S3xC3^2:2C8 - GroupNames

G = S₃×C₃²⋊₂C₈ order 432 = 2⁴·3³

Direct product of S₃ and C₃²⋊₂C₈

direct product, metabelian, soluble, monomial, A-group

Aliases: S₃×C₃²⋊₂C₈, C₃³⋊₄(C₂×C₈), (S₃×C₃²)⋊₂C₈, C₃²⋊₁₀(S₃×C₈), C₃³⋊₄C₈⋊₄C₂, C₃⋊Dic₃.₂₃D₆, C₃³⋊₅C₄.₁C₄, D₆.₂(C₃²⋊C₄), (S₃×C₃×C₆).₁C₄, C₆.₄(C₂×C₃²⋊C₄), C₂.₂(S₃×C₃²⋊C₄), (C₃×C₆).₂₉(C₄×S₃), C₃⋊₁(C₂×C₃²⋊₂C₈), (C₃×C₃²⋊₂C₈)⋊₄C₂, (S₃×C₃⋊Dic₃).₃C₂, (C₃²×C₆).₄(C₂×C₄), (C₃×C₃⋊Dic₃).₂₆C₂², SmallGroup(432,570)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — S₃×C₃²⋊₂C₈

Chief series

C₁ — C₃ — C₃³ — C₃²×C₆ — C₃×C₃⋊Dic₃ — S₃×C₃⋊Dic₃ — S₃×C₃²⋊₂C₈

Lower central

C₃³ — S₃×C₃²⋊₂C₈

Upper central

C₁ — C₂

Generators and relations for S₃×C₃²⋊₂C₈
G = < a,b,c,d,e | a³=b²=c³=d³=e⁸=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede^-1=cd=dc, ece^-1=c^-1d >

Subgroups: 544 in 88 conjugacy classes, 22 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₃², C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₈, C₃×S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, C₄×S₃, C₂×Dic₃, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃⋊Dic₃, S₃×C₆, C₆², S₃×C₈, S₃×C₃², C₃²×C₆, C₃²⋊₂C₈, C₃²⋊₂C₈, S₃×Dic₃, C₂×C₃⋊Dic₃, C₃×C₃⋊Dic₃, C₃³⋊₅C₄, S₃×C₃×C₆, C₂×C₃²⋊₂C₈, C₃×C₃²⋊₂C₈, C₃³⋊₄C₈, S₃×C₃⋊Dic₃, S₃×C₃²⋊₂C₈
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂×C₄, D₆, C₂×C₈, C₄×S₃, C₃²⋊C₄, S₃×C₈, C₃²⋊₂C₈, C₂×C₃²⋊C₄, C₂×C₃²⋊₂C₈, S₃×C₃²⋊C₄, S₃×C₃²⋊₂C₈

Smallest permutation representation of S₃×C₃²⋊₂C₈
►On 48 points

Generators in S₄₈

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

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

(1 25 35)(3 37 27)(5 29 39)(7 33 31)(9 23 44)(11 46 17)(13 19 48)(15 42 21)

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

(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)

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	6H	6I	8A	8B	8C	8D	8E	8F	8G	8H	12A	12B	24A	24B	24C	24D
order	1	2	2	2	3	3	3	3	3	4	4	4	4	6	6	6	6	6	6	6	6	6	8	8	8	8	8	8	8	8	12	12	24	24	24	24
size	1	1	3	3	2	4	4	8	8	9	9	27	27	2	4	4	8	8	12	12	12	12	9	9	9	9	27	27	27	27	18	18	18	18	18	18

36 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	4	4	4	8	8
type	+	+	+	+				+	+			+	-	+	+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	S₃	D₆	C₄×S₃	S₃×C₈	C₃²⋊C₄	C₃²⋊₂C₈	C₂×C₃²⋊C₄	S₃×C₃²⋊C₄	S₃×C₃²⋊₂C₈
kernel	S₃×C₃²⋊₂C₈	C₃×C₃²⋊₂C₈	C₃³⋊₄C₈	S₃×C₃⋊Dic₃	C₃³⋊₅C₄	S₃×C₃×C₆	S₃×C₃²	C₃²⋊₂C₈	C₃⋊Dic₃	C₃×C₆	C₃²	D₆	S₃	C₆	C₂	C₁
# reps	1	1	1	1	2	2	8	1	1	2	4	2	4	2	2	2

Matrix representation of S₃×C₃²⋊₂C₈ ►in GL₆(𝔽₇₃)

72	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

0	1	0	0	0	0
1	0	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	72

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

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	0	51	0
0	0	51	0	0	0
0	0	0	0	0	51
0	0	0	51	0	0

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,72,0,0,0,0,1,0,0,0,0,72,72,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,51,0,0,0,0,0,0,0,51,0,0,51,0,0,0,0,0,0,0,51,0] >;

S₃×C₃²⋊₂C₈ in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes_2C_8

% in TeX

G:=Group("S3xC3^2:2C8");

// GroupNames label

G:=SmallGroup(432,570);

// by ID

G=gap.SmallGroup(432,570);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,36,58,1411,298,1356,1027,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^3=e^8=1,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,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;

// generators/relations

G = S3×C32⋊2C8 order 432 = 24·33

Direct product of S3 and C32⋊2C8

G = S₃×C₃²⋊₂C₈ order 432 = 2⁴·3³

Direct product of S₃ and C₃²⋊₂C₈