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G = C11×D12order 264 = 23·3·11

Direct product of C11 and D12

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

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

C1C6 — C11×D12
C1C3C6C66S3×C22 — C11×D12
C3C6 — C11×D12
C1C22C44

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

6C2
6C2
3C22
3C22
2S3
2S3
6C22
6C22
3D4
3C2×C22
3C2×C22
2S3×C11
2S3×C11
3D4×C11

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 2A2B2C 3  4  6 11A···11J12A12B22A···22J22K···22AD33A···33J44A···44J66A···66J132A···132T
order122234611···11121222···2222···2233···3344···4466···66132···132
size11662221···1221···16···62···22···22···22···2

99 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C11C22C22S3D4D6D12S3×C11D4×C11S3×C22C11×D12
kernelC11×D12C132S3×C22D12C12D6C44C33C22C11C4C3C2C1
# reps112101020111210101020

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

130
013
,
76
190
,
017
190
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

Export

Subgroup lattice of C11×D12 in TeX

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