direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C6×D11, C33⋊3C23, C66⋊3C22, C22⋊(C2×C6), C11⋊(C22×C6), (C2×C22)⋊5C6, (C2×C66)⋊5C2, SmallGroup(264,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C2×C6×D11 |
Generators and relations for C2×C6×D11
G = < a,b,c,d | a2=b6=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 284 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C23, C11, C2×C6, C2×C6, D11, C22, C22×C6, C33, D22, C2×C22, C3×D11, C66, C22×D11, C6×D11, C2×C66, C2×C6×D11
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, D11, C22×C6, D22, C3×D11, C22×D11, C6×D11, C2×C6×D11
►On 132 points
Generators in S132
(1 109)(2 110)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 120)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 89)(57 90)(58 91)(59 92)(60 93)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)
(1 54 32 43 21 65)(2 55 33 44 22 66)(3 45 23 34 12 56)(4 46 24 35 13 57)(5 47 25 36 14 58)(6 48 26 37 15 59)(7 49 27 38 16 60)(8 50 28 39 17 61)(9 51 29 40 18 62)(10 52 30 41 19 63)(11 53 31 42 20 64)(67 111 89 100 78 122)(68 112 90 101 79 123)(69 113 91 102 80 124)(70 114 92 103 81 125)(71 115 93 104 82 126)(72 116 94 105 83 127)(73 117 95 106 84 128)(74 118 96 107 85 129)(75 119 97 108 86 130)(76 120 98 109 87 131)(77 121 99 110 88 132)
(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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 55)(20 54)(21 53)(22 52)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 56)(30 66)(31 65)(32 64)(33 63)(67 106)(68 105)(69 104)(70 103)(71 102)(72 101)(73 100)(74 110)(75 109)(76 108)(77 107)(78 117)(79 116)(80 115)(81 114)(82 113)(83 112)(84 111)(85 121)(86 120)(87 119)(88 118)(89 128)(90 127)(91 126)(92 125)(93 124)(94 123)(95 122)(96 132)(97 131)(98 130)(99 129)
G:=sub<Sym(132)| (1,109)(2,110)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,111)(13,112)(14,113)(15,114)(16,115)(17,116)(18,117)(19,118)(20,119)(21,120)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,89)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99), (1,54,32,43,21,65)(2,55,33,44,22,66)(3,45,23,34,12,56)(4,46,24,35,13,57)(5,47,25,36,14,58)(6,48,26,37,15,59)(7,49,27,38,16,60)(8,50,28,39,17,61)(9,51,29,40,18,62)(10,52,30,41,19,63)(11,53,31,42,20,64)(67,111,89,100,78,122)(68,112,90,101,79,123)(69,113,91,102,80,124)(70,114,92,103,81,125)(71,115,93,104,82,126)(72,116,94,105,83,127)(73,117,95,106,84,128)(74,118,96,107,85,129)(75,119,97,108,86,130)(76,120,98,109,87,131)(77,121,99,110,88,132), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,44)(11,43)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,55)(20,54)(21,53)(22,52)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,66)(31,65)(32,64)(33,63)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,110)(75,109)(76,108)(77,107)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,121)(86,120)(87,119)(88,118)(89,128)(90,127)(91,126)(92,125)(93,124)(94,123)(95,122)(96,132)(97,131)(98,130)(99,129)>;
G:=Group( (1,109)(2,110)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,111)(13,112)(14,113)(15,114)(16,115)(17,116)(18,117)(19,118)(20,119)(21,120)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,89)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99), (1,54,32,43,21,65)(2,55,33,44,22,66)(3,45,23,34,12,56)(4,46,24,35,13,57)(5,47,25,36,14,58)(6,48,26,37,15,59)(7,49,27,38,16,60)(8,50,28,39,17,61)(9,51,29,40,18,62)(10,52,30,41,19,63)(11,53,31,42,20,64)(67,111,89,100,78,122)(68,112,90,101,79,123)(69,113,91,102,80,124)(70,114,92,103,81,125)(71,115,93,104,82,126)(72,116,94,105,83,127)(73,117,95,106,84,128)(74,118,96,107,85,129)(75,119,97,108,86,130)(76,120,98,109,87,131)(77,121,99,110,88,132), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,44)(11,43)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,55)(20,54)(21,53)(22,52)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,66)(31,65)(32,64)(33,63)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,110)(75,109)(76,108)(77,107)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,121)(86,120)(87,119)(88,118)(89,128)(90,127)(91,126)(92,125)(93,124)(94,123)(95,122)(96,132)(97,131)(98,130)(99,129) );
G=PermutationGroup([[(1,109),(2,110),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,107),(11,108),(12,111),(13,112),(14,113),(15,114),(16,115),(17,116),(18,117),(19,118),(20,119),(21,120),(22,121),(23,122),(24,123),(25,124),(26,125),(27,126),(28,127),(29,128),(30,129),(31,130),(32,131),(33,132),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,81),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,88),(56,89),(57,90),(58,91),(59,92),(60,93),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99)], [(1,54,32,43,21,65),(2,55,33,44,22,66),(3,45,23,34,12,56),(4,46,24,35,13,57),(5,47,25,36,14,58),(6,48,26,37,15,59),(7,49,27,38,16,60),(8,50,28,39,17,61),(9,51,29,40,18,62),(10,52,30,41,19,63),(11,53,31,42,20,64),(67,111,89,100,78,122),(68,112,90,101,79,123),(69,113,91,102,80,124),(70,114,92,103,81,125),(71,115,93,104,82,126),(72,116,94,105,83,127),(73,117,95,106,84,128),(74,118,96,107,85,129),(75,119,97,108,86,130),(76,120,98,109,87,131),(77,121,99,110,88,132)], [(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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,44),(11,43),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,55),(20,54),(21,53),(22,52),(23,62),(24,61),(25,60),(26,59),(27,58),(28,57),(29,56),(30,66),(31,65),(32,64),(33,63),(67,106),(68,105),(69,104),(70,103),(71,102),(72,101),(73,100),(74,110),(75,109),(76,108),(77,107),(78,117),(79,116),(80,115),(81,114),(82,113),(83,112),(84,111),(85,121),(86,120),(87,119),(88,118),(89,128),(90,127),(91,126),(92,125),(93,124),(94,123),(95,122),(96,132),(97,131),(98,130),(99,129)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | ··· | 6F | 6G | ··· | 6N | 11A | ··· | 11E | 22A | ··· | 22O | 33A | ··· | 33J | 66A | ··· | 66AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | 1 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D11 | D22 | C3×D11 | C6×D11 |
| kernel | C2×C6×D11 | C6×D11 | C2×C66 | C22×D11 | D22 | C2×C22 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 12 | 2 | 5 | 15 | 10 | 30 |
Matrix representation of C2×C6×D11 ►in GL3(𝔽67) generated by
| 66 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 38 | 0 | 0 |
| 0 | 38 | 0 |
| 0 | 0 | 38 |
| 1 | 0 | 0 |
| 0 | 43 | 10 |
| 0 | 66 | 59 |
| 1 | 0 | 0 |
| 0 | 47 | 20 |
| 0 | 37 | 20 |
G:=sub<GL(3,GF(67))| [66,0,0,0,1,0,0,0,1],[38,0,0,0,38,0,0,0,38],[1,0,0,0,43,66,0,10,59],[1,0,0,0,47,37,0,20,20] >;
C2×C6×D11 in GAP, Magma, Sage, TeX
C_2\times C_6\times D_{11} % in TeX
G:=Group("C2xC6xD11"); // GroupNames label
G:=SmallGroup(264,36);
// by ID
G=gap.SmallGroup(264,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations