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G = C3×Dic11order 132 = 22·3·11

Direct product of C3 and Dic11

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

Aliases: C3×Dic11, C11⋊C12, C332C4, C22.C6, C66.2C2, C6.2D11, C2.(C3×D11), SmallGroup(132,2)

Series: Derived Chief Lower central Upper central

C1C11 — C3×Dic11
C1C11C22C66 — C3×Dic11
C11 — C3×Dic11
C1C6

Generators and relations for C3×Dic11
 G = < a,b,c | a3=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >

11C4
11C12

Smallest permutation representation of C3×Dic11
Regular action on 132 points
Generators in S132
(1 62 23)(2 63 24)(3 64 25)(4 65 26)(5 66 27)(6 45 28)(7 46 29)(8 47 30)(9 48 31)(10 49 32)(11 50 33)(12 51 34)(13 52 35)(14 53 36)(15 54 37)(16 55 38)(17 56 39)(18 57 40)(19 58 41)(20 59 42)(21 60 43)(22 61 44)(67 111 89)(68 112 90)(69 113 91)(70 114 92)(71 115 93)(72 116 94)(73 117 95)(74 118 96)(75 119 97)(76 120 98)(77 121 99)(78 122 100)(79 123 101)(80 124 102)(81 125 103)(82 126 104)(83 127 105)(84 128 106)(85 129 107)(86 130 108)(87 131 109)(88 132 110)
(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 12 67)(2 77 13 88)(3 76 14 87)(4 75 15 86)(5 74 16 85)(6 73 17 84)(7 72 18 83)(8 71 19 82)(9 70 20 81)(10 69 21 80)(11 68 22 79)(23 100 34 89)(24 99 35 110)(25 98 36 109)(26 97 37 108)(27 96 38 107)(28 95 39 106)(29 94 40 105)(30 93 41 104)(31 92 42 103)(32 91 43 102)(33 90 44 101)(45 117 56 128)(46 116 57 127)(47 115 58 126)(48 114 59 125)(49 113 60 124)(50 112 61 123)(51 111 62 122)(52 132 63 121)(53 131 64 120)(54 130 65 119)(55 129 66 118)

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

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

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

C3×Dic11 is a maximal subgroup of   D33⋊C4  C11⋊D12  C33⋊Q8  C12×D11

42 conjugacy classes

class 1  2 3A3B4A4B6A6B11A···11E12A12B12C12D22A···22E33A···33J66A···66J
order1233446611···111212121222···2233···3366···66
size11111111112···2111111112···22···22···2

42 irreducible representations

dim1111112222
type+++-
imageC1C2C3C4C6C12D11Dic11C3×D11C3×Dic11
kernelC3×Dic11C66Dic11C33C22C11C6C3C2C1
# reps112224551010

Matrix representation of C3×Dic11 in GL2(𝔽43) generated by

360
036
,
122
2133
,
2316
2620
G:=sub<GL(2,GF(43))| [36,0,0,36],[1,21,22,33],[23,26,16,20] >;

C3×Dic11 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{11}
% in TeX

G:=Group("C3xDic11");
// GroupNames label

G:=SmallGroup(132,2);
// by ID

G=gap.SmallGroup(132,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-11,24,1923]);
// Polycyclic

G:=Group<a,b,c|a^3=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic11 in TeX

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