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## G = C11×D12order 264 = 23·3·11

### Direct product of C11 and D12

Aliases: C11×D12, C336D4, C443S3, C121C22, C1325C2, D61C22, C22.15D6, C66.20C22, C4⋊(S3×C11), C31(D4×C11), (S3×C22)⋊4C2, C2.4(S3×C22), C6.3(C2×C22), SmallGroup(264,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C11×D12
 Chief series C1 — C3 — C6 — C66 — S3×C22 — C11×D12
 Lower central C3 — C6 — C11×D12
 Upper central C1 — C22 — C44

Generators and relations for C11×D12
G = < a,b,c | a11=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C11×D12
On 132 points
Generators in S132
(1 28 67 79 20 57 47 97 122 117 86)(2 29 68 80 21 58 48 98 123 118 87)(3 30 69 81 22 59 37 99 124 119 88)(4 31 70 82 23 60 38 100 125 120 89)(5 32 71 83 24 49 39 101 126 109 90)(6 33 72 84 13 50 40 102 127 110 91)(7 34 61 73 14 51 41 103 128 111 92)(8 35 62 74 15 52 42 104 129 112 93)(9 36 63 75 16 53 43 105 130 113 94)(10 25 64 76 17 54 44 106 131 114 95)(11 26 65 77 18 55 45 107 132 115 96)(12 27 66 78 19 56 46 108 121 116 85)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 52)(50 51)(53 60)(54 59)(55 58)(56 57)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 86)(87 96)(88 95)(89 94)(90 93)(91 92)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 112)(110 111)(113 120)(114 119)(115 118)(116 117)(121 122)(123 132)(124 131)(125 130)(126 129)(127 128)

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

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

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

99 conjugacy classes

 class 1 2A 2B 2C 3 4 6 11A ··· 11J 12A 12B 22A ··· 22J 22K ··· 22AD 33A ··· 33J 44A ··· 44J 66A ··· 66J 132A ··· 132T order 1 2 2 2 3 4 6 11 ··· 11 12 12 22 ··· 22 22 ··· 22 33 ··· 33 44 ··· 44 66 ··· 66 132 ··· 132 size 1 1 6 6 2 2 2 1 ··· 1 2 2 1 ··· 1 6 ··· 6 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

99 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C11 C22 C22 S3 D4 D6 D12 S3×C11 D4×C11 S3×C22 C11×D12 kernel C11×D12 C132 S3×C22 D12 C12 D6 C44 C33 C22 C11 C4 C3 C2 C1 # reps 1 1 2 10 10 20 1 1 1 2 10 10 10 20

Matrix representation of C11×D12 in GL2(𝔽23) generated by

 13 0 0 13
,
 7 6 19 0
,
 0 17 19 0
G:=sub<GL(2,GF(23))| [13,0,0,13],[7,19,6,0],[0,19,17,0] >;

C11×D12 in GAP, Magma, Sage, TeX

C_{11}\times D_{12}
% in TeX

G:=Group("C11xD12");
// GroupNames label

G:=SmallGroup(264,20);
// by ID

G=gap.SmallGroup(264,20);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-3,461,226,4404]);
// Polycyclic

G:=Group<a,b,c|a^11=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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