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## G = (C3×C9)⋊D9order 486 = 2·35

### 2nd semidirect product of C3×C9 and D9 acting via D9/C3=S3

Aliases: (C3×C9)⋊2D9, C32⋊C9.5C6, (C32×C9).2S3, C32.6(C3×D9), C32.3(C9⋊C6), C33.27(C3×S3), C322D9.4C3, C3.7(C32⋊D9), C32.19He32C2, C32.37(C32⋊C6), C3.4(He3.C6), SmallGroup(486,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — (C3×C9)⋊D9
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32.19He3 — (C3×C9)⋊D9
 Lower central C32⋊C9 — (C3×C9)⋊D9
 Upper central C1 — C3

Generators and relations for (C3×C9)⋊D9
G = < a,b,c,d | a3=b9=c9=d2=1, ab=ba, cac-1=ab6, dad=a-1b6, cbc-1=a-1b, bd=db, dcd=c-1 >

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

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

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

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

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.C6 C32⋊C6 C9⋊C6 (C3×C9)⋊D9 kernel (C3×C9)⋊D9 C32.19He3 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)⋊D9 in GL5(𝔽19)

 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 17 0 0 0 0 0 5 0 0 0 0 0 5
,
 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 2 0 0 0 10 0 0 0 0 0 0 18 0 0 0 0 0 0 15 0 0 0 14 0

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,17,0,0,0,0,0,5,0,0,0,0,0,5],[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],[0,10,0,0,0,2,0,0,0,0,0,0,18,0,0,0,0,0,0,14,0,0,0,15,0] >;

(C3×C9)⋊D9 in GAP, Magma, Sage, TeX

(C_3\times C_9)\rtimes D_9
% in TeX

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

G:=SmallGroup(486,21);
// by ID

G=gap.SmallGroup(486,21);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,224,338,8643,1383,3244]);
// 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^6,d*a*d=a^-1*b^6,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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