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G = Dic3×C33order 396 = 22·32·11

Direct product of C33 and Dic3

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

Aliases: Dic3×C33, C3⋊C132, C6.C66, C333C12, C66.5C6, C66.8S3, C322C44, (C3×C33)⋊6C4, C2.(S3×C33), C6.4(S3×C11), C22.2(C3×S3), (C3×C6).1C22, (C3×C66).4C2, SmallGroup(396,12)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C33
C1C3C6C66C3×C66 — Dic3×C33
C3 — Dic3×C33
C1C66

Generators and relations for Dic3×C33
 G = < a,b,c | a33=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

2C3
3C4
2C6
2C33
3C12
3C44
2C66
3C132

Smallest permutation representation of Dic3×C33
On 132 points
Generators in S132
(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 62 23 51 12 40)(2 63 24 52 13 41)(3 64 25 53 14 42)(4 65 26 54 15 43)(5 66 27 55 16 44)(6 34 28 56 17 45)(7 35 29 57 18 46)(8 36 30 58 19 47)(9 37 31 59 20 48)(10 38 32 60 21 49)(11 39 33 61 22 50)(67 112 78 123 89 101)(68 113 79 124 90 102)(69 114 80 125 91 103)(70 115 81 126 92 104)(71 116 82 127 93 105)(72 117 83 128 94 106)(73 118 84 129 95 107)(74 119 85 130 96 108)(75 120 86 131 97 109)(76 121 87 132 98 110)(77 122 88 100 99 111)
(1 126 51 70)(2 127 52 71)(3 128 53 72)(4 129 54 73)(5 130 55 74)(6 131 56 75)(7 132 57 76)(8 100 58 77)(9 101 59 78)(10 102 60 79)(11 103 61 80)(12 104 62 81)(13 105 63 82)(14 106 64 83)(15 107 65 84)(16 108 66 85)(17 109 34 86)(18 110 35 87)(19 111 36 88)(20 112 37 89)(21 113 38 90)(22 114 39 91)(23 115 40 92)(24 116 41 93)(25 117 42 94)(26 118 43 95)(27 119 44 96)(28 120 45 97)(29 121 46 98)(30 122 47 99)(31 123 48 67)(32 124 49 68)(33 125 50 69)

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

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

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

198 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E11A···11J12A12B12C12D22A···22J33A···33T33U···33AX44A···44T66A···66T66U···66AX132A···132AN
order1233333446666611···111212121222···2233···3333···3344···4466···6666···66132···132
size111122233112221···133331···11···12···23···31···12···23···3

198 irreducible representations

dim11111111111122222222
type+++-
imageC1C2C3C4C6C11C12C22C33C44C66C132S3Dic3C3×S3C3×Dic3S3×C11C11×Dic3S3×C33Dic3×C33
kernelDic3×C33C3×C66C11×Dic3C3×C33C66C3×Dic3C33C3×C6Dic3C32C6C3C66C33C22C11C6C3C2C1
# reps112221041020202040112210102020

Matrix representation of Dic3×C33 in GL3(𝔽397) generated by

36200
01470
00147
,
39600
03620
0034
,
33400
001
010
G:=sub<GL(3,GF(397))| [362,0,0,0,147,0,0,0,147],[396,0,0,0,362,0,0,0,34],[334,0,0,0,0,1,0,1,0] >;

Dic3×C33 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{33}
% in TeX

G:=Group("Dic3xC33");
// GroupNames label

G:=SmallGroup(396,12);
// by ID

G=gap.SmallGroup(396,12);
# by ID

G:=PCGroup([5,-2,-3,-11,-2,-3,330,6604]);
// Polycyclic

G:=Group<a,b,c|a^33=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Dic3×C33 in TeX

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