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, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C4⋊Dic3, C3×C4⋊C4, C2×He3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C32⋊C12, C4×He3, C22×He3, C3×C4⋊Dic3, C12⋊Dic3, C2×C32⋊C12, C2×C4×He3, C62.20D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3, S3×C6, C4⋊Dic3, C3×C4⋊C4, C32⋊C6, C3×Dic6, C3×D12, C6×Dic3, C32⋊C12, C2×C32⋊C6, C3×C4⋊Dic3, He3⋊3Q8, He3⋊4D4, C2×C32⋊C12, C62.20D6
►On 144 points
Generators in S144
(1 114 70 5 80 127)(2 111 67 6 77 124)(3 120 64 7 74 121)(4 117 61 8 83 130)(9 73 128 37 119 71)(10 82 125 38 116 68)(11 79 122 39 113 65)(12 76 131 40 110 62)(13 144 102 17 92 57)(14 141 99 18 89 54)(15 138 108 19 86 51)(16 135 105 20 95 60)(21 85 58 32 137 103)(22 94 55 29 134 100)(23 91 52 30 143 97)(24 88 49 31 140 106)(25 133 107 35 93 50)(26 142 104 36 90 59)(27 139 101 33 87 56)(28 136 98 34 96 53)(41 115 63 47 81 132)(42 112 72 48 78 129)(43 109 69 45 75 126)(44 118 66 46 84 123)
(1 42 40 3 44 38)(2 43 37 4 41 39)(5 48 12 7 46 10)(6 45 9 8 47 11)(13 26 31 15 28 29)(14 27 32 16 25 30)(17 36 24 19 34 22)(18 33 21 20 35 23)(49 51 53 55 57 59)(50 52 54 56 58 60)(61 63 65 67 69 71)(62 64 66 68 70 72)(73 83 81 79 77 75)(74 84 82 80 78 76)(85 95 93 91 89 87)(86 96 94 92 90 88)(97 99 101 103 105 107)(98 100 102 104 106 108)(109 119 117 115 113 111)(110 120 118 116 114 112)(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 16 5 20)(2 15 6 19)(3 14 7 18)(4 13 8 17)(9 36 37 26)(10 35 38 25)(11 34 39 28)(12 33 40 27)(21 42 32 48)(22 41 29 47)(23 44 30 46)(24 43 31 45)(49 65 106 122)(50 64 107 121)(51 63 108 132)(52 62 97 131)(53 61 98 130)(54 72 99 129)(55 71 100 128)(56 70 101 127)(57 69 102 126)(58 68 103 125)(59 67 104 124)(60 66 105 123)(73 86 119 138)(74 85 120 137)(75 96 109 136)(76 95 110 135)(77 94 111 134)(78 93 112 133)(79 92 113 144)(80 91 114 143)(81 90 115 142)(82 89 116 141)(83 88 117 140)(84 87 118 139)
G:=sub<Sym(144)| (1,114,70,5,80,127)(2,111,67,6,77,124)(3,120,64,7,74,121)(4,117,61,8,83,130)(9,73,128,37,119,71)(10,82,125,38,116,68)(11,79,122,39,113,65)(12,76,131,40,110,62)(13,144,102,17,92,57)(14,141,99,18,89,54)(15,138,108,19,86,51)(16,135,105,20,95,60)(21,85,58,32,137,103)(22,94,55,29,134,100)(23,91,52,30,143,97)(24,88,49,31,140,106)(25,133,107,35,93,50)(26,142,104,36,90,59)(27,139,101,33,87,56)(28,136,98,34,96,53)(41,115,63,47,81,132)(42,112,72,48,78,129)(43,109,69,45,75,126)(44,118,66,46,84,123), (1,42,40,3,44,38)(2,43,37,4,41,39)(5,48,12,7,46,10)(6,45,9,8,47,11)(13,26,31,15,28,29)(14,27,32,16,25,30)(17,36,24,19,34,22)(18,33,21,20,35,23)(49,51,53,