S3xC3^2:2Q8 - GroupNames

G = S₃×C₃²⋊₂Q₈ order 432 = 2⁴·3³

Direct product of S₃ and C₃²⋊₂Q₈

direct product, metabelian, supersoluble, monomial

Aliases: S₃×C₃²⋊₂Q₈, D₆.₉S₃², C₃³⋊₄(C₂×Q₈), C₃⋊₁(S₃×Dic₆), Dic₃.₁S₃², (S₃×Dic₃).S₃, (C₃×S₃)⋊₁Dic₆, (S₃×C₆).₁₉D₆, (S₃×C₃²)⋊₂Q₈, C₃²⋊₁₁(S₃×Q₈), C₃³⋊₄Q₈⋊₁C₂, C₃³⋊₅Q₈⋊₃C₂, C₃⋊Dic₃.₂₇D₆, C₃²⋊₆(C₂×Dic₆), (C₃×Dic₃).₁₉D₆, (C₃²×C₆).₁₀C₂³, C₃³⋊₅C₄.₁C₂², (C₃²×Dic₃).₁C₂², C₂.₁₀S₃³, C₆.₁₀(C₂×S₃²), C₃⋊₁(C₂×C₃²⋊₂Q₈), (S₃×C₃×C₆).₃C₂², (C₃×S₃×Dic₃).₁C₂, (C₃×C₃²⋊₂Q₈)⋊₂C₂, (S₃×C₃⋊Dic₃).₁C₂, (C₃×C₆).₅₉(C₂²×S₃), (C₃×C₃⋊Dic₃).₄C₂², SmallGroup(432,603)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₆ — S₃×C₃²⋊₂Q₈

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — S₃×C₃×C₆ — C₃×S₃×Dic₃ — S₃×C₃²⋊₂Q₈

Lower central

C₃³ — C₃²×C₆ — S₃×C₃²⋊₂Q₈

Upper central

C₁ — C₂

Generators and relations for S₃×C₃²⋊₂Q₈
G = < a,b,c,d,e,f | a³=b²=c³=d³=e⁴=1, f²=e², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece^-1=c^-1, cf=fc, de=ed, fdf^-1=d^-1, fef^-1=e^-1 >

Subgroups: 980 in 198 conjugacy classes, 54 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₆, C₂×C₄, Q₈, C₃², C₃², C₃², Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₂×Q₈, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₆, C₃×C₆, Dic₆, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₃×Q₈, C₃³, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃⋊Dic₃, C₃⋊Dic₃, C₃×C₁₂, S₃×C₆, S₃×C₆, C₆², C₂×Dic₆, S₃×Q₈, S₃×C₃², C₃²×C₆, S₃×Dic₃, S₃×Dic₃, C₃²⋊₂Q₈, C₃²⋊₂Q₈, C₃×Dic₆, S₃×C₁₂, C₆×Dic₃, C₃²⋊₄Q₈, C₂×C₃⋊Dic₃, C₃²×Dic₃, C₃×C₃⋊Dic₃, C₃×C₃⋊Dic₃, C₃³⋊₅C₄, S₃×C₃×C₆, S₃×Dic₆, C₂×C₃²⋊₂Q₈, C₃×S₃×Dic₃, C₃×C₃²⋊₂Q₈, S₃×C₃⋊Dic₃, C₃³⋊₄Q₈, C₃³⋊₅Q₈, S₃×C₃²⋊₂Q₈
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, C₂×Q₈, Dic₆, C₂²×S₃, S₃², C₂×Dic₆, S₃×Q₈, C₃²⋊₂Q₈, C₂×S₃², S₃×Dic₆, C₂×C₃²⋊₂Q₈, S₃³, S₃×C₃²⋊₂Q₈

Smallest permutation representation of S₃×C₃²⋊₂Q₈
►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 14 19)(2 20 15)(3 16 17)(4 18 13)(5 10 45)(6 46 11)(7 12 47)(8 48 9)(21 25 30)(22 31 26)(23 27 32)(24 29 28)(33 42 37)(34 38 43)(35 44 39)(36 40 41)

