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## G = C3×Dic33order 396 = 22·32·11

### Direct product of C3 and Dic33

Aliases: C3×Dic33, C331C12, C66.1C6, C66.4S3, C6.4D33, C332Dic3, C322Dic11, (C3×C33)⋊4C4, C22.(C3×S3), C2.(C3×D33), C6.(C3×D11), C11⋊(C3×Dic3), C3⋊(C3×Dic11), (C3×C66).2C2, (C3×C6).1D11, SmallGroup(396,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3×Dic33
 Chief series C1 — C11 — C33 — C66 — C3×C66 — C3×Dic33
 Lower central C33 — C3×Dic33
 Upper central C1 — C6

Generators and relations for C3×Dic33
G = < a,b,c | a3=b66=1, c2=b33, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic33
On 132 points
Generators in S132
(1 45 23)(2 46 24)(3 47 25)(4 48 26)(5 49 27)(6 50 28)(7 51 29)(8 52 30)(9 53 31)(10 54 32)(11 55 33)(12 56 34)(13 57 35)(14 58 36)(15 59 37)(16 60 38)(17 61 39)(18 62 40)(19 63 41)(20 64 42)(21 65 43)(22 66 44)(67 89 111)(68 90 112)(69 91 113)(70 92 114)(71 93 115)(72 94 116)(73 95 117)(74 96 118)(75 97 119)(76 98 120)(77 99 121)(78 100 122)(79 101 123)(80 102 124)(81 103 125)(82 104 126)(83 105 127)(84 106 128)(85 107 129)(86 108 130)(87 109 131)(88 110 132)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132)
(1 78 34 111)(2 77 35 110)(3 76 36 109)(4 75 37 108)(5 74 38 107)(6 73 39 106)(7 72 40 105)(8 71 41 104)(9 70 42 103)(10 69 43 102)(11 68 44 101)(12 67 45 100)(13 132 46 99)(14 131 47 98)(15 130 48 97)(16 129 49 96)(17 128 50 95)(18 127 51 94)(19 126 52 93)(20 125 53 92)(21 124 54 91)(22 123 55 90)(23 122 56 89)(24 121 57 88)(25 120 58 87)(26 119 59 86)(27 118 60 85)(28 117 61 84)(29 116 62 83)(30 115 63 82)(31 114 64 81)(32 113 65 80)(33 112 66 79)

G:=sub<Sym(132)| (1,45,23)(2,46,24)(3,47,25)(4,48,26)(5,49,27)(6,50,28)(7,51,29)(8,52,30)(9,53,31)(10,54,32)(11,55,33)(12,56,34)(13,57,35)(14,58,36)(15,59,37)(16,60,38)(17,61,39)(18,62,40)(19,63,41)(20,64,42)(21,65,43)(22,66,44)(67,89,111)(68,90,112)(69,91,113)(70,92,114)(71,93,115)(72,94,116)(73,95,117)(74,96,118)(75,97,119)(76,98,120)(77,99,121)(78,100,122)(79,101,123)(80,102,124)(81,103,125)(82,104,126)(83,105,127)(84,106,128)(85,107,129)(86,108,130)(87,109,131)(88,110,132), (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)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132), (1,78,34,111)(2,77,35,110)(3,76,36,109)(4,75,37,108)(5,74,38,107)(6,73,39,106)(7,72,40,105)(8,71,41,104)(9,70,42,103)(10,69,43,102)(11,68,44,101)(12,67,45,100)(13,132,46,99)(14,131,47,98)(15,130,48,97)(16,129,49,96)(17,128,50,95)(18,127,51,94)(19,126,52,93)(20,125,53,92)(21,124,54,91)(22,123,55,90)(23,122,56,89)(24,121,57,88)(25,120,58,87)(26,119,59,86)(27,118,60,85)(28,117,61,84)(29,116,62,83)(30,115,63,82)(31,114,64,81)(32,113,65,80)(33,112,66,79)>;

