D9xES-(3,1) - GroupNames

G = D₉×3_-¹⁺² order 486 = 2·3⁵

Direct product of D₉ and 3_-¹⁺²

direct product, metabelian, supersoluble, monomial

Aliases: D₉×3_-¹⁺², C₉²⋊₁₃C₆, C₉⋊₂(C₃×D₉), (C₉×D₉)⋊₂C₃, (C₃²×C₉).₆C₆, C₃.₆(C₃²×D₉), C₃³.₅₄(C₃×S₃), (C₃²×D₉).₁C₃, (C₃×D₉).₉C₃², C₃².₁₀(C₃×D₉), (C₉×3_-¹⁺²)⋊₃C₂, C₉⋊₇(C₂×3_-¹⁺²), C₃².₃₄(S₃×C₃²), C₃.₁(S₃×3_-¹⁺²), (C₃×3_-¹⁺²).₁₄S₃, (C₃×C₉).₄₁(C₃×C₆), (C₃×C₉).₄₃(C₃×S₃), SmallGroup(486,101)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₉ — D₉×3_-¹⁺²

Chief series

C₁ — C₃ — C₉ — C₃×C₉ — C₉² — C₉×3_-¹⁺² — D₉×3_-¹⁺²

Lower central

C₉ — C₃×C₉ — D₉×3_-¹⁺²

Upper central

C₁ — C₃ — 3_-¹⁺²

Generators and relations for D₉×3_-¹⁺²
G = < a,b,c,d | a⁹=b²=c⁹=d³=1, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c⁴ >

Subgroups: 238 in 73 conjugacy classes, 28 normal (16 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₉, C₃², C₃², D₉, C₁₈, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, C₃×C₉, 3_-¹⁺², 3_-¹⁺², C₃³, C₃×D₉, C₃×D₉, S₃×C₉, C₂×3_-¹⁺², S₃×C₃², C₉², C₃²⋊C₉, C₉⋊C₉, C₃²×C₉, C₃×3_-¹⁺², C₉×D₉, C₃²×D₉, S₃×3_-¹⁺², C₉×3_-¹⁺², D₉×3_-¹⁺²
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², D₉, C₃×S₃, C₃×C₆, 3_-¹⁺², C₃×D₉, C₂×3_-¹⁺², S₃×C₃², C₃²×D₉, S₃×3_-¹⁺², D₉×3_-¹⁺²

Smallest permutation representation of D₉×3_-¹⁺²
►On 54 points

Generators in S₅₄

(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)

(1 30)(2 29)(3 28)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 54)(27 53)

(1 23 14 4 26 17 7 20 11)(2 24 15 5 27 18 8 21 12)(3 25 16 6 19 10 9 22 13)(28 46 37 34 52 43 31 49 40)(29 47 38 35 53 44 32 50 41)(30 48 39 36 54 45 33 51 42)

(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)

G:=sub<Sym(54)| (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,30)(2,29)(3,28)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,54)(27,53), (1,23,14,4,26,17,7,20,11)(2,24,15,5,27,18,8,21,12)(3,25,16,6,19,10,9,22,13)(28,46,37,34,52,43,31,49,40)(29,47,38,35,53,44,32,50,41)(30,48,39,36,54,45,33,51,42), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54)>;

G:=Group( (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,30)(2,29)(3,28)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,54)(27,53), (1,23,14,4,26,17,7,20,11)(2,24,15,5,27,18,8,21,12)(3,25,16,6,19,10,9,22,13)(28,46,37,34,52,43,31,49,40)(29,47,38,35,53,44,32,50,41)(30,48,39,36,54,45,33,51,42), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54) );

G=PermutationGroup([[(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,30),(2,29),(3,28),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,45),(18,44),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,54),(27,53)], [(1,23,14,4,26,17,7,20,11),(2,24,15,5,27,18,8,21,12),(3,25,16,6,19,10,9,22,13),(28,46,37,34,52,43,31,49,40),(29,47,38,35,53,44,32,50,41),(30,48,39,36,54,45,33,51,42)], [(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(37,43,40),(38,44,41),(39,45,42),(46,49,52),(47,50,53),(48,51,54)]])

66 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	3I	6A	6B	6C	6D	9A	···	9I	9J	···	9O	9P	···	9AS	18A	···	18F
order	1	2	3	3	3	3	3	3	3	3	3	6	6	6	6	9	···	9	9	···	9	9	···	9	18	···	18
size	1	9	1	1	2	2	2	3	3	6	6	9	9	27	27	2	···	2	3	···	3	6	···	6	27	···	27

66 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	3	3	6	6
type	+	+					+		+
image	C₁	C₂	C₃	C₃	C₆	C₆	S₃	C₃×S₃	D₉	C₃×S₃	C₃×D₉	C₃×D₉	3_-¹⁺²	C₂×3_-¹⁺²	S₃×3_-¹⁺²	D₉×3_-¹⁺²
kernel	D₉×3_-¹⁺²	C₉×3_-¹⁺²	C₉×D₉	C₃²×D₉	C₉²	C₃²×C₉	C₃×3_-¹⁺²	C₃×C₉	3_-¹⁺²	C₃³	C₉	C₃²	D₉	C₉	C₃	C₁
# reps	1	1	6	2	6	2	1	6	3	2	18	6	2	2	2	6

Matrix representation of D₉×3_-¹⁺² ►in GL₅(𝔽₁₉)

0	1	0	0	0
18	9	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

18	9	0	0	0
0	1	0	0	0
0	0	18	0	0
0	0	0	18	0
0	0	0	0	18

1	0	0	0	0
0	1	0	0	0
0	0	0	0	7
0	0	7	0	0
0	0	0	11	0

7	0	0	0	0
0	7	0	0	0
0	0	1	0	0
0	0	0	11	0
0	0	0	0	7

G:=sub<GL(5,GF(19))| [0,18,0,0,0,1,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,0,0,0,0,9,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,11,0,0,7,0,0],[7,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7] >;

D₉×3_-¹⁺² in GAP, Magma, Sage, TeX

D_9\times 3_-^{1+2}

% in TeX

G:=Group("D9xES-(3,1)");

// GroupNames label

G:=SmallGroup(486,101);

// by ID

G=gap.SmallGroup(486,101);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,68,8104,208,11669]);

// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^9=d^3=1,b*a*b=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 = D9×3- 1+2 order 486 = 2·35

Direct product of D9 and 3- 1+2

G = D₉×3_-¹⁺² order 486 = 2·3⁵

Direct product of D₉ and 3_-¹⁺²