C3^4:6S3 - GroupNames

G = C₃⁴⋊₆S₃ order 486 = 2·3⁵

6^th semidirect product of C₃⁴ and S₃ acting faithfully

Aliases: C₃⁴⋊₆S₃, C₃²⋊He₃⋊₅C₂, C₃³⋊₂(C₃⋊S₃), (C₃×He₃)⋊₁₄S₃, C₃.₈(He₃⋊₅S₃), C₃²⋊₂(He₃⋊C₂), C₃².₁₅(C₃³⋊C₂), SmallGroup(486,183)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃²⋊He₃ — C₃⁴⋊₆S₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×He₃ — C₃²⋊He₃ — C₃⁴⋊₆S₃

Lower central

C₃²⋊He₃ — C₃⁴⋊₆S₃

Upper central

C₁ — C₃²

Generators and relations for C₃⁴⋊₆S₃
G = < a,b,c,d,e,f | a³=b³=c³=d³=e³=f²=1, ab=ba, ac=ca, ad=da, eae^-1=ad^-1, faf=a^-1, bc=cb, bd=db, ebe^-1=bc^-1, fbf=b^-1, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1486 in 267 conjugacy classes, 38 normal (5 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₃², C₃², C₃², C₃×S₃, C₃⋊S₃, C₃×C₆, He₃, C₃³, C₃³, He₃⋊C₂, S₃×C₃², C₃×C₃⋊S₃, C₃×He₃, C₃⁴, C₃×He₃⋊C₂, C₃²×C₃⋊S₃, C₃²⋊He₃, C₃⁴⋊₆S₃
Quotients: C₁, C₂, S₃, C₃⋊S₃, He₃⋊C₂, C₃³⋊C₂, He₃⋊₅S₃, C₃⁴⋊₆S₃

Permutation representations of C₃⁴⋊₆S₃
►On 27 points - transitive group 27T154

Generators in S₂₇

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(27,154);

39 conjugacy classes

class	1	2	3A	···	3H	3I	···	3T	3U	···	3AC	6A	···	6H
order	1	2	3	···	3	3	···	3	3	···	3	6	···	6
size	1	27	1	···	1	6	···	6	18	···	18	27	···	27

39 irreducible representations

dim	1	1	2	2	3	6
type	+	+	+	+
image	C₁	C₂	S₃	S₃	He₃⋊C₂	He₃⋊₅S₃
kernel	C₃⁴⋊₆S₃	C₃²⋊He₃	C₃×He₃	C₃⁴	C₃²	C₃
# reps	1	1	12	1	16	8

Matrix representation of C₃⁴⋊₆S₃ ►in GL₆(𝔽₇)

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

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	4	0
0	0	0	0	0	2

2	0	0	0	0	0
0	2	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	2	0
0	0	0	0	0	2

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

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

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

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

C₃⁴⋊₆S₃ in GAP, Magma, Sage, TeX

C_3^4\rtimes_6S_3

% in TeX

G:=Group("C3^4:6S3");

// GroupNames label

G:=SmallGroup(486,183);

// by ID

G=gap.SmallGroup(486,183);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,867,2169,303]);

// 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*d^-1,f*a*f=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^-1,f*b*f=b^-1,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C34⋊6S3 order 486 = 2·35

6th semidirect product of C34 and S3 acting faithfully

G = C₃⁴⋊₆S₃ order 486 = 2·3⁵

6^th semidirect product of C₃⁴ and S₃ acting faithfully