G:=Group( (1,45,23)(2,46,24)(3,47,25)(4,48,26)(5,49,27)(6,50,28)(7,51,29)(8,52,30)(9,53,31)(10,54,32)(11,55,33)(12,56,34)(13,57,35)(14,58,36)(15,59,37)(16,60,38)(17,61,39)(18,62,40)(19,63,41)(20,64,42)(21,65,43)(22,66,44)(67,89,111)(68,90,112)(69,91,113)(70,92,114)(71,93,115)(72,94,116)(73,95,117)(74,96,118)(75,97,119)(76,98,120)(77,99,121)(78,100,122)(79,101,123)(80,102,124)(81,103,125)(82,104,126)(83,105,127)(84,106,128)(85,107,129)(86,108,130)(87,109,131)(88,110,132), (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)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132), (1,78,34,111)(2,77,35,110)(3,76,36,109)(4,75,37,108)(5,74,38,107)(6,73,39,106)(7,72,40,105)(8,71,41,104)(9,70,42,103)(10,69,43,102)(11,68,44,101)(12,67,45,100)(13,132,46,99)(14,131,47,98)(15,130,48,97)(16,129,49,96)(17,128,50,95)(18,127,51,94)(19,126,52,93)(20,125,53,92)(21,124,54,91)(22,123,55,90)(23,122,56,89)(24,121,57,88)(25,120,58,87)(26,119,59,86)(27,118,60,85)(28,117,61,84)(29,116,62,83)(30,115,63,82)(31,114,64,81)(32,113,65,80)(33,112,66,79) );

G=PermutationGroup([(1,45,23),(2,46,24),(3,47,25),(4,48,26),(5,49,27),(6,50,28),(7,51,29),(8,52,30),(9,53,31),(10,54,32),(11,55,33),(12,56,34),(13,57,35),(14,58,36),(15,59,37),(16,60,38),(17,61,39),(18,62,40),(19,63,41),(20,64,42),(21,65,43),(22,66,44),(67,89,111),(68,90,112),(69,91,113),(70,92,114),(71,93,115),(72,94,116),(73,95,117),(74,96,118),(75,97,119),(76,98,120),(77,99,121),(78,100,122),(79,101,123),(80,102,124),(81,103,125),(82,104,126),(83,105,127),(84,106,128),(85,107,129),(86,108,130),(87,109,131),(88,110,132)], [(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),(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132)], [(1,78,34,111),(2,77,35,110),(3,76,36,109),(4,75,37,108),(5,74,38,107),(6,73,39,106),(7,72,40,105),(8,71,41,104),(9,70,42,103),(10,69,43,102),(11,68,44,101),(12,67,45,100),(13,132,46,99),(14,131,47,98),(15,130,48,97),(16,129,49,96),(17,128,50,95),(18,127,51,94),(19,126,52,93),(20,125,53,92),(21,124,54,91),(22,123,55,90),(23,122,56,89),(24,121,57,88),(25,120,58,87),(26,119,59,86),(27,118,60,85),(28,117,61,84),(29,116,62,83),(30,115,63,82),(31,114,64,81),(32,113,65,80),(33,112,66,79)])

108 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 11A ··· 11E 12A 12B 12C 12D 22A ··· 22E 33A ··· 33AN 66A ··· 66AN order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 11 ··· 11 12 12 12 12 22 ··· 22 33 ··· 33 66 ··· 66 size 1 1 1 1 2 2 2 33 33 1 1 2 2 2 2 ··· 2 33 33 33 33 2 ··· 2 2 ··· 2 2 ··· 2

108 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 Dic3 C3×S3 D11 C3×Dic3 Dic11 C3×D11 D33 C3×Dic11 Dic33 C3×D33 C3×Dic33 kernel C3×Dic33 C3×C66 Dic33 C3×C33 C66 C33 C66 C33 C22 C3×C6 C11 C32 C6 C6 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 2 5 2 5 10 10 10 10 20 20

Matrix representation of C3×Dic33 in GL3(𝔽397) generated by

 362 0 0 0 34 0 0 0 34
,
 396 0 0 0 110 0 0 0 314
,
 63 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(397))| [362,0,0,0,34,0,0,0,34],[396,0,0,0,110,0,0,0,314],[63,0,0,0,0,1,0,1,0] >;

C3×Dic33 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{33}
% in TeX

G:=Group("C3xDic33");
// GroupNames label

G:=SmallGroup(396,13);
// by ID

G=gap.SmallGroup(396,13);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-11,30,483,9004]);
// Polycyclic

G:=Group<a,b,c|a^3=b^66=1,c^2=b^33,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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