D9xHe3 - GroupNames

G = D₉×He₃ order 486 = 2·3⁵

Direct product of D₉ and He₃

direct product, metabelian, supersoluble, monomial

Aliases: D₉×He₃, C₉⋊₃(C₂×He₃), (C₉×He₃)⋊₃C₂, (C₃²×C₉)⋊₇C₆, C₃²⋊₄(C₃×D₉), C₃.₁(S₃×He₃), (C₃²×D₉)⋊₁C₃, C₃.₅(C₃²×D₉), (C₃×He₃).₂₁S₃, C₃³.₅₃(C₃×S₃), (C₃×D₉).₈C₃², C₃².₃₂(S₃×C₃²), (C₃×C₉).₄₀(C₃×C₆), SmallGroup(486,99)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₉ — D₉×He₃

Chief series

C₁ — C₃ — C₉ — C₃×C₉ — C₃²×C₉ — C₉×He₃ — D₉×He₃

Lower central

C₉ — C₃×C₉ — D₉×He₃

Upper central

C₁ — C₃ — He₃

Generators and relations for D₉×He₃
G = < a,b,c,d,e | a⁹=b²=c³=d³=e³=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=cd^-1, de=ed >

Subgroups: 436 in 100 conjugacy classes, 28 normal (12 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², C₃², D₉, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, He₃, He₃, C₃³, C₃×D₉, C₃×D₉, C₂×He₃, S₃×C₃², C₃²⋊C₉, C₃²×C₉, C₃×He₃, C₃²×D₉, S₃×He₃, C₉×He₃, D₉×He₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², D₉, C₃×S₃, C₃×C₆, He₃, C₃×D₉, C₂×He₃, S₃×C₃², C₃²×D₉, S₃×He₃, D₉×He₃

Smallest permutation representation of D₉×He₃
►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)(2 24 15)(3 25 16)(4 26 17)(5 27 18)(6 19 10)(7 20 11)(8 21 12)(9 22 13)(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)

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

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

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

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

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

66 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3M	3N	···	3U	6A	6B	6C	···	6J	9A	···	9I	9J	···	9AG
order	1	2	3	3	3	3	3	3	···	3	3	···	3	6	6	6	···	6	9	···	9	9	···	9
size	1	9	1	1	2	2	2	3	···	3	6	···	6	9	9	27	···	27	2	···	2	6	···	6

66 irreducible representations

dim	1	1	1	1	2	2	2	2	3	3	6	6
type	+	+			+	+
image	C₁	C₂	C₃	C₆	S₃	D₉	C₃×S₃	C₃×D₉	He₃	C₂×He₃	S₃×He₃	D₉×He₃
kernel	D₉×He₃	C₉×He₃	C₃²×D₉	C₃²×C₉	C₃×He₃	He₃	C₃³	C₃²	D₉	C₉	C₃	C₁
# reps	1	1	8	8	1	3	8	24	2	2	2	6

Matrix representation of D₉×He₃ ►in GL₅(𝔽₁₉)

0	1	0	0	0
18	3	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

18	3	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	7	0	0
0	0	0	11	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	11	0	0
0	0	0	11	0
0	0	0	0	11

7	0	0	0	0
0	7	0	0	0
0	0	0	0	1
0	0	1	0	0
0	0	0	1	0

G:=sub<GL(5,GF(19))| [0,18,0,0,0,1,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,0,0,0,0,3,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[7,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;

D₉×He₃ in GAP, Magma, Sage, TeX

D_9\times {\rm He}_3

% in TeX

G:=Group("D9xHe3");

// GroupNames label

G:=SmallGroup(486,99);

// by ID

G=gap.SmallGroup(486,99);

# by ID

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

// Polycyclic

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

// generators/relations

G = D9×He3 order 486 = 2·35

Direct product of D9 and He3

G = D₉×He₃ order 486 = 2·3⁵

Direct product of D₉ and He₃