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## G = D9×He3order 486 = 2·35

### Direct product of D9 and He3

Aliases: D9×He3, C93(C2×He3), (C9×He3)⋊3C2, (C32×C9)⋊7C6, C324(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

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