C3^4:3S3 - GroupNames

G = C₃⁴⋊₃S₃ order 486 = 2·3⁵

3^rd semidirect product of C₃⁴ and S₃ acting faithfully

Aliases: C₃⁴⋊₃S₃, (C₃×He₃)⋊₉C₆, (C₃×He₃)⋊₇S₃, C₃³⋊₃(C₃×S₃), C₃²⋊He₃⋊₃C₂, He₃⋊₅S₃⋊₂C₃, C₃³.₂₉(C₃⋊S₃), C₃²⋊₂(C₃²⋊C₆), C₃²⋊₁(He₃⋊C₂), C₃.₁₁(He₃⋊₄S₃), C₃².₃₅(C₃×C₃⋊S₃), C₃.₂(C₃×He₃⋊C₂), SmallGroup(486,145)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₃×He₃ — C₃⁴⋊₃S₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×He₃ — C₃²⋊He₃ — C₃⁴⋊₃S₃

Lower central

C₃×He₃ — C₃⁴⋊₃S₃

Upper central

C₁ — C₃

Generators and relations for C₃⁴⋊₃S₃
G = < a,b,c,d,e,f | a³=b³=c³=d³=e³=f²=1, 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=e^-1 >

Subgroups: 1172 in 192 conjugacy classes, 23 normal (11 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₃², C₃², C₃², C₃×S₃, C₃⋊S₃, C₃×C₆, He₃, C₃³, C₃³, C₃³, C₃²⋊C₆, He₃⋊C₂, S₃×C₃², C₃×C₃⋊S₃, C₃×He₃, C₃×He₃, C₃×He₃, C₃⁴, C₃×C₃²⋊C₆, He₃⋊₅S₃, C₃²×C₃⋊S₃, C₃²⋊He₃, C₃⁴⋊₃S₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃×S₃, C₃⋊S₃, C₃²⋊C₆, He₃⋊C₂, C₃×C₃⋊S₃, He₃⋊₄S₃, C₃×He₃⋊C₂, C₃⁴⋊₃S₃

Permutation representations of C₃⁴⋊₃S₃
►On 18 points - transitive group 18T165

Generators in S₁₈

(7 8 9)(10 11 12)(13 14 15)(16 17 18)

(1 2 3)(4 5 6)(10 12 11)(13 14 15)

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

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

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

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

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

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

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

G:=TransitiveGroup(18,165);

►On 18 points - transitive group 18T168

Generators in S₁₈

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(18,168);

►On 27 points - transitive group 27T186

Generators in S₂₇

(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

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

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

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

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

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

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

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

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

G:=TransitiveGroup(27,186);

39 conjugacy classes

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

39 irreducible representations

dim	1	1	1	1	2	2	2	3	6	6
type	+	+			+	+			+
image	C₁	C₂	C₃	C₆	S₃	S₃	C₃×S₃	He₃⋊C₂	C₃²⋊C₆	C₃⁴⋊₃S₃
kernel	C₃⁴⋊₃S₃	C₃²⋊He₃	He₃⋊₅S₃	C₃×He₃	C₃×He₃	C₃⁴	C₃³	C₃²	C₃²	C₁
# reps	1	1	2	2	3	1	8	12	3	6

Matrix representation of C₃⁴⋊₃S₃ ►in GL₆(𝔽₇)

4	0	0	0	0	0
0	2	0	0	0	0
0	0	1	0	0	0
0	0	0	4	0	0
0	0	0	0	2	0
0	0	0	0	0	1

1	0	0	0	0	0
0	4	0	0	0	0
0	0	2	0	0	0
0	0	0	1	0	0
0	0	0	0	2	0
0	0	0	0	0	4

2	0	0	0	0	0
0	2	0	0	0	0
0	0	2	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

0	0	1	0	0	0
1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

0	0	0	1	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

G:=sub<GL(6,GF(7))| [4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

C₃⁴⋊₃S₃ in GAP, Magma, Sage, TeX

C_3^4\rtimes_3S_3

% in TeX

G:=Group("C3^4:3S3");

// GroupNames label

G:=SmallGroup(486,145);

// by ID

G=gap.SmallGroup(486,145);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,548,867,735,3244]);

// Polycyclic

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

// generators/relations

G = C34⋊3S3 order 486 = 2·35

3rd semidirect product of C34 and S3 acting faithfully

G = C₃⁴⋊₃S₃ order 486 = 2·3⁵

3^rd semidirect product of C₃⁴ and S₃ acting faithfully