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## G = D9⋊He3order 486 = 2·35

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

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

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