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G = Dic33order 132 = 22·3·11

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic33, C331C4, C22.S3, C2.D33, C6.D11, C3⋊Dic11, C11⋊Dic3, C66.1C2, SmallGroup(132,3)

Series: Derived Chief Lower central Upper central

C1C33 — Dic33
C1C11C33C66 — Dic33
C33 — Dic33
C1C2

Generators and relations for Dic33
 G = < a,b | a66=1, b2=a33, bab-1=a-1 >

33C4
11Dic3
3Dic11

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

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

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

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

Dic33 is a maximal subgroup of
Dic3×D11  S3×Dic11  C33⋊D4  C33⋊Q8  Dic66  C4×D33  C337D4  Dic99  C3⋊Dic33
Dic33 is a maximal quotient of
C33⋊C8  Dic99  C3⋊Dic33

36 conjugacy classes

class 1  2  3 4A4B 6 11A···11E22A···22E33A···33J66A···66J
order12344611···1122···2233···3366···66
size112333322···22···22···22···2

36 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D11Dic11D33Dic33
kernelDic33C66C33C22C11C6C3C2C1
# reps11211551010

Matrix representation of Dic33 in GL2(𝔽397) generated by

21287
110388
,
45348
252352
G:=sub<GL(2,GF(397))| [21,110,287,388],[45,252,348,352] >;

Dic33 in GAP, Magma, Sage, TeX

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

G:=Group("Dic33");
// GroupNames label

G:=SmallGroup(132,3);
// by ID

G=gap.SmallGroup(132,3);
# by ID

G:=PCGroup([4,-2,-2,-3,-11,8,98,1923]);
// Polycyclic

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

Export

Subgroup lattice of Dic33 in TeX

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