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G = C3×Dic33order 396 = 22·32·11

Direct product of C3 and Dic33

direct product, metacyclic, supersoluble, monomial, A-group

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

C1C33 — C3×Dic33
C1C11C33C66C3×C66 — C3×Dic33
C33 — C3×Dic33
C1C6

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

2C3
33C4
2C6
2C33
11Dic3
33C12
3Dic11
2C66
11C3×Dic3
3C3×Dic11

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 3A3B3C3D3E4A4B6A6B6C6D6E11A···11E12A12B12C12D22A···22E33A···33AN66A···66AN
order1233333446666611···111212121222···2233···3366···66
size11112223333112222···2333333332···22···22···2

108 irreducible representations

dim111111222222222222
type+++-+-+-
imageC1C2C3C4C6C12S3Dic3C3×S3D11C3×Dic3Dic11C3×D11D33C3×Dic11Dic33C3×D33C3×Dic33
kernelC3×Dic33C3×C66Dic33C3×C33C66C33C66C33C22C3×C6C11C32C6C6C3C3C2C1
# reps112224112525101010102020

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

36200
0340
0034
,
39600
01100
00314
,
6300
001
010
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

Export

Subgroup lattice of C3×Dic33 in TeX

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