S3^2xQ8 - GroupNames

G = S₃²×Q₈ order 288 = 2⁵·3²

Direct product of S₃, S₃ and Q₈

direct product, metabelian, supersoluble, monomial, rational

Aliases: S₃²×Q₈, Dic₆⋊₁₄D₆, (C₃×Q₈)⋊₁₀D₆, (C₄×S₃).₂₆D₆, (S₃×Dic₆)⋊₁₂C₂, C₃²⋊₄(C₂²×Q₈), (C₃×C₆).₂₄C₂⁴, C₆.₂₄(S₃×C₂³), (S₃×C₆).₂₆C₂³, (C₃×C₁₂).₃₆C₂³, C₁₂.₃₆(C₂²×S₃), D₆.₂₆(C₂²×S₃), C₃²⋊₂Q₈⋊₅C₂², Dic₃.D₆⋊₁₄C₂, (Q₈×C₃²)⋊₉C₂², (S₃×C₁₂).₃₅C₂², (C₃×Dic₆)⋊₁₆C₂², C₃⋊Dic₃.₂₅C₂³, (S₃×Dic₃).₆C₂², C₃²⋊₄Q₈⋊₁₁C₂², Dic₃.₁₃(C₂²×S₃), (C₃×Dic₃).₁₇C₂³, C₆.D₆.₁₀C₂², C₃⋊₃(C₂×S₃×Q₈), (C₃×S₃×Q₈)⋊₉C₂, (C₄×S₃²).₄C₂, C₄.₃₆(C₂×S₃²), (Q₈×C₃⋊S₃)⋊₇C₂, C₃⋊S₃⋊₃(C₂×Q₈), (C₃×S₃)⋊₂(C₂×Q₈), C₂.₂₆(C₂²×S₃²), (C₂×S₃²).₁₆C₂², (C₄×C₃⋊S₃).₄₆C₂², (C₂×C₃⋊S₃).₄₇C₂³, SmallGroup(288,965)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — S₃²×Q₈

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — S₃×C₆ — C₂×S₃² — C₄×S₃² — S₃²×Q₈

Lower central

C₃² — C₃×C₆ — S₃²×Q₈

Upper central

C₁ — C₂ — Q₈

Generators and relations for S₃²×Q₈
G = < a,b,c,d,e,f | a³=b²=c³=d²=e⁴=1, f²=e², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 1058 in 331 conjugacy classes, 122 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, S₃, C₆, C₆, C₂×C₄, Q₈, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂×C₆, C₂²×C₄, C₂×Q₈, C₃×S₃, C₃⋊S₃, C₃×C₆, Dic₆, Dic₆, C₄×S₃, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₃×Q₈, C₃×Q₈, C₂²×S₃, C₂²×Q₈, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, S₃², S₃×C₆, C₂×C₃⋊S₃, C₂×Dic₆, S₃×C₂×C₄, S₃×Q₈, S₃×Q₈, C₆×Q₈, S₃×Dic₃, C₆.D₆, C₃²⋊₂Q₈, C₃×Dic₆, S₃×C₁₂, C₃²⋊₄Q₈, C₄×C₃⋊S₃, Q₈×C₃², C₂×S₃², C₂×S₃×Q₈, S₃×Dic₆, Dic₃.D₆, C₄×S₃², C₃×S₃×Q₈, Q₈×C₃⋊S₃, S₃²×Q₈
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, C₂×Q₈, C₂⁴, C₂²×S₃, C₂²×Q₈, S₃², S₃×Q₈, S₃×C₂³, C₂×S₃², C₂×S₃×Q₈, C₂²×S₃², S₃²×Q₈

Smallest permutation representation of S₃²×Q₈
►On 48 points

Generators in S₄₈

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

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

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

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

45 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	4A	4B	4C	4D	···	4I	4J	4K	4L	6A	6B	6C	6D	6E	6F	6G	12A	···	12F	12G	12H	12I	12J	···	12O
order	1	2	2	2	2	2	2	2	3	3	3	4	4	4	4	···	4	4	4	4	6	6	6	6	6	6	6	12	···	12	12	12	12	12	···	12
size	1	1	3	3	3	3	9	9	2	2	4	2	2	2	6	···	6	18	18	18	2	2	4	6	6	6	6	4	···	4	8	8	8	12	···	12

45 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	4	4	4	8
type	+	+	+	+	+	+	+	-	+	+	+	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	Q₈	D₆	D₆	D₆	S₃²	S₃×Q₈	C₂×S₃²	S₃²×Q₈
kernel	S₃²×Q₈	S₃×Dic₆	Dic₃.D₆	C₄×S₃²	C₃×S₃×Q₈	Q₈×C₃⋊S₃	S₃×Q₈	S₃²	Dic₆	C₄×S₃	C₃×Q₈	Q₈	S₃	C₄	C₁
# reps	1	6	3	3	2	1	2	4	6	6	2	1	4	3	1

Matrix representation of S₃²×Q₈ ►in GL₆(𝔽₁₃)

12	1	0	0	0	0
12	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

0	1	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	12	0
0	0	0	0	0	12

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	12	12

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	11	0	0
0	0	1	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	10	0	0
0	0	0	5	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,10,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S₃²×Q₈ in GAP, Magma, Sage, TeX

S_3^2\times Q_8

% in TeX

G:=Group("S3^2xQ8");

// GroupNames label

G:=SmallGroup(288,965);

// by ID

G=gap.SmallGroup(288,965);

# by ID

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

// Polycyclic

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

// generators/relations

G = S32×Q8 order 288 = 25·32

Direct product of S3, S3 and Q8

G = S₃²×Q₈ order 288 = 2⁵·3²

Direct product of S₃, S₃ and Q₈