D9:ES-(3,1) - GroupNames

G = D₉⋊3_-¹⁺² order 486 = 2·3⁵

The semidirect product of D₉ and 3_-¹⁺² acting via 3_-¹⁺²/C₃²=C₃

Aliases: D₉⋊₃3_-¹⁺², C₉⋊C₉⋊₁C₆, C₉⋊C₁₈⋊₁C₃, (C₃²×C₉).₈C₆, C₃².₇(C₉⋊C₆), C₃³.₅₆(C₃×S₃), (C₃×D₉).₄C₃², (C₃²×D₉).₃C₃, C₉⋊3_-¹⁺²⋊C₂, C₉⋊₃(C₂×3_-¹⁺²), C₃².₄₁(S₃×C₃²), C₃.₄(S₃×3_-¹⁺²), (C₃×3_-¹⁺²).₁₀S₃, C₃.₆(C₃×C₉⋊C₆), (C₃×C₉).₁₀(C₃×S₃), (C₃×C₉).₁₃(C₃×C₆), SmallGroup(486,108)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₉ — D₉⋊3_-¹⁺²

Chief series

C₁ — C₃ — C₉ — C₃×C₉ — C₃²×C₉ — C₉⋊3_-¹⁺² — D₉⋊3_-¹⁺²

Lower central

C₉ — C₃×C₉ — D₉⋊3_-¹⁺²

Upper central

C₁ — C₃ — C₃²

Generators and relations for D₉⋊3_-¹⁺²
G = < a,b,c,d | a⁹=b²=c⁹=d³=1, bab=a^-1, cac^-1=a⁴, ad=da, cbc^-1=a³b, bd=db, dcd^-1=c⁴ >

Subgroups: 238 in 67 conjugacy classes, 24 normal (15 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², D₉, C₁₈, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, C₃×C₉, 3_-¹⁺², C₃³, C₃×D₉, C₃×D₉, S₃×C₉, C₂×3_-¹⁺², S₃×C₃², C₃²⋊C₉, C₉⋊C₉, C₉⋊C₉, C₃²×C₉, C₃×3_-¹⁺², C₉⋊C₁₈, C₃²×D₉, S₃×3_-¹⁺², C₉⋊3_-¹⁺², D₉⋊3_-¹⁺²
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, 3_-¹⁺², C₉⋊C₆, C₂×3_-¹⁺², S₃×C₃², C₃×C₉⋊C₆, 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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(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 13 24 7 10 21 4 16 27)(2 11 19 8 17 25 5 14 22)(3 18 23 9 15 20 6 12 26)(28 41 48 31 44 51 34 38 54)(29 39 52 32 42 46 35 45 49)(30 37 47 33 40 50 36 43 53)

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

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

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

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

42 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	3I	6A	6B	6C	6D	9A	···	9I	9J	···	9O	9P	···	9U	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	6	···	6	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	C₉⋊C₆	C₃×C₉⋊C₆	S₃×3_-¹⁺²	D₉⋊3_-¹⁺²
kernel	D₉⋊3_-¹⁺²	C₉⋊3_-¹⁺²	C₉⋊C₁₈	C₃²×D₉	C₉⋊C₉	C₃²×C₉	C₃×3_-¹⁺²	C₃×C₉	C₃³	D₉	C₉	C₃²	C₃	C₃	C₁
# reps	1	1	6	2	6	2	1	6	2	2	2	1	2	2	6

Matrix representation of D₉⋊3_-¹⁺² ►in GL₆(𝔽₁₉)

0	7	0	0	0	0
0	0	7	0	0	0
11	0	0	0	0	0
0	0	0	0	0	7
0	0	0	11	0	0
0	0	0	0	11	0

0	0	0	0	0	7
0	0	0	11	0	0
0	0	0	0	11	0
0	7	0	0	0	0
0	0	7	0	0	0
11	0	0	0	0	0

0	4	0	0	0	0
0	0	9	0	0	0
4	0	0	0	0	0
0	0	0	0	4	0
0	0	0	0	0	9
0	0	0	4	0	0

0	16	0	0	0	0
0	0	16	0	0	0
17	0	0	0	0	0
0	0	0	0	16	0
0	0	0	0	0	16
0	0	0	17	0	0

G:=sub<GL(6,GF(19))| [0,0,11,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,7,0,0],[0,0,0,0,0,11,0,0,0,7,0,0,0,0,0,0,7,0,0,11,0,0,0,0,0,0,11,0,0,0,7,0,0,0,0,0],[0,0,4,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,9,0],[0,0,17,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,17,0,0,0,16,0,0,0,0,0,0,16,0] >;

D₉⋊3_-¹⁺² in GAP, Magma, Sage, TeX

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

% in TeX

G:=Group("D9:ES-(3,1)");

// GroupNames label

G:=SmallGroup(486,108);

// by ID

G=gap.SmallGroup(486,108);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,224,8104,3250,208,11669]);

// Polycyclic

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

// generators/relations

G = D9⋊3- 1+2 order 486 = 2·35

The semidirect product of D9 and 3- 1+2 acting via 3- 1+2/C32=C3

G = D₉⋊3_-¹⁺² order 486 = 2·3⁵

The semidirect product of D₉ and 3_-¹⁺² acting via 3_-¹⁺²/C₃²=C₃