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## G = C9×C3⋊Dic3order 324 = 22·34

### Direct product of C9 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C9×C3⋊Dic3
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C18 — C9×C3⋊Dic3
 Lower central C32 — C9×C3⋊Dic3
 Upper central C1 — C18

Generators and relations for C9×C3⋊Dic3
G = < a,b,c,d | a9=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 166 in 86 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C9, C9, C32, C32, C32, Dic3, C12, C18, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C33, C36, C3×Dic3, C3⋊Dic3, C3×C18, C3×C18, C32×C6, C32×C9, C9×Dic3, C3×C3⋊Dic3, C32×C18, C9×C3⋊Dic3
Quotients: C1, C2, C3, C4, S3, C6, C9, Dic3, C12, C18, C3×S3, C3⋊S3, C36, C3×Dic3, C3⋊Dic3, S3×C9, C3×C3⋊S3, C9×Dic3, C3×C3⋊Dic3, C9×C3⋊S3, C9×C3⋊Dic3

Smallest permutation representation of C9×C3⋊Dic3
On 108 points
Generators in S108
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)
(1 50 16)(2 51 17)(3 52 18)(4 53 10)(5 54 11)(6 46 12)(7 47 13)(8 48 14)(9 49 15)(19 70 104)(20 71 105)(21 72 106)(22 64 107)(23 65 108)(24 66 100)(25 67 101)(26 68 102)(27 69 103)(28 80 59)(29 81 60)(30 73 61)(31 74 62)(32 75 63)(33 76 55)(34 77 56)(35 78 57)(36 79 58)(37 91 85)(38 92 86)(39 93 87)(40 94 88)(41 95 89)(42 96 90)(43 97 82)(44 98 83)(45 99 84)
(1 31 53 77 13 59)(2 32 54 78 14 60)(3 33 46 79 15 61)(4 34 47 80 16 62)(5 35 48 81 17 63)(6 36 49 73 18 55)(7 28 50 74 10 56)(8 29 51 75 11 57)(9 30 52 76 12 58)(19 88 67 37 107 97)(20 89 68 38 108 98)(21 90 69 39 100 99)(22 82 70 40 101 91)(23 83 71 41 102 92)(24 84 72 42 103 93)(25 85 64 43 104 94)(26 86 65 44 105 95)(27 87 66 45 106 96)
(1 37 77 19)(2 38 78 20)(3 39 79 21)(4 40 80 22)(5 41 81 23)(6 42 73 24)(7 43 74 25)(8 44 75 26)(9 45 76 27)(10 94 28 64)(11 95 29 65)(12 96 30 66)(13 97 31 67)(14 98 32 68)(15 99 33 69)(16 91 34 70)(17 92 35 71)(18 93 36 72)(46 90 61 100)(47 82 62 101)(48 83 63 102)(49 84 55 103)(50 85 56 104)(51 86 57 105)(52 87 58 106)(53 88 59 107)(54 89 60 108)

G:=sub<Sym(108)| (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,50,16)(2,51,17)(3,52,18)(4,53,10)(5,54,11)(6,46,12)(7,47,13)(8,48,14)(9,49,15)(19,70,104)(20,71,105)(21,72,106)(22,64,107)(23,65,108)(24,66,100)(25,67,101)(26,68,102)(27,69,103)(28,80,59)(29,81,60)(30,73,61)(31,74,62)(32,75,63)(33,76,55)(34,77,56)(35,78,57)(36,79,58)(37,91,85)(38,92,86)(39,93,87)(40,94,88)(41,95,89)(42,96,90)(43,97,82)(44,98,83)(45,99,84), (1,31,53,77,13,59)(2,32,54,78,14,60)(3,33,46,79,15,61)(4,34,47,80,16,62)(5,35,48,81,17,63)(6,36,49,73,18,55)(7,28,50,74,10,56)(8,29,51,75,11,57)(9,30,52,76,12,58)(19,88,67,37,107,97)(20,89,68,38,108,98)(21,90,69,39,100,99)(22,82,70,40,101,91)(23,83,71,41,102,92)(24,84,72,42,103,93)(25,85,64,43,104,94)(26,86,65,44,105,95)(27,87,66,45,106,96), (1,37,77,19)(2,38,78,20)(3,39,79,21)(4,40,80,22)(5,41,81,23)(6,42,73,24)(7,43,74,25)(8,44,75,26)(9,45,76,27)(10,94,28,64)(11,95,29,65)(12,96,30,66)(13,97,31,67)(14,98,32,68)(15,99,33,69)(16,91,34,70)(17,92,35,71)(18,93,36,72)(46,90,61,100)(47,82,62,101)(48,83,63,102)(49,84,55,103)(50,85,56,104)(51,86,57,105)(52,87,58,106)(53,88,59,107)(54,89,60,108)>;