(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 25 30)(22 26 31)(23 27 32)(24 28 29)(33 37 42)(34 38 43)(35 39 44)(36 40 41)

(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)(5,45,10)(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), (1,14,19)(2,20,15)(3,16,17)(4,18,13)(5,10,45)(6,46,11)(7,12,47)(8,48,9)(21,25,30)(22,31,26)(23,27,32)(24,29,28)(33,42,37)(34,38,43)(35,44,39)(36,40,41), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,45)(6,11,46)(7,12,47)(8,9,48)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (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)(5,45,10)(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), (1,14,19)(2,20,15)(3,16,17)(4,18,13)(5,10,45)(6,46,11)(7,12,47)(8,48,9)(21,25,30)(22,31,26)(23,27,32)(24,29,28)(33,42,37)(34,38,43)(35,44,39)(36,40,41), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,45)(6,11,46)(7,12,47)(8,9,48)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (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),(5,45,10),(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)], [(1,14,19),(2,20,15),(3,16,17),(4,18,13),(5,10,45),(6,46,11),(7,12,47),(8,48,9),(21,25,30),(22,31,26),(23,27,32),(24,29,28),(33,42,37),(34,38,43),(35,44,39),(36,40,41)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,10,45),(6,11,46),(7,12,47),(8,9,48),(21,25,30),(22,26,31),(23,27,32),(24,28,29),(33,37,42),(34,38,43),(35,39,44),(36,40,41)], [(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	3A	3B	3C	3D	3E	3F	3G	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	6M	12A	12B	12C	12D	12E	···	12J	12K	12L	12M	12N	12O
order	1	2	2	2	3	3	3	3	3	3	3	4	4	4	4	4	4	6	6	6	6	6	6	6	6	6	6	6	6	6	12	12	12	12	12	···	12	12	12	12	12	12
size	1	1	3	3	2	2	2	4	4	4	8	6	6	18	18	18	54	2	2	2	4	4	4	6	6	6	6	8	12	12	6	6	6	6	12	···	12	18	18	18	18	36

45 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	4	8	8
type	+	+	+	+	+	+	+	+	-	+	+	+	-	+	+	-	-	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	S₃	Q₈	D₆	D₆	D₆	Dic₆	S₃²	S₃²	S₃×Q₈	C₃²⋊₂Q₈	C₂×S₃²	S₃×Dic₆	S₃³	S₃×C₃²⋊₂Q₈
kernel	S₃×C₃²⋊₂Q₈	C₃×S₃×Dic₃	C₃×C₃²⋊₂Q₈	S₃×C₃⋊Dic₃	C₃³⋊₄Q₈	C₃³⋊₅Q₈	S₃×Dic₃	C₃²⋊₂Q₈	S₃×C₃²	C₃×Dic₃	C₃⋊Dic₃	S₃×C₆	C₃×S₃	Dic₃	D₆	C₃²	S₃	C₆	C₃	C₂	C₁
# reps	1	2	1	1	2	1	2	1	2	4	3	2	8	2	1	1	2	3	4	1	1

Matrix representation of S₃×C₃²⋊₂Q₈ ►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	12	0	0	0	0
12	0	0	0	0	0
0	0	12	0	0	0
0	0	0	12	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	12	1
0	0	0	0	12	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	9	0	0	0
0	0	0	3	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	8	0	0	0
0	0	0	5	0	0
0	0	0	0	0	1
0	0	0	0	1	0

12	0	0	0	0	0
0	12	0	0	0	0
0	0	0	1	0	0
0	0	12	0	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],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S₃×C₃²⋊₂Q₈ in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes_2Q_8

% in TeX

G:=Group("S3xC3^2:2Q8");

// GroupNames label

G:=SmallGroup(432,603);

// by ID

G=gap.SmallGroup(432,603);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,298,2028,14118]);

// Polycyclic

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

// generators/relations

G = S3×C32⋊2Q8 order 432 = 24·33

Direct product of S3 and C32⋊2Q8

G = S₃×C₃²⋊₂Q₈ order 432 = 2⁴·3³

Direct product of S₃ and C₃²⋊₂Q₈