metabelian, supersoluble, monomial
Aliases: C62.20D6, (C3×C12)⋊1C12, C4⋊(C32⋊C12), He3⋊7(C4⋊C4), (C6×C12).9S3, (C6×C12).5C6, (C4×He3)⋊3C4, C12⋊Dic3⋊C3, (C3×C12)⋊1Dic3, (C3×C6).19D12, C6.12(C3×D12), C62.6(C2×C6), (C2×He3).5Q8, C6.6(C3×Dic6), (C3×C6).6Dic6, (C2×He3).20D4, C6.13(C6×Dic3), C12.4(C3×Dic3), C2.1(He3⋊4D4), C32⋊4(C4⋊Dic3), C2.2(He3⋊3Q8), (C22×He3).18C22, C32⋊4(C3×C4⋊C4), (C3×C6).9(C3×D4), (C2×C4×He3).6C2, (C3×C6).3(C3×Q8), (C3×C6).8(C2×C12), (C2×C6).40(S3×C6), C3.2(C3×C4⋊Dic3), (C2×C12).12(C3×S3), C2.4(C2×C32⋊C12), (C2×C3⋊Dic3).2C6, (C3×C6).9(C2×Dic3), (C2×C32⋊C12).2C2, (C2×C4).3(C32⋊C6), (C2×He3).29(C2×C4), C22.5(C2×C32⋊C6), SmallGroup(432,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.20D6
G = < a,b,c,d | a6=b6=1, c6=b3, d2=a3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b3c5 >
Subgroups: 397 in 107 conjugacy classes, 46 normal (32 characteristic)
C1, C2 [×3], C3, C3 [×3], C4 [×2], C4 [×2], C22, C6 [×3], C6 [×9], C2×C4, C2×C4 [×2], C32 [×2], C32, Dic3 [×4], C12 [×2], C12 [×8], C2×C6, C2×C6 [×3], C4⋊C4, C3×C6 [×6], C3×C6 [×3], C2×Dic3 [×4], C2×C12, C2×C12 [×5], He3, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×4], C3×C12 [×2], C62 [×2], C62, C4⋊Dic3 [×2], C3×C4⋊C4, C2×He3 [×3], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12 [×2], C6×C12, C32⋊C12 [×2], C4×He3 [×2], C22×He3, C3×C4⋊Dic3, C12⋊Dic3, C2×C32⋊C12 [×2], C2×C4×He3, C62.20D6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3 [×2], S3×C6, C4⋊Dic3, C3×C4⋊C4, C32⋊C6, C3×Dic6, C3×D12, C6×Dic3, C32⋊C12 [×2], C2×C32⋊C6, C3×C4⋊Dic3, He3⋊3Q8, He3⋊4D4, C2×C32⋊C12, C62.20D6
►On 144 points
Generators in S144
(1 66 81 15 98 111)(2 63 78 16 107 120)(3 72 75 13 104 117)(4 69 84 14 101 114)(5 106 115 41 62 73)(6 103 112 42 71 82)(7 100 109 43 68 79)(8 97 118 44 65 76)(9 99 116 38 67 74)(10 108 113 39 64 83)(11 105 110 40 61 80)(12 102 119 37 70 77)(17 91 121 27 136 49)(18 88 130 28 133 58)(19 85 127 25 142 55)(20 94 124 26 139 52)(21 92 126 45 137 54)(22 89 123 46 134 51)(23 86 132 47 143 60)(24 95 129 48 140 57)(29 96 122 35 141 50)(30 93 131 36 138 59)(31 90 128 33 135 56)(32 87 125 34 144 53)
(1 39 41 3 37 43)(2 40 42 4 38 44)(5 13 12 7 15 10)(6 14 9 8 16 11)(17 22 32 19 24 30)(18 23 29 20 21 31)(25 48 36 27 46 34)(26 45 33 28 47 35)(49 51 53 55 57 59)(50 52 54 56 58 60)(61 71 69 67 65 63)(62 72 70 68 66 64)(73 75 77 79 81 83)(74 76 78 80 82 84)(85 95 93 91 89 87)(86 96 94 92 90 88)(97 107 105 103 101 99)(98 108 106 104 102 100)(109 111 113 115 117 119)(110 112 114 116 118 120)(121 123 125 127 129 131)(122 124 126 128 130 132)(133 143 141 139 137 135)(134 144 142 140 138 136)
(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)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 17 15 27)(2 20 16 26)(3 19 13 25)(4 18 14 28)(5 48 41 24)(6 47 42 23)(7 46 43 22)(8 45 44 21)(9 33 38 31)(10 36 39 30)(11 35 40 29)(12 34 37 32)(49 77 121 119)(50 76 122 118)(51 75 123 117)(52 74 124 116)(53 73 125 115)(54 84 126 114)(55 83 127 113)(56 82 128 112)(57 81 129 111)(58 80 130 110)(59 79 131 109)(60 78 132 120)(61 92 105 137)(62 91 106 136)(63 90 107 135)(64 89 108 134)(65 88 97 133)(66 87 98 144)(67 86 99 143)(68 85 100 142)(69 96 101 141)(70 95 102 140)(71 94 103 139)(72 93 104 138)
