C3^4.7S3 - GroupNames

G = C₃⁴.₇S₃ order 486 = 2·3⁵

7^th non-split extension by C₃⁴ of S₃ acting faithfully

Aliases: C₃⁴.₇S₃, C₃²:C₉:₅C₆, C₃²:₃(C₉:C₆), C₃²:₂D₉:₅C₃, C₃³.₄₁(C₃xS₃), C₃⁴.C₃:₂C₂, C₃³.₃₀(C₃:S₃), (C₃x3_-¹⁺²):₂S₃, C₃.₈(C₃³.S₃), C₃².₂(He₃:C₂), (C₃xC₉):₁(C₃xS₃), C₃².₃₇(C₃xC₃:S₃), C₃.₃(C₃xHe₃:C₂), SmallGroup(486,147)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₃²:C₉ — C₃⁴.₇S₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²:C₉ — C₃⁴.C₃ — C₃⁴.₇S₃

Lower central

C₃²:C₉ — C₃⁴.₇S₃

Upper central

C₁ — C₃

Generators and relations for C₃⁴.₇S₃
G = < a,b,c,d,e,f | a³=b³=c³=d³=f²=1, e³=c, ab=ba, ac=ca, ad=da, eae^-1=ac^-1, af=fa, bc=cb, ebe^-1=bd=db, fbf=b^-1, cd=dc, ce=ec, fcf=c^-1, de=ed, df=fd, fef=c^-1e² >

Subgroups: 740 in 147 conjugacy classes, 23 normal (11 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₃², D₉, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₉, C₃xC₉, 3_-¹⁺², C₃³, C₃³, C₃xD₉, C₉:C₆, S₃xC₃², C₃xC₃:S₃, C₃²:C₉, C₃²:C₉, C₃x3_-¹⁺², C₃⁴, C₃²:₂D₉, C₃xC₉:C₆, C₃²xC₃:S₃, C₃⁴.C₃, C₃⁴.₇S₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃xS₃, C₃:S₃, C₉:C₆, He₃:C₂, C₃xC₃:S₃, C₃xHe₃:C₂, C₃³.S₃, C₃⁴.₇S₃

Permutation representations of C₃⁴.₇S₃
►On 18 points - transitive group 18T171

Generators in S₁₈

(1 7 4)(2 5 8)(10 16 13)(11 14 17)

(2 8 5)(3 6 9)(10 16 13)(12 15 18)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)

(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)

(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(9 18)

G:=sub<Sym(18)| (1,7,4)(2,5,8)(10,16,13)(11,14,17), (2,8,5)(3,6,9)(10,16,13)(12,15,18), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(9,18)>;

G:=Group( (1,7,4)(2,5,8)(10,16,13)(11,14,17), (2,8,5)(3,6,9)(10,16,13)(12,15,18), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(9,18) );

G=PermutationGroup([[(1,7,4),(2,5,8),(10,16,13),(11,14,17)], [(2,8,5),(3,6,9),(10,16,13),(12,15,18)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(9,18)]])

G:=TransitiveGroup(18,171);

►On 27 points - transitive group 27T146

Generators in S₂₇

(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)

(1 7 4)(2 12 21)(3 25 10)(5 15 24)(6 19 13)(8 18 27)(9 22 16)(11 17 14)(20 26 23)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)

(1 14 26)(2 15 27)(3 16 19)(4 17 20)(5 18 21)(6 10 22)(7 11 23)(8 12 24)(9 13 25)

(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)

(2 9)(3 8)(4 7)(5 6)(10 18)(11 17)(12 16)(13 15)(19 24)(20 23)(21 22)(25 27)

G:=sub<Sym(27)| (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24), (1,7,4)(2,12,21)(3,25,10)(5,15,24)(6,19,13)(8,18,27)(9,22,16)(11,17,14)(20,26,23), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (1,14,26)(2,15,27)(3,16,19)(4,17,20)(5,18,21)(6,10,22)(7,11,23)(8,12,24)(9,13,25), (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), (2,9)(3,8)(4,7)(5,6)(10,18)(11,17)(12,16)(13,15)(19,24)(20,23)(21,22)(25,27)>;

