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## G = C3×C9⋊Dic3order 324 = 22·34

### Direct product of C3 and C9⋊Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C9⋊Dic3
G = < a,b,c,d | a3=b9=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 252 in 90 conjugacy classes, 42 normal (18 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, C3×C9, C33, Dic9, C3×Dic3, C3⋊Dic3, C3×C18, C3×C18, C3×C18, C32×C6, C32×C9, C3×Dic9, C9⋊Dic3, C3×C3⋊Dic3, C32×C18, C3×C9⋊Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, D9, C3×S3, C3⋊S3, Dic9, C3×Dic3, C3⋊Dic3, C3×D9, C9⋊S3, C3×C3⋊S3, C3×Dic9, C9⋊Dic3, C3×C3⋊Dic3, C3×C9⋊S3, C3×C9⋊Dic3

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

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

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

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

90 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 4A 4B 6A 6B 6C ··· 6N 9A ··· 9AA 12A 12B 12C 12D 18A ··· 18AA order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 9 ··· 9 12 12 12 12 18 ··· 18 size 1 1 1 1 2 ··· 2 27 27 1 1 2 ··· 2 2 ··· 2 27 27 27 27 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - - + - image C1 C2 C3 C4 C6 C12 S3 S3 Dic3 Dic3 C3×S3 D9 C3×S3 C3×Dic3 Dic9 C3×Dic3 C3×D9 C3×Dic9 kernel C3×C9⋊Dic3 C32×C18 C9⋊Dic3 C32×C9 C3×C18 C3×C9 C3×C18 C32×C6 C3×C9 C33 C18 C3×C6 C3×C6 C9 C32 C32 C6 C3 # reps 1 1 2 2 2 4 3 1 3 1 6 9 2 6 9 2 18 18

Matrix representation of C3×C9⋊Dic3 in GL5(𝔽37)

 10 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 26 0 0 0 0 0 26
,
 1 0 0 0 0 0 16 0 0 0 0 0 7 0 0 0 0 0 9 0 0 0 0 0 33
,
 36 0 0 0 0 0 10 0 0 0 0 0 26 0 0 0 0 0 1 0 0 0 0 0 1
,
 31 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(37))| [10,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,26,0,0,0,0,0,26],[1,0,0,0,0,0,16,0,0,0,0,0,7,0,0,0,0,0,9,0,0,0,0,0,33],[36,0,0,0,0,0,10,0,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,0,1],[31,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3×C9⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC9:Dic3");
// GroupNames label

G:=SmallGroup(324,96);
// by ID

G=gap.SmallGroup(324,96);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,3171,453,2164,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=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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