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## G = (C3×C9)⋊3D9order 486 = 2·35

### 3rd semidirect product of C3×C9 and D9 acting via D9/C3=S3

Aliases: (C3×C9)⋊3D9, C32⋊C9.6C6, (C32×C9).3S3, C32.7(C3×D9), C32.4(C9⋊C6), C33.28(C3×S3), C322D9.5C3, C3.8(C32⋊D9), C32.20He32C2, C32.39(C32⋊C6), C3.4(He3.2C6), SmallGroup(486,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — (C3×C9)⋊3D9
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32.20He3 — (C3×C9)⋊3D9
 Lower central C32⋊C9 — (C3×C9)⋊3D9
 Upper central C1 — C3

Generators and relations for (C3×C9)⋊3D9
G = < a,b,c,d | a3=b9=c9=d2=1, ab=ba, cac-1=ab3, dad=a-1b3, cbc-1=a-1b, bd=db, dcd=c-1 >

Smallest permutation representation of (C3×C9)⋊3D9
On 54 points
Generators in S54
(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(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 40 32 4 43 35 7 37 29)(2 41 36 5 44 30 8 38 33)(3 42 31 6 45 34 9 39 28)(10 20 52 16 26 49 13 23 46)(11 21 47 17 27 53 14 24 50)(12 22 51 18 19 48 15 25 54)
(1 26)(2 27)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 43)(11 44)(12 45)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 46)(36 47)

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

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

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

39 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 9A ··· 9F 9G ··· 9L 9M ··· 9U 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 2 2 6 6 6 27 27 3 ··· 3 6 ··· 6 18 ··· 18 27 ··· 27

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 6 6 6 type + + + + + + image C1 C2 C3 C6 S3 D9 C3×S3 C3×D9 He3.2C6 C32⋊C6 C9⋊C6 (C3×C9)⋊3D9 kernel (C3×C9)⋊3D9 C32.20He3 C32⋊2D9 C32⋊C9 C32×C9 C3×C9 C33 C32 C3 C32 C32 C1 # reps 1 1 2 2 1 3 2 6 12 1 2 6

Matrix representation of (C3×C9)⋊3D9 in GL5(𝔽19)

 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 11
,
 11 0 0 0 0 0 11 0 0 0 0 0 5 0 0 0 0 0 17 0 0 0 0 0 17
,
 17 4 0 0 0 18 11 0 0 0 0 0 0 0 16 0 0 5 0 0 0 0 0 5 0
,
 0 17 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 5 0

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,11],[11,0,0,0,0,0,11,0,0,0,0,0,5,0,0,0,0,0,17,0,0,0,0,0,17],[17,18,0,0,0,4,11,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,16,0,0],[0,9,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,4,0] >;

(C3×C9)⋊3D9 in GAP, Magma, Sage, TeX

(C_3\times C_9)\rtimes_3D_9
% in TeX

G:=Group("(C3xC9):3D9");
// GroupNames label

G:=SmallGroup(486,23);
// by ID

G=gap.SmallGroup(486,23);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,224,338,4755,735,3244]);
// Polycyclic

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

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