C3^4.C6 - GroupNames

G = C₃⁴.C₆ order 486 = 2·3⁵

4^th non-split extension by C₃⁴ of C₆ acting faithfully

Aliases: C₃⁴.₄C₆, C₃²⋊C₉⋊₇C₆, C₃²⋊C₁₈⋊₁C₃, C₃³.₂₃(C₃×C₆), C₃³.₅₅(C₃×S₃), C₃⁴.C₃⋊₃C₂, C₃⋊S₃⋊₂3_-¹⁺², C₃².₃₇(S₃×C₃²), C₃.₂(S₃×3_-¹⁺²), C₃².₂₀(C₃²⋊C₆), (C₃×3_-¹⁺²)⋊₁₃S₃, C₃²⋊₂(C₂×3_-¹⁺²), (C₃×C₉)⋊₅(C₃×S₃), (C₃²×C₃⋊S₃).₂C₃, (C₃×C₃⋊S₃).₄C₃², C₃.₁₃(C₃×C₃²⋊C₆), SmallGroup(486,104)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃⁴.C₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃⁴ — C₃⁴.C₃ — 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=1, e⁶=b, 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: 472 in 108 conjugacy classes, 24 normal (15 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₁₈, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, C₃×C₉, 3_-¹⁺², C₃³, C₃³, S₃×C₉, C₂×3_-¹⁺², S₃×C₃², C₃×C₃⋊S₃, C₃×C₃⋊S₃, C₃²⋊C₉, C₃²⋊C₉, C₃×3_-¹⁺², C₃×3_-¹⁺², C₃⁴, C₃²⋊C₁₈, S₃×3_-¹⁺², C₃²×C₃⋊S₃, C₃⁴.C₃, C₃⁴.C₆
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, 3_-¹⁺², C₃²⋊C₆, C₂×3_-¹⁺², S₃×C₃², C₃×C₃²⋊C₆, S₃×3_-¹⁺², C₃⁴.C₆

Permutation representations of C₃⁴.C₆
►On 18 points - transitive group 18T160

Generators in S₁₈

(2 8 14)(3 15 9)(5 11 17)(6 18 12)

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

(2 8 14)(3 9 15)(5 17 11)(6 18 12)

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

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

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

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

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

G:=TransitiveGroup(18,160);

►On 27 points - transitive group 27T208

Generators in S₂₇

(1 7 4)(2 5 8)(11 17 23)(12 24 18)(14 20 26)(15 27 21)

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

(1 26 17)(2 27 18)(4 20 11)(5 21 12)(7 14 23)(8 15 24)

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

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

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

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

G:=TransitiveGroup(27,208);

42 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	···	3R	6A	6B	6C	6D	9A	···	9F	9G	···	9L	18A	···	18F
order	1	2	3	3	3	3	3	3	3	3	···	3	6	6	6	6	9	···	9	9	···	9	18	···	18
size	1	9	1	1	2	2	2	3	3	6	···	6	9	9	27	27	9	···	9	18	···	18	27	···	27

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	3	3	6	6	6	6
type	+	+					+					+
image	C₁	C₂	C₃	C₃	C₆	C₆	S₃	C₃×S₃	C₃×S₃	3_-¹⁺²	C₂×3_-¹⁺²	C₃²⋊C₆	C₃×C₃²⋊C₆	S₃×3_-¹⁺²	C₃⁴.C₆
kernel	C₃⁴.C₆	C₃⁴.C₃	C₃²⋊C₁₈	C₃²×C₃⋊S₃	C₃²⋊C₉	C₃⁴	C₃×3_-¹⁺²	C₃×C₉	C₃³	C₃⋊S₃	C₃²	C₃²	C₃	C₃	C₁
# reps	1	1	6	2	6	2	1	6	2	2	2	1	2	2	6

Matrix representation of C₃⁴.C₆ ►in GL₆(𝔽₁₉)

11	0	0	0	0	0
0	1	0	0	0	0
18	1	7	0	0	0
0	6	0	11	0	0
4	0	0	0	1	0
14	14	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

1	0	0	0	0	0
0	11	0	0	0	0
11	12	7	0	0	0
0	13	0	1	0	0
9	17	0	0	7	0
3	0	0	0	0	11

11	0	0	0	0	0
0	11	0	0	0	0
0	0	11	0	0	0
2	9	0	7	0	0
13	17	0	0	7	0
14	16	0	0	0	7

14	16	0	0	0	15
14	6	0	10	0	0
2	5	0	11	1	1
8	17	11	13	0	17
12	11	0	14	0	6
10	6	0	2	0	5

G:=sub<GL(6,GF(19))| [11,0,18,0,4,14,0,1,1,6,0,14,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7],[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],[1,0,11,0,9,3,0,11,12,13,17,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[11,0,0,2,13,14,0,11,0,9,17,16,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[14,14,2,8,12,10,16,6,5,17,11,6,0,0,0,11,0,0,0,10,11,13,14,2,0,0,1,0,0,0,15,0,1,17,6,5] >;

C₃⁴.C₆ in GAP, Magma, Sage, TeX

C_3^4.C_6

% in TeX

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

// GroupNames label

G:=SmallGroup(486,104);

// by ID

G=gap.SmallGroup(486,104);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,224,3244,3250,11669]);

// Polycyclic

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

4th non-split extension by C34 of C6 acting faithfully

G = C₃⁴.C₆ order 486 = 2·3⁵

4^th non-split extension by C₃⁴ of C₆ acting faithfully