G:=sub<Sym(144)| (1,66,81,15,98,111)(2,63,78,16,107,120)(3,72,75,13,104,117)(4,69,84,14,101,114)(5,106,115,41,62,73)(6,103,112,42,71,82)(7,100,109,43,68,79)(8,97,118,44,65,76)(9,99,116,38,67,74)(10,108,113,39,64,83)(11,105,110,40,61,80)(12,102,119,37,70,77)(17,91,121,27,136,49)(18,88,130,28,133,58)(19,85,127,25,142,55)(20,94,124,26,139,52)(21,92,126,45,137,54)(22,89,123,46,134,51)(23,86,132,47,143,60)(24,95,129,48,140,57)(29,96,122,35,141,50)(30,93,131,36,138,59)(31,90,128,33,135,56)(32,87,125,34,144,53), (1,39,41,3,37,43)(2,40,42,4,38,44)(5,13,12,7,15,10)(6,14,9,8,16,11)(17,22,32,19,24,30)(18,23,29,20,21,31)(25,48,36,27,46,34)(26,45,33,28,47,35)(49,51,53,55,57,59)(50,52,54,56,58,60)(61,71,69,67,65,63)(62,72,70,68,66,64)(73,75,77,79,81,83)(74,76,78,80,82,84)(85,95,93,91,89,87)(86,96,94,92,90,88)(97,107,105,103,101,99)(98,108,106,104,102,100)(109,111,113,115,117,119)(110,112,114,116,118,120)(121,123,125,127,129,131)(122,124,126,128,130,132)(133,143,141,139,137,135)(134,144,142,140,138,136), (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)(133,134,135,136,137,138,139,140,141,142,143,144), (1,17,15,27)(2,20,16,26)(3,19,13,25)(4,18,14,28)(5,48,41,24)(6,47,42,23)(7,46,43,22)(8,45,44,21)(9,33,38,31)(10,36,39,30)(11,35,40,29)(12,34,37,32)(49,77,121,119)(50,76,122,118)(51,75,123,117)(52,74,124,116)(53,73,125,115)(54,84,126,114)(55,83,127,113)(56,82,128,112)(57,81,129,111)(58,80,130,110)(59,79,131,109)(60,78,132,120)(61,92,105,137)(62,91,106,136)(63,90,107,135)(64,89,108,134)(65,88,97,133)(66,87,98,144)(67,86,99,143)(68,85,100,142)(69,96,101,141)(70,95,102,140)(71,94,103,139)(72,93,104,138)>;
G:=Group( (1,66,81,15,98,111)(2,63,78,16,107,120)(3,72,75,13,104,117)(4,69,84,14,101,114)(5,106,115,41,62,73)(6,103,112,42,71,82)(7,100,109,43,68,79)(8,97,118,44,65,76)(9,99,116,38,67,74)(10,108,113,39,64,83)(11,105,110,40,61,80)(12,102,119,37,70,77)(17,91,121,27,136,49)(18,88,130,28,133,58)(19,85,127,25,142,55)(20,94,124,26,139,52)(21,92,126,45,137,54)(22,89,123,46,134,51)(23,86,132,47,143,60)(24,95,129,48,140,57)(29,96,122,35,141,50)(30,93,131,36,138,59)(31,90,128,33,135,56)(32,87,125,34,144,53), (1,39,41,3,37,43)(2,40,42,4,38,44)(5,13,12,7,15,10)(6,14,9,8,16,11)(17,22,32,19,24,30)(18,23,29,20,21,31)(25,48,36,27,46,34)(26,45,33,28,47,35)(49,51,53,55,57,59)(50,52,54,56,58,60)(61,71,69,67,65,63)(62,72,70,68,66,64)(73,75,77,79,81,83)(74,76,78,80,82,84)(85,95,93,91,89,87)(86,96,94,92,90,88)(97,107,105,103,101,99)(98,108,106,104,102,100)(109,111,113,115,117,119)(110,112,114,116,118,120)(121,123,125,127,129,131)(122,124,126,128,130,132)(133,143,141,139,137,135)(134,144,142,140,138,136), (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)(133,134,135,136,137,138,139,140,141,142,143,144), (1,17,15,27)(2,20,16,26)(3,19,13,25)(4,18,14,28)(5,48,41,24)(6,47,42,23)(7,46,43,22)(8,45,44,21)(9,33,38,31)(10,36,39,30)(11,35,40,29)(12,34,37,32)(49,77,121,119)(50,76,122,118)(51,75,123,117)(52,74,124,116)(53,73,125,115)(54,84,126,114)(55,83,127,113)(56,82,128,112)(57,81,129,111)(58,80,130,110)(59,79,131,109)(60,78,132,120)(61,92,105,137)(62,91,106,136)(63,90,107,135)(64,89,108,134)(65,88,97,133)(66,87,98,144)(67,86,99,143)(68,85,100,142)(69,96,101,141)(70,95,102,140)(71,94,103,139)(72,93,104,138) );
