Dic3xES-(3,1) - GroupNames

G = Dic₃×3_-¹⁺² order 324 = 2²·3⁴

Direct product of Dic₃ and 3_-¹⁺²

direct product, metabelian, supersoluble, monomial

Aliases: Dic₃×3_-¹⁺², C₃³.₂C₁₂, (C₃×C₉)⋊₇C₁₂, (C₉×Dic₃)⋊C₃, C₁₈.₂(C₃×S₃), (C₃×C₁₈).₉C₆, C₉⋊₂(C₃×Dic₃), (C₃²×C₆).₆C₆, C₃⋊(C₄×3_-¹⁺²), C₃².₈(C₃×C₁₂), C₆.₁₀(S₃×C₃²), (C₃²×Dic₃).C₃, C₆.(C₂×3_-¹⁺²), C₂.(S₃×3_-¹⁺²), (C₃×3_-¹⁺²)⋊₃C₄, C₃.₆(C₃²×Dic₃), (C₃×Dic₃).₃C₃², C₃².₁₀(C₃×Dic₃), (C₂×3_-¹⁺²).₇S₃, (C₆×3_-¹⁺²).₃C₂, (C₃×C₆).₁₇(C₃×C₆), (C₃×C₆).₂₃(C₃×S₃), SmallGroup(324,95)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — Dic₃×3_-¹⁺²

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₈ — C₆×3_-¹⁺² — Dic₃×3_-¹⁺²

Lower central

C₃ — C₃² — Dic₃×3_-¹⁺²

Upper central

C₁ — C₆ — C₂×3_-¹⁺²

Generators and relations for Dic₃×3_-¹⁺²
G = < a,b,c,d | a⁶=c⁹=d³=1, b²=a³, bab^-1=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c⁴ >

Subgroups: 140 in 70 conjugacy classes, 35 normal (20 characteristic)
C₁, C₂, C₃, C₃, C₄, C₆, C₆, C₉, C₉, C₃², C₃², Dic₃, C₁₂, C₁₈, C₁₈, C₃×C₆, C₃×C₆, C₃×C₉, 3_-¹⁺², 3_-¹⁺², C₃³, C₃₆, C₃×Dic₃, C₃×Dic₃, C₃×C₁₂, C₃×C₁₈, C₂×3_-¹⁺², C₂×3_-¹⁺², C₃²×C₆, C₃×3_-¹⁺², C₉×Dic₃, C₄×3_-¹⁺², C₃²×Dic₃, C₆×3_-¹⁺², Dic₃×3_-¹⁺²
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, C₃², Dic₃, C₁₂, C₃×S₃, C₃×C₆, 3_-¹⁺², C₃×Dic₃, C₃×C₁₂, C₂×3_-¹⁺², S₃×C₃², C₄×3_-¹⁺², C₃²×Dic₃, S₃×3_-¹⁺², Dic₃×3_-¹⁺²

Smallest permutation representation of Dic₃×3_-¹⁺²
►On 36 points

Generators in S₃₆

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

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

(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 25 22)(20 23 26)(29 35 32)(30 33 36)

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

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

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

66 conjugacy classes

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

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	6	6
type	+	+								+	-
image	C₁	C₂	C₃	C₃	C₄	C₆	C₆	C₁₂	C₁₂	S₃	Dic₃	C₃×S₃	C₃×S₃	C₃×Dic₃	C₃×Dic₃	3_-¹⁺²	C₂×3_-¹⁺²	C₄×3_-¹⁺²	S₃×3_-¹⁺²	Dic₃×3_-¹⁺²
kernel	Dic₃×3_-¹⁺²	C₆×3_-¹⁺²	C₉×Dic₃	C₃²×Dic₃	C₃×3_-¹⁺²	C₃×C₁₈	C₃²×C₆	C₃×C₉	C₃³	C₂×3_-¹⁺²	3_-¹⁺²	C₁₈	C₃×C₆	C₉	C₃²	Dic₃	C₆	C₃	C₂	C₁
# reps	1	1	6	2	2	6	2	12	4	1	1	6	2	6	2	2	2	4	2	2

Matrix representation of Dic₃×3_-¹⁺² ►in GL₅(𝔽₃₇)

0	1	0	0	0
36	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	6	0	0	0
6	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

26	0	0	0	0
0	26	0	0	0
0	0	0	0	26
0	0	26	0	0
0	0	0	1	0

10	0	0	0	0
0	10	0	0	0
0	0	1	0	0
0	0	0	26	0
0	0	0	0	10

G:=sub<GL(5,GF(37))| [0,36,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,6,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[26,0,0,0,0,0,26,0,0,0,0,0,0,26,0,0,0,0,0,1,0,0,26,0,0],[10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,10] >;

Dic₃×3_-¹⁺² in GAP, Magma, Sage, TeX

{\rm Dic}_3\times 3_-^{1+2}

% in TeX

G:=Group("Dic3xES-(3,1)");

// GroupNames label

G:=SmallGroup(324,95);

// by ID

G=gap.SmallGroup(324,95);

# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,386,122,7781]);

// Polycyclic

G:=Group<a,b,c,d|a^6=c^9=d^3=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;

// generators/relations

G = Dic3×3- 1+2 order 324 = 22·34

Direct product of Dic3 and 3- 1+2

G = Dic₃×3_-¹⁺² order 324 = 2²·3⁴

Direct product of Dic₃ and 3_-¹⁺²