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G = C11×Dic3order 132 = 22·3·11

Direct product of C11 and Dic3

Aliases: C11×Dic3, C3⋊C44, C333C4, C6.C22, C66.3C2, C22.2S3, C2.(S3×C11), SmallGroup(132,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C11×Dic3
 Chief series C1 — C3 — C6 — C66 — C11×Dic3
 Lower central C3 — C11×Dic3
 Upper central C1 — C22

Generators and relations for C11×Dic3
G = < a,b,c | a11=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C11×Dic3
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 88 116 99 101 67)(2 78 117 89 102 68)(3 79 118 90 103 69)(4 80 119 91 104 70)(5 81 120 92 105 71)(6 82 121 93 106 72)(7 83 111 94 107 73)(8 84 112 95 108 74)(9 85 113 96 109 75)(10 86 114 97 110 76)(11 87 115 98 100 77)(12 123 50 24 34 61)(13 124 51 25 35 62)(14 125 52 26 36 63)(15 126 53 27 37 64)(16 127 54 28 38 65)(17 128 55 29 39 66)(18 129 45 30 40 56)(19 130 46 31 41 57)(20 131 47 32 42 58)(21 132 48 33 43 59)(22 122 49 23 44 60)
(1 23 99 22)(2 24 89 12)(3 25 90 13)(4 26 91 14)(5 27 92 15)(6 28 93 16)(7 29 94 17)(8 30 95 18)(9 31 96 19)(10 32 97 20)(11 33 98 21)(34 117 123 68)(35 118 124 69)(36 119 125 70)(37 120 126 71)(38 121 127 72)(39 111 128 73)(40 112 129 74)(41 113 130 75)(42 114 131 76)(43 115 132 77)(44 116 122 67)(45 108 56 84)(46 109 57 85)(47 110 58 86)(48 100 59 87)(49 101 60 88)(50 102 61 78)(51 103 62 79)(52 104 63 80)(53 105 64 81)(54 106 65 82)(55 107 66 83)

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

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

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

C11×Dic3 is a maximal subgroup of   D33⋊C4  C3⋊D44  C33⋊Q8  S3×C44

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C11 C22 C44 S3 Dic3 S3×C11 C11×Dic3 kernel C11×Dic3 C66 C33 Dic3 C6 C3 C22 C11 C2 C1 # reps 1 1 2 10 10 20 1 1 10 10

Matrix representation of C11×Dic3 in GL2(𝔽23) generated by

 13 0 0 13
,
 1 21 12 0
,
 12 12 9 11
G:=sub<GL(2,GF(23))| [13,0,0,13],[1,12,21,0],[12,9,12,11] >;

C11×Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C11xDic3");
// GroupNames label

G:=SmallGroup(132,1);
// by ID

G=gap.SmallGroup(132,1);
# by ID

G:=PCGroup([4,-2,-11,-2,-3,88,1411]);
// Polycyclic

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

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