D9:He3 - GroupNames

G = D₉⋊He₃ order 486 = 2·3⁵

The semidirect product of D₉ and He₃ acting via He₃/C₃²=C₃

Aliases: D₉⋊He₃, C₉⋊(C₂×He₃), C₉⋊He₃⋊₁C₂, C₃.₃(S₃×He₃), C₃²⋊₄(C₉⋊C₆), (C₃²×C₉)⋊₁₅C₆, (C₃²×D₉)⋊₂C₃, (C₃×He₃).₇S₃, C₃³.₁₀(C₃×S₃), (C₃×D₉).₃C₃², C₃².₃₉(S₃×C₃²), (C₃×3_-¹⁺²)⋊₂C₆, (C₃×C₉⋊C₆)⋊C₃, C₃.₅(C₃×C₉⋊C₆), (C₃×C₉).₁₂(C₃×C₆), SmallGroup(486,106)

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 — C₃²

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, eae^-1=a⁴, bc=cb, bd=db, ebe^-1=a³b, cd=dc, ece^-1=cd^-1, de=ed >

Subgroups: 436 in 91 conjugacy classes, 24 normal (15 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², D₉, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, He₃, 3_-¹⁺², C₃³, C₃³, C₃×D₉, C₃×D₉, C₉⋊C₆, C₂×He₃, S₃×C₃², C₃²⋊C₉, C₃²×C₉, C₃×He₃, C₃×3_-¹⁺², C₃²×D₉, S₃×He₃, C₃×C₉⋊C₆, C₉⋊He₃, D₉⋊He₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, He₃, C₉⋊C₆, C₂×He₃, S₃×C₃², S₃×He₃, C₃×C₉⋊C₆, 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 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 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)

(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)

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

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

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

42 conjugacy classes

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

42 irreducible representations

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

Matrix representation of D₉⋊He₃ ►in GL₆(𝔽₁₉)

11	12	12	13	9	7
0	0	7	0	0	0
9	8	8	13	11	14
0	0	0	0	0	7
0	0	0	11	0	0
0	0	0	0	11	0

5	7	17	8	9	10
0	0	0	11	0	0
0	0	0	0	11	0
0	7	0	0	0	0
0	0	7	0	0	0
9	8	8	13	11	14

4	0	13	3	5	6
5	15	15	3	4	12
0	4	0	0	0	0
0	0	0	0	0	6
0	0	0	4	0	0
0	0	0	0	4	0

7	0	0	0	0	0
0	7	0	0	0	0
0	0	7	0	0	0
0	0	0	7	0	0
0	0	0	0	7	0
0	0	0	0	0	7

17	0	17	15	11	14
0	0	5	0	0	0
7	2	2	8	17	13
0	0	0	0	17	0
0	0	0	0	0	5
0	0	0	17	0	0

G:=sub<GL(6,GF(19))| [11,0,9,0,0,0,12,0,8,0,0,0,12,7,8,0,0,0,13,0,13,0,11,0,9,0,11,0,0,11,7,0,14,7,0,0],[5,0,0,0,0,9,7,0,0,7,0,8,17,0,0,0,7,8,8,11,0,0,0,13,9,0,11,0,0,11,10,0,0,0,0,14],[4,5,0,0,0,0,0,15,4,0,0,0,13,15,0,0,0,0,3,3,0,0,4,0,5,4,0,0,0,4,6,12,0,6,0,0],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[17,0,7,0,0,0,0,0,2,0,0,0,17,5,2,0,0,0,15,0,8,0,0,17,11,0,17,17,0,0,14,0,13,0,5,0] >;

D₉⋊He₃ in GAP, Magma, Sage, TeX

D_9\rtimes {\rm He}_3

% in TeX

G:=Group("D9:He3");

// GroupNames label

G:=SmallGroup(486,106);

// by ID

G=gap.SmallGroup(486,106);

# by ID

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

// 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,e*a*e^-1=a^4,b*c=c*b,b*d=d*b,e*b*e^-1=a^3*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

The semidirect product of D9 and He3 acting via He3/C32=C3

G = D₉⋊He₃ order 486 = 2·3⁵

The semidirect product of D₉ and He₃ acting via He₃/C₃²=C₃