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G = C12×D11order 264 = 23·3·11

Direct product of C12 and D11

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

Aliases: C12×D11, C442C6, D22.C6, C1324C2, C6.14D22, Dic112C6, C66.14C22, C335(C2×C4), C111(C2×C12), C22.2(C2×C6), C2.1(C6×D11), (C6×D11).2C2, (C3×Dic11)⋊5C2, SmallGroup(264,14)

Series: Derived Chief Lower central Upper central

C1C11 — C12×D11
C1C11C22C66C6×D11 — C12×D11
C11 — C12×D11
C1C12

Generators and relations for C12×D11
 G = < a,b,c | a12=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C2
11C4
11C22
11C6
11C6
11C2×C4
11C12
11C2×C6
11C2×C12

Smallest permutation representation of C12×D11
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 13 93 58 70 103 31 47 115 130 82)(2 14 94 59 71 104 32 48 116 131 83)(3 15 95 60 72 105 33 37 117 132 84)(4 16 96 49 61 106 34 38 118 121 73)(5 17 85 50 62 107 35 39 119 122 74)(6 18 86 51 63 108 36 40 120 123 75)(7 19 87 52 64 97 25 41 109 124 76)(8 20 88 53 65 98 26 42 110 125 77)(9 21 89 54 66 99 27 43 111 126 78)(10 22 90 55 67 100 28 44 112 127 79)(11 23 91 56 68 101 29 45 113 128 80)(12 24 92 57 69 102 30 46 114 129 81)
(1 82)(2 83)(3 84)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 130)(14 131)(15 132)(16 121)(17 122)(18 123)(19 124)(20 125)(21 126)(22 127)(23 128)(24 129)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 61)(35 62)(36 63)(37 60)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(85 119)(86 120)(87 109)(88 110)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)

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

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

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

84 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F11A···11E12A12B12C12D12E12F12G12H22A···22E33A···33J44A···44J66A···66J132A···132T
order122233444466666611···11121212121212121222···2233···3344···4466···66132···132
size1111111111111111111111112···21111111111112···22···22···22···22···2

84 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C3C4C6C6C6C12D11D22C3×D11C4×D11C6×D11C12×D11
kernelC12×D11C3×Dic11C132C6×D11C4×D11C3×D11Dic11C44D22D11C12C6C4C3C2C1
# reps11112422285510101020

Matrix representation of C12×D11 in GL3(𝔽397) generated by

33400
03620
00362
,
100
039697
039696
,
39600
0104293
062293
G:=sub<GL(3,GF(397))| [334,0,0,0,362,0,0,0,362],[1,0,0,0,396,396,0,97,96],[396,0,0,0,104,62,0,293,293] >;

C12×D11 in GAP, Magma, Sage, TeX

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

G:=Group("C12xD11");
// GroupNames label

G:=SmallGroup(264,14);
// by ID

G=gap.SmallGroup(264,14);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-11,66,6004]);
// Polycyclic

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

Export

Subgroup lattice of C12×D11 in TeX

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