C3^4:C6 - GroupNames

G = C₃⁴⋊C₆ order 486 = 2·3⁵

1^st semidirect product of C₃⁴ and C₆ acting faithfully

Aliases: C₃⁴⋊₁C₆, C₃⋊S₃⋊He₃, C₃²⋊(C₂×He₃), (C₃×He₃)⋊₂C₆, (C₃×He₃)⋊₃S₃, C₃³⋊₁(C₃×S₃), C₃²⋊He₃⋊₁C₂, C₃.₂(S₃×He₃), C₃³.₂₂(C₃×C₆), C₃²⋊₄(C₃²⋊C₆), C₃².₃₅(S₃×C₃²), (C₃×C₃²⋊C₆)⋊C₃, (C₃²×C₃⋊S₃)⋊₁C₃, (C₃×C₃⋊S₃).₃C₃², C₃.₁₂(C₃×C₃²⋊C₆), SmallGroup(486,102)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃⁴⋊C₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃⁴ — C₃²⋊He₃ — C₃⁴⋊C₆

Lower central

C₃² — C₃³ — C₃⁴⋊C₆

Upper central

C₁ — C₃ — C₃²

Generators and relations for C₃⁴⋊C₆
G = < a,b,c,d,e | a³=b³=c³=d³=e⁶=1, eae^-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ece^-1=c^-1d^-1, ede^-1=d^-1 >

Subgroups: 832 in 153 conjugacy classes, 24 normal (15 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₃², C₃², C₃×S₃, C₃⋊S₃, C₃×C₆, He₃, C₃³, C₃³, C₃³, C₃²⋊C₆, C₂×He₃, S₃×C₃², C₃×C₃⋊S₃, C₃×C₃⋊S₃, C₃×He₃, C₃×He₃, C₃×He₃, C₃⁴, C₃×C₃²⋊C₆, S₃×He₃, C₃²×C₃⋊S₃, C₃²⋊He₃, C₃⁴⋊C₆
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, He₃, C₃²⋊C₆, C₂×He₃, S₃×C₃², C₃×C₃²⋊C₆, S₃×He₃, C₃⁴⋊C₆

Permutation representations of C₃⁴⋊C₆
►On 18 points - transitive group 18T161

Generators in S₁₈

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

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

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

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

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

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

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

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

G:=TransitiveGroup(18,161);

►On 18 points - transitive group 18T164

Generators in S₁₈

(1 8 14)(2 15 9)(4 11 17)(5 18 12)

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

(2 9 15)(3 10 16)(5 18 12)(6 13 7)

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

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

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

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

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

G:=TransitiveGroup(18,164);

►On 27 points - transitive group 27T180

Generators in S₂₇

(1 7 4)(2 5 8)(10 22 20)(11 21 23)(13 25 17)(14 18 26)

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

(1 22 25)(2 23 26)(4 10 13)(5 11 14)(7 20 17)(8 21 18)

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

(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)

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

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

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

G:=TransitiveGroup(27,180);

42 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	···	3R	3S	···	3X	3Y	···	3AD	6A	6B	6C	···	6J
order	1	2	3	3	3	3	3	3	3	3	···	3	3	···	3	3	···	3	6	6	6	···	6
size	1	9	1	1	2	2	2	3	3	6	···	6	9	···	9	18	···	18	9	9	27	···	27

42 irreducible representations

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

Matrix representation of C₃⁴⋊C₆ ►in GL₆(𝔽₇)

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

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

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

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

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

C₃⁴⋊C₆ in GAP, Magma, Sage, TeX

C_3^4\rtimes C_6

% in TeX

G:=Group("C3^4:C6");

// GroupNames label

G:=SmallGroup(486,102);

// by ID

G=gap.SmallGroup(486,102);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,3244,3250,11669]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^6=1,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1*d^-1,e*d*e^-1=d^-1>;

// generators/relations

G = C34⋊C6 order 486 = 2·35

1st semidirect product of C34 and C6 acting faithfully

G = C₃⁴⋊C₆ order 486 = 2·3⁵

1^st semidirect product of C₃⁴ and C₆ acting faithfully