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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,50,16)(2,51,17)(3,52,18)(4,53,10)(5,54,11)(6,46,12)(7,47,13)(8,48,14)(9,49,15)(19,70,104)(20,71,105)(21,72,106)(22,64,107)(23,65,108)(24,66,100)(25,67,101)(26,68,102)(27,69,103)(28,80,59)(29,81,60)(30,73,61)(31,74,62)(32,75,63)(33,76,55)(34,77,56)(35,78,57)(36,79,58)(37,91,85)(38,92,86)(39,93,87)(40,94,88)(41,95,89)(42,96,90)(43,97,82)(44,98,83)(45,99,84), (1,31,53,77,13,59)(2,32,54,78,14,60)(3,33,46,79,15,61)(4,34,47,80,16,62)(5,35,48,81,17,63)(6,36,49,73,18,55)(7,28,50,74,10,56)(8,29,51,75,11,57)(9,30,52,76,12,58)(19,88,67,37,107,97)(20,89,68,38,108,98)(21,90,69,39,100,99)(22,82,70,40,101,91)(23,83,71,41,102,92)(24,84,72,42,103,93)(25,85,64,43,104,94)(26,86,65,44,105,95)(27,87,66,45,106,96), (1,37,77,19)(2,38,78,20)(3,39,79,21)(4,40,80,22)(5,41,81,23)(6,42,73,24)(7,43,74,25)(8,44,75,26)(9,45,76,27)(10,94,28,64)(11,95,29,65)(12,96,30,66)(13,97,31,67)(14,98,32,68)(15,99,33,69)(16,91,34,70)(17,92,35,71)(18,93,36,72)(46,90,61,100)(47,82,62,101)(48,83,63,102)(49,84,55,103)(50,85,56,104)(51,86,57,105)(52,87,58,106)(53,88,59,107)(54,89,60,108) );

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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)], [(1,50,16),(2,51,17),(3,52,18),(4,53,10),(5,54,11),(6,46,12),(7,47,13),(8,48,14),(9,49,15),(19,70,104),(20,71,105),(21,72,106),(22,64,107),(23,65,108),(24,66,100),(25,67,101),(26,68,102),(27,69,103),(28,80,59),(29,81,60),(30,73,61),(31,74,62),(32,75,63),(33,76,55),(34,77,56),(35,78,57),(36,79,58),(37,91,85),(38,92,86),(39,93,87),(40,94,88),(41,95,89),(42,96,90),(43,97,82),(44,98,83),(45,99,84)], [(1,31,53,77,13,59),(2,32,54,78,14,60),(3,33,46,79,15,61),(4,34,47,80,16,62),(5,35,48,81,17,63),(6,36,49,73,18,55),(7,28,50,74,10,56),(8,29,51,75,11,57),(9,30,52,76,12,58),(19,88,67,37,107,97),(20,89,68,38,108,98),(21,90,69,39,100,99),(22,82,70,40,101,91),(23,83,71,41,102,92),(24,84,72,42,103,93),(25,85,64,43,104,94),(26,86,65,44,105,95),(27,87,66,45,106,96)], [(1,37,77,19),(2,38,78,20),(3,39,79,21),(4,40,80,22),(5,41,81,23),(6,42,73,24),(7,43,74,25),(8,44,75,26),(9,45,76,27),(10,94,28,64),(11,95,29,65),(12,96,30,66),(13,97,31,67),(14,98,32,68),(15,99,33,69),(16,91,34,70),(17,92,35,71),(18,93,36,72),(46,90,61,100),(47,82,62,101),(48,83,63,102),(49,84,55,103),(50,85,56,104),(51,86,57,105),(52,87,58,106),(53,88,59,107),(54,89,60,108)]])

108 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 4A 4B 6A 6B 6C ··· 6N 9A ··· 9F 9G ··· 9AD 12A 12B 12C 12D 18A ··· 18F 18G ··· 18AD 36A ··· 36L order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 ··· 2 9 9 1 1 2 ··· 2 1 ··· 1 2 ··· 2 9 9 9 9 1 ··· 1 2 ··· 2 9 ··· 9

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C9 C12 C18 C36 S3 Dic3 C3×S3 C3×Dic3 S3×C9 C9×Dic3 kernel C9×C3⋊Dic3 C32×C18 C3×C3⋊Dic3 C32×C9 C32×C6 C3⋊Dic3 C33 C3×C6 C32 C3×C18 C3×C9 C3×C6 C32 C6 C3 # reps 1 1 2 2 2 6 4 6 12 4 4 8 8 24 24

Matrix representation of C9×C3⋊Dic3 in GL4(𝔽37) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 26
,
 27 0 0 0 0 11 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 36 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(37))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,26],[27,0,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[0,36,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C9×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_9\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C9xC3:Dic3");
// GroupNames label

G:=SmallGroup(324,97);
// by ID

G=gap.SmallGroup(324,97);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,2164,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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