GroupNames label

G:=SmallGroup(288,586);

// by ID

G=gap.SmallGroup(288,586);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,100,346,185,80,1356,9414]);

// Polycyclic

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

// generators/relations

G = S3×Q8⋊2S3 order 288 = 25·32

Direct product of S3 and Q8⋊2S3

G = S₃×Q₈⋊₂S₃ order 288 = 2⁵·3²

Direct product of S₃ and Q₈⋊₂S₃