direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C12×D11, C44⋊2C6, D22.C6, C132⋊4C2, C6.14D22, Dic11⋊2C6, C66.14C22, C33⋊5(C2×C4), C11⋊1(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
| C11 — C12×D11 |
Generators and relations for C12×D11
G = < a,b,c | a12=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
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 76 121 47 60 29 22 118 70 96 97)(2 77 122 48 49 30 23 119 71 85 98)(3 78 123 37 50 31 24 120 72 86 99)(4 79 124 38 51 32 13 109 61 87 100)(5 80 125 39 52 33 14 110 62 88 101)(6 81 126 40 53 34 15 111 63 89 102)(7 82 127 41 54 35 16 112 64 90 103)(8 83 128 42 55 36 17 113 65 91 104)(9 84 129 43 56 25 18 114 66 92 105)(10 73 130 44 57 26 19 115 67 93 106)(11 74 131 45 58 27 20 116 68 94 107)(12 75 132 46 59 28 21 117 69 95 108)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(37 120)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 121)(71 122)(72 123)(73 93)(74 94)(75 95)(76 96)(77 85)(78 86)(79 87)(80 88)(81 89)(82 90)(83 91)(84 92)
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), (1,76,121,47,60,29,22,118,70,96,97)(2,77,122,48,49,30,23,119,71,85,98)(3,78,123,37,50,31,24,120,72,86,99)(4,79,124,38,51,32,13,109,61,87,100)(5,80,125,39,52,33,14,110,62,88,101)(6,81,126,40,53,34,15,111,63,89,102)(7,82,127,41,54,35,16,112,64,90,103)(8,83,128,42,55,36,17,113,65,91,104)(9,84,129,43,56,25,18,114,66,92,105)(10,73,130,44,57,26,19,115,67,93,106)(11,74,131,45,58,27,20,116,68,94,107)(12,75,132,46,59,28,21,117,69,95,108), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(37,120)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,121)(71,122)(72,123)(73,93)(74,94)(75,95)(76,96)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92)>;
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), (1,76,121,47,60,29,22,118,70,96,97)(2,77,122,48,49,30,23,119,71,85,98)(3,78,123,37,50,31,24,120,72,86,99)(4,79,124,38,51,32,13,109,61,87,100)(5,80,125,39,52,33,14,110,62,88,101)(6,81,126,40,53,34,15,111,63,89,102)(7,82,127,41,54,35,16,112,64,90,103)(8,83,128,42,55,36,17,113,65,91,104)(9,84,129,43,56,25,18,114,66,92,105)(10,73,130,44,57,26,19,115,67,93,106)(11,74,131,45,58,27,20,116,68,94,107)(12,75,132,46,59,28,21,117,69,95,108), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(37,120)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,121)(71,122)(72,123)(73,93)(74,94)(75,95)(76,96)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92) );
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)], [(1,76,121,47,60,29,22,118,70,96,97),(2,77,122,48,49,30,23,119,71,85,98),(3,78,123,37,50,31,24,120,72,86,99),(4,79,124,38,51,32,13,109,61,87,100),(5,80,125,39,52,33,14,110,62,88,101),(6,81,126,40,53,34,15,111,63,89,102),(7,82,127,41,54,35,16,112,64,90,103),(8,83,128,42,55,36,17,113,65,91,104),(9,84,129,43,56,25,18,114,66,92,105),(10,73,130,44,57,26,19,115,67,93,106),(11,74,131,45,58,27,20,116,68,94,107),(12,75,132,46,59,28,21,117,69,95,108)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,105),(10,106),(11,107),(12,108),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(37,120),(38,109),(39,110),(40,111),(41,112),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,121),(71,122),(72,123),(73,93),(74,94),(75,95),(76,96),(77,85),(78,86),(79,87),(80,88),(81,89),(82,90),(83,91),(84,92)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 11A | ··· | 11E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 22A | ··· | 22E | 33A | ··· | 33J | 44A | ··· | 44J | 66A | ··· | 66J | 132A | ··· | 132T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 11 | 11 | 1 | 1 | 1 | 1 | 11 | 11 | 1 | 1 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D11 | D22 | C3×D11 | C4×D11 | C6×D11 | C12×D11 |
| kernel | C12×D11 | C3×Dic11 | C132 | C6×D11 | C4×D11 | C3×D11 | Dic11 | C44 | D22 | D11 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 5 | 5 | 10 | 10 | 10 | 20 |
Matrix representation of C12×D11 ►in GL3(𝔽397) generated by
| 334 | 0 | 0 |
| 0 | 362 | 0 |
| 0 | 0 | 362 |
| 1 | 0 | 0 |
| 0 | 396 | 97 |
| 0 | 396 | 96 |
| 396 | 0 | 0 |
| 0 | 104 | 293 |
| 0 | 62 | 293 |
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
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