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## G = D9×C3⋊S3order 324 = 22·34

### Direct product of D9 and C3⋊S3

Aliases: D9×C3⋊S3, C327D18, C33.8D6, (C3×D9)⋊S3, C31(S3×D9), (C3×C9)⋊13D6, C32.8S32, (C32×D9)⋊2C2, (C32×C9)⋊3C22, C324D91C2, C91(C2×C3⋊S3), (C9×C3⋊S3)⋊2C2, C3.2(S3×C3⋊S3), (C3×C3⋊S3).4S3, SmallGroup(324,119)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — D9×C3⋊S3
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C32×D9 — D9×C3⋊S3
 Lower central C32×C9 — D9×C3⋊S3
 Upper central C1

Generators and relations for D9×C3⋊S3
G = < a,b,c,d,e | a9=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1150 in 122 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C3, C3, C3, C22, S3, C6, C9, C9, C32, C32, C32, D6, D9, D9, C18, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C33, D18, S32, C2×C3⋊S3, C3×D9, S3×C9, C9⋊S3, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C9, S3×D9, S3×C3⋊S3, C32×D9, C9×C3⋊S3, C324D9, D9×C3⋊S3
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, D18, S32, C2×C3⋊S3, S3×D9, S3×C3⋊S3, D9×C3⋊S3

Smallest permutation representation of D9×C3⋊S3
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 31)(2 30)(3 29)(4 28)(5 36)(6 35)(7 34)(8 33)(9 32)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 54)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 45)
(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 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 54)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 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), (1,31)(2,30)(3,29)(4,28)(5,36)(6,35)(7,34)(8,33)(9,32)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,54)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,45), (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)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (1,32)(2,33)(3,34)(4,35)(5,36)(6,28)(7,29)(8,30)(9,31)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)>;

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 6A ··· 6E 9A 9B 9C 9D ··· 9O 18A 18B 18C order 1 2 2 2 3 ··· 3 3 3 3 3 6 ··· 6 9 9 9 9 ··· 9 18 18 18 size 1 9 9 81 2 ··· 2 4 4 4 4 18 ··· 18 2 2 2 4 ··· 4 18 18 18

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D6 D6 D9 D18 S32 S3×D9 kernel D9×C3⋊S3 C32×D9 C9×C3⋊S3 C32⋊4D9 C3×D9 C3×C3⋊S3 C3×C9 C33 C3⋊S3 C32 C32 C3 # reps 1 1 1 1 4 1 4 1 3 3 4 12

Matrix representation of D9×C3⋊S3 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 7 0 0 0 0 10 2
,
 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 2 6 0 0 0 0 9 17
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 18 1 0 0 0 0 18 0 0 0 0 0 0 0 0 1 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 18 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 0 0 0 1 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,10,0,0,0,0,7,2],[18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,2,9,0,0,0,0,6,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,18,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,18,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18] >;

D9×C3⋊S3 in GAP, Magma, Sage, TeX

D_9\times C_3\rtimes S_3
% in TeX

G:=Group("D9xC3:S3");
// GroupNames label

G:=SmallGroup(324,119);
// by ID

G=gap.SmallGroup(324,119);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,80,297,5404,208,7781]);
// Polycyclic

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

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