C4xES-(3,1) - GroupNames

G = C₄x3_-¹⁺² order 108 = 2²·3³

Direct product of C₄ and 3_-¹⁺²

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C₄x3_-¹⁺², C₃₆:C₃, C₉:₂C₁₂, C₁₈.₂C₆, C₃².C₁₂, C₁₂.₂C₃², (C₃xC₁₂).C₃, C₆.₃(C₃xC₆), (C₃xC₆).₃C₆, C₃.₂(C₃xC₁₂), C₂.(C₂x3_-¹⁺²), C₁₂ o(C₂x3_-¹⁺²), (C₂x3_-¹⁺²).₂C₂, SmallGroup(108,14)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₄x3_-¹⁺²

Chief series

C₁ — C₃ — C₆ — C₃xC₆ — C₂x3_-¹⁺² — C₄x3_-¹⁺²

Lower central

C₁ — C₃ — C₄x3_-¹⁺²

Upper central

C₁ — C₁₂ — C₄x3_-¹⁺²

Generators and relations for C₄x3_-¹⁺²
G = < a,b,c | a⁴=b⁹=c³=1, ab=ba, ac=ca, cbc^-1=b⁴ >

Subgroups: 30 in 24 conjugacy classes, 21 normal (12 characteristic)
Quotients: C₁, C₂, C₃, C₄, C₆, C₃², C₁₂, C₃xC₆, 3_-¹⁺², C₃xC₁₂, C₂x3_-¹⁺², C₄x3_-¹⁺²

C₁

C₂

C₃

₃C₃

C₄

C₆

₃C₆

C₃²

C₉

C₁₂

₃C₁₂

C₁₈

C₃xC₆

C₁₈

3_-¹⁺²

C₃₆

C₃xC₁₂

C₃₆

C₂x3_-¹⁺²

C₄x3_-¹⁺²

Smallest permutation representation of C₄x3_-¹⁺²
►On 36 points

Generators in S₃₆

(1 18 20 34)(2 10 21 35)(3 11 22 36)(4 12 23 28)(5 13 24 29)(6 14 25 30)(7 15 26 31)(8 16 27 32)(9 17 19 33)

(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)(28 29 30 31 32 33 34 35 36)

(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 22 25)(21 27 24)(29 35 32)(30 33 36)

G:=sub<Sym(36)| (1,18,20,34)(2,10,21,35)(3,11,22,36)(4,12,23,28)(5,13,24,29)(6,14,25,30)(7,15,26,31)(8,16,27,32)(9,17,19,33), (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)(28,29,30,31,32,33,34,35,36), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24)(29,35,32)(30,33,36)>;

G:=Group( (1,18,20,34)(2,10,21,35)(3,11,22,36)(4,12,23,28)(5,13,24,29)(6,14,25,30)(7,15,26,31)(8,16,27,32)(9,17,19,33), (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)(28,29,30,31,32,33,34,35,36), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24)(29,35,32)(30,33,36) );

G=PermutationGroup([[(1,18,20,34),(2,10,21,35),(3,11,22,36),(4,12,23,28),(5,13,24,29),(6,14,25,30),(7,15,26,31),(8,16,27,32),(9,17,19,33)], [(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),(28,29,30,31,32,33,34,35,36)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,22,25),(21,27,24),(29,35,32),(30,33,36)]])

C₄x3_-¹⁺² is a maximal subgroup of C₉:C₂₄ C₃₆.C₆ D₃₆:C₃ C₃₆.A₄ Q₈:C₉:₄C₆

44 conjugacy classes

class	1	2	3A	3B	3C	3D	4A	4B	6A	6B	6C	6D	9A	···	9F	12A	12B	12C	12D	12E	12F	12G	12H	18A	···	18F	36A	···	36L
order	1	2	3	3	3	3	4	4	6	6	6	6	9	···	9	12	12	12	12	12	12	12	12	18	···	18	36	···	36
size	1	1	1	1	3	3	1	1	1	1	3	3	3	···	3	1	1	1	1	3	3	3	3	3	···	3	3	···	3

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	3	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₄	C₆	C₆	C₁₂	C₁₂	3_-¹⁺²	C₂x3_-¹⁺²	C₄x3_-¹⁺²
kernel	C₄x3_-¹⁺²	C₂x3_-¹⁺²	C₃₆	C₃xC₁₂	3_-¹⁺²	C₁₈	C₃xC₆	C₉	C₃²	C₄	C₂	C₁
# reps	1	1	6	2	2	6	2	12	4	2	2	4

Matrix representation of C₄x3_-¹⁺² ►in GL₃(F₁₃) generated by

8	0	0
0	8	0
0	0	8

0	0	8
3	0	0
0	2	0

3	0	0
0	9	0
0	0	1

G:=sub<GL(3,GF(13))| [8,0,0,0,8,0,0,0,8],[0,3,0,0,0,2,8,0,0],[3,0,0,0,9,0,0,0,1] >;

C₄x3_-¹⁺² in GAP, Magma, Sage, TeX

C_4\times 3_-^{1+2}

% in TeX

G:=Group("C4xES-(3,1)");

// GroupNames label

G:=SmallGroup(108,14);

// by ID

G=gap.SmallGroup(108,14);

# by ID

G:=PCGroup([5,-2,-3,-3,-2,-3,90,186,322]);

// Polycyclic

G:=Group<a,b,c|a^4=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;

// generators/relations

Export

Subgroup lattice of C₄x3_-¹⁺² in TeX

G = C4x3- 1+2 order 108 = 22·33

Direct product of C4 and 3- 1+2

G = C₄x3_-¹⁺² order 108 = 2²·3³

Direct product of C₄ and 3_-¹⁺²