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## G = Dic3×D11order 264 = 23·3·11

### Direct product of Dic3 and D11

Aliases: Dic3×D11, D22.S3, C6.1D22, C22.1D6, Dic332C2, C66.1C22, C331(C2×C4), (C3×D11)⋊C4, C33(C4×D11), (C6×D11).C2, C2.1(S3×D11), C111(C2×Dic3), (C11×Dic3)⋊1C2, SmallGroup(264,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — Dic3×D11
 Chief series C1 — C11 — C33 — C66 — C6×D11 — Dic3×D11
 Lower central C33 — Dic3×D11
 Upper central C1 — C2

Generators and relations for Dic3×D11
G = < a,b,c,d | a6=c11=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of Dic3×D11
On 132 points
Generators in S132
(1 65 21 43 32 54)(2 66 22 44 33 55)(3 56 12 34 23 45)(4 57 13 35 24 46)(5 58 14 36 25 47)(6 59 15 37 26 48)(7 60 16 38 27 49)(8 61 17 39 28 50)(9 62 18 40 29 51)(10 63 19 41 30 52)(11 64 20 42 31 53)(67 111 89 100 78 122)(68 112 90 101 79 123)(69 113 91 102 80 124)(70 114 92 103 81 125)(71 115 93 104 82 126)(72 116 94 105 83 127)(73 117 95 106 84 128)(74 118 96 107 85 129)(75 119 97 108 86 130)(76 120 98 109 87 131)(77 121 99 110 88 132)
(1 109 43 76)(2 110 44 77)(3 100 34 67)(4 101 35 68)(5 102 36 69)(6 103 37 70)(7 104 38 71)(8 105 39 72)(9 106 40 73)(10 107 41 74)(11 108 42 75)(12 111 45 78)(13 112 46 79)(14 113 47 80)(15 114 48 81)(16 115 49 82)(17 116 50 83)(18 117 51 84)(19 118 52 85)(20 119 53 86)(21 120 54 87)(22 121 55 88)(23 122 56 89)(24 123 57 90)(25 124 58 91)(26 125 59 92)(27 126 60 93)(28 127 61 94)(29 128 62 95)(30 129 63 96)(31 130 64 97)(32 131 65 98)(33 132 66 99)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)(78 84)(79 83)(80 82)(85 88)(86 87)(89 95)(90 94)(91 93)(96 99)(97 98)(100 106)(101 105)(102 104)(107 110)(108 109)(111 117)(112 116)(113 115)(118 121)(119 120)(122 128)(123 127)(124 126)(129 132)(130 131)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 11A ··· 11E 22A ··· 22E 33A ··· 33E 44A ··· 44J 66A ··· 66E order 1 2 2 2 3 4 4 4 4 6 6 6 11 ··· 11 22 ··· 22 33 ··· 33 44 ··· 44 66 ··· 66 size 1 1 11 11 2 3 3 33 33 2 22 22 2 ··· 2 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + - + + + + - image C1 C2 C2 C2 C4 S3 Dic3 D6 D11 D22 C4×D11 S3×D11 Dic3×D11 kernel Dic3×D11 C11×Dic3 Dic33 C6×D11 C3×D11 D22 D11 C22 Dic3 C6 C3 C2 C1 # reps 1 1 1 1 4 1 2 1 5 5 10 5 5

Matrix representation of Dic3×D11 in GL4(𝔽397) generated by

 1 0 0 0 0 1 0 0 0 0 396 147 0 0 81 2
,
 1 0 0 0 0 1 0 0 0 0 256 226 0 0 302 141
,
 104 1 0 0 43 260 0 0 0 0 1 0 0 0 0 1
,
 260 396 0 0 109 137 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(397))| [1,0,0,0,0,1,0,0,0,0,396,81,0,0,147,2],[1,0,0,0,0,1,0,0,0,0,256,302,0,0,226,141],[104,43,0,0,1,260,0,0,0,0,1,0,0,0,0,1],[260,109,0,0,396,137,0,0,0,0,1,0,0,0,0,1] >;

Dic3×D11 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times D_{11}
% in TeX

G:=Group("Dic3xD11");
// GroupNames label

G:=SmallGroup(264,5);
// by ID

G=gap.SmallGroup(264,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-11,26,168,6004]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^11=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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