G:=Group( (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24), (1,7,4)(2,12,21)(3,25,10)(5,15,24)(6,19,13)(8,18,27)(9,22,16)(11,17,14)(20,26,23), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (1,14,26)(2,15,27)(3,16,19)(4,17,20)(5,18,21)(6,10,22)(7,11,23)(8,12,24)(9,13,25), (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), (2,9)(3,8)(4,7)(5,6)(10,18)(11,17)(12,16)(13,15)(19,24)(20,23)(21,22)(25,27) );

G=PermutationGroup([[(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,22,25),(21,27,24)], [(1,7,4),(2,12,21),(3,25,10),(5,15,24),(6,19,13),(8,18,27),(9,22,16),(11,17,14),(20,26,23)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27)], [(1,14,26),(2,15,27),(3,16,19),(4,17,20),(5,18,21),(6,10,22),(7,11,23),(8,12,24),(9,13,25)], [(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)], [(2,9),(3,8),(4,7),(5,6),(10,18),(11,17),(12,16),(13,15),(19,24),(20,23),(21,22),(25,27)]])

G:=TransitiveGroup(27,146);

39 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3K	3L	···	3T	6A	···	6H	9A	···	9I
order	1	2	3	3	3	3	3	3	···	3	3	···	3	6	···	6	9	···	9
size	1	27	1	1	2	2	2	3	···	3	6	···	6	27	···	27	18	···	18

39 irreducible representations

dim	1	1	1	1	2	2	2	2	3	6	6
type	+	+			+	+				+
image	C₁	C₂	C₃	C₆	S₃	S₃	C₃xS₃	C₃xS₃	He₃:C₂	C₉:C₆	C₃⁴.₇S₃
kernel	C₃⁴.₇S₃	C₃⁴.C₃	C₃²:₂D₉	C₃²:C₉	C₃x3_-¹⁺²	C₃⁴	C₃xC₉	C₃³	C₃²	C₃²	C₁
# reps	1	1	2	2	3	1	6	2	12	3	6

Matrix representation of C₃⁴.₇S₃ ►in GL₆(F₁₉)

7	0	0	0	0	12
0	11	0	0	0	1
0	0	1	0	0	0
0	0	0	7	0	12
0	0	0	0	11	1
0	0	0	0	0	1

1	0	0	0	0	7
0	7	0	0	0	0
0	0	11	0	0	8
0	0	0	1	0	7
0	0	0	0	11	8
0	0	0	0	0	7

11	0	0	0	0	8
0	11	0	0	0	8
0	0	11	0	0	8
0	0	0	7	0	0
0	0	0	0	7	0
0	0	0	0	0	7

7	0	0	0	0	0
0	7	0	0	0	0
0	0	7	0	0	0
0	0	0	7	0	0
0	0	0	0	7	0
0	0	0	0	0	7

0	0	11	12	0	8
1	0	0	12	0	7
0	1	0	12	0	7
0	0	0	12	1	7
0	0	0	12	0	7
0	0	0	13	0	7

0	0	18	1	0	0
0	0	18	0	1	0
0	0	18	0	0	0
1	0	18	0	0	0
0	1	18	0	0	0
0	0	10	0	0	1

G:=sub<GL(6,GF(19))| [7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,12,1,0,12,1,1],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,7,0,8,7,8,7],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,8,8,8,0,0,7],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,1,0,0,0,0,0,0,1,0,0,0,11,0,0,0,0,0,12,12,12,12,12,13,0,0,0,1,0,0,8,7,7,7,7,7],[0,0,0,1,0,0,0,0,0,0,1,0,18,18,18,18,18,10,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1] >;

C₃⁴.₇S₃ in GAP, Magma, Sage, TeX

C_3^4._7S_3

% in TeX

G:=Group("C3^4.7S3");

// GroupNames label

G:=SmallGroup(486,147);

// by ID

G=gap.SmallGroup(486,147);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,548,338,867,735,3244]);

// Polycyclic

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

// generators/relations

G = C34.7S3 order 486 = 2·35

7th non-split extension by C34 of S3 acting faithfully

G = C₃⁴.₇S₃ order 486 = 2·3⁵

7^th non-split extension by C₃⁴ of S₃ acting faithfully