G=PermutationGroup([(1,66,81,15,98,111),(2,63,78,16,107,120),(3,72,75,13,104,117),(4,69,84,14,101,114),(5,106,115,41,62,73),(6,103,112,42,71,82),(7,100,109,43,68,79),(8,97,118,44,65,76),(9,99,116,38,67,74),(10,108,113,39,64,83),(11,105,110,40,61,80),(12,102,119,37,70,77),(17,91,121,27,136,49),(18,88,130,28,133,58),(19,85,127,25,142,55),(20,94,124,26,139,52),(21,92,126,45,137,54),(22,89,123,46,134,51),(23,86,132,47,143,60),(24,95,129,48,140,57),(29,96,122,35,141,50),(30,93,131,36,138,59),(31,90,128,33,135,56),(32,87,125,34,144,53)], [(1,39,41,3,37,43),(2,40,42,4,38,44),(5,13,12,7,15,10),(6,14,9,8,16,11),(17,22,32,19,24,30),(18,23,29,20,21,31),(25,48,36,27,46,34),(26,45,33,28,47,35),(49,51,53,55,57,59),(50,52,54,56,58,60),(61,71,69,67,65,63),(62,72,70,68,66,64),(73,75,77,79,81,83),(74,76,78,80,82,84),(85,95,93,91,89,87),(86,96,94,92,90,88),(97,107,105,103,101,99),(98,108,106,104,102,100),(109,111,113,115,117,119),(110,112,114,116,118,120),(121,123,125,127,129,131),(122,124,126,128,130,132),(133,143,141,139,137,135),(134,144,142,140,138,136)], [(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),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,17,15,27),(2,20,16,26),(3,19,13,25),(4,18,14,28),(5,48,41,24),(6,47,42,23),(7,46,43,22),(8,45,44,21),(9,33,38,31),(10,36,39,30),(11,35,40,29),(12,34,37,32),(49,77,121,119),(50,76,122,118),(51,75,123,117),(52,74,124,116),(53,73,125,115),(54,84,126,114),(55,83,127,113),(56,82,128,112),(57,81,129,111),(58,80,130,110),(59,79,131,109),(60,78,132,120),(61,92,105,137),(62,91,106,136),(63,90,107,135),(64,89,108,134),(65,88,97,133),(66,87,98,144),(67,86,99,143),(68,85,100,142),(69,96,101,141),(70,95,102,140),(71,94,103,139),(72,93,104,138)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | ··· | 6I | 6J | ··· | 6R | 12A | 12B | 12C | 12D | 12E | ··· | 12T | 12U | ··· | 12AB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 18 | ··· | 18 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | - | - | + | - | + | + | - | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Q8 | Dic3 | D6 | C3×S3 | Dic6 | D12 | C3×D4 | C3×Q8 | C3×Dic3 | S3×C6 | C3×Dic6 | C3×D12 | C32⋊C6 | C32⋊C12 | C2×C32⋊C6 | He3⋊3Q8 | He3⋊4D4 |
| kernel | C62.20D6 | C2×C32⋊C12 | C2×C4×He3 | C12⋊Dic3 | C4×He3 | C2×C3⋊Dic3 | C6×C12 | C3×C12 | C6×C12 | C2×He3 | C2×He3 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C62.20D6 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 6 |
| 3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1],[6,10,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,7,10,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,10,0,0,0,0,0,0,0,0,3,10,0,0,0,0,0,0,3,6],[3,3,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,8] >;
C62.20D6 in GAP, Magma, Sage, TeX
C_6^2._{20}D_6 % in TeX
G:=Group("C6^2.20D6"); // GroupNames label
G:=SmallGroup(432,140);
// by ID
G=gap.SmallGroup(432,140);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,176,4037,2035,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;
// generators/relations