55,57,59)(50,52,54,56,58,60)(61,63,65,67,69,71)(62,64,66,68,70,72)(73,83,81,79,77,75)(74,84,82,80,78,76)(85,95,93,91,89,87)(86,96,94,92,90,88)(97,99,101,103,105,107)(98,100,102,104,106,108)(109,119,117,115,113,111)(110,120,118,116,114,112)(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), (1,16,5,20)(2,15,6,19)(3,14,7,18)(4,13,8,17)(9,36,37,26)(10,35,38,25)(11,34,39,28)(12,33,40,27)(21,42,32,48)(22,41,29,47)(23,44,30,46)(24,43,31,45)(49,65,106,122)(50,64,107,121)(51,63,108,132)(52,62,97,131)(53,61,98,130)(54,72,99,129)(55,71,100,128)(56,70,101,127)(57,69,102,126)(58,68,103,125)(59,67,104,124)(60,66,105,123)(73,86,119,138)(74,85,120,137)(75,96,109,136)(76,95,110,135)(77,94,111,134)(78,93,112,133)(79,92,113,144)(80,91,114,143)(81,90,115,142)(82,89,116,141)(83,88,117,140)(84,87,118,139)>;
G:=Group( (1,114,70,5,80,127)(2,111,67,6,77,124)(3,120,64,7,74,121)(4,117,61,8,83,130)(9,73,128,37,119,71)(10,82,125,38,116,68)(11,79,122,39,113,65)(12,76,131,40,110,62)(13,144,102,17,92,57)(14,141,99,18,89,54)(15,138,108,19,86,51)(16,135,105,20,95,60)(21,85,58,32,137,103)(22,94,55,29,134,100)(23,91,52,30,143,97)(24,88,49,31,140,106)(25,133,107,35,93,50)(26,142,104,36,90,59)(27,139,101,33,87,56)(28,136,98,34,96,53)(41,115,63,47,81,132)(42,112,72,48,78,129)(43,109,69,45,75,126)(44,118,66,46,84,123), (1,42,40,3,44,38)(2,43,37,4,41,39)(5,48,12,7,46,10)(6,45,9,8,47,11)(13,26,31,15,28,29)(14,27,32,16,25,30)(17,36,24,19,34,22)(18,33,21,20,35,23)(49,51,53,55,57,59)(50,52,54,56,58,60)(61,63,65,67,69,71)(62,64,66,68,70,72)(73,83,81,79,77,75)(74,84,82,80,78,76)(85,95,93,91,89,87)(86,96,94,92,90,88)(97,99,101,103,105,107)(98,100,102,104,106,108)(109,119,117,115,113,111)(110,120,118,116,114,112)(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), (1,16,5,20)(2,15,6,19)(3,14,7,18)(4,13,8,17)(9,36,37,26)(10,35,38,25)(11,34,39,28)(12,33,40,27)(21,42,32,48)(22,41,29,47)(23,44,30,46)(24,43,31,45)(49,65,106,122)(50,64,107,121)(51,63,108,132)(52,62,97,131)(53,61,98,130)(54,72,99,129)(55,71,100,128)(56,70,101,127)(57,69,102,126)(58,68,103,125)(59,67,104,124)(60,66,105,123)(73,86,119,138)(74,85,120,137)(75,96,109,136)(76,95,110,135)(77,94,111,134)(78,93,112,133)(79,92,113,144)(80,91,114,143)(81,90,115,142)(82,89,116,141)(83,88,117,140)(84,87,118,139) );
G=PermutationGroup([[(1,114,70,5,80,127),(2,111,67,6,77,124),(3,120,64,7,74,121),(4,117,61,8,83,130),(9,73,128,37,119,71),(10,82,125,38,116,68),(11,79,122,39,113,65),(12,76,131,40,110,62),(13,144,102,17,92,57),(14,141,99,18,89,54),(15,138,108,19,86,51),(16,135,105,20,95,60),(21,85,58,32,137,103),(22,94,55,29,134,100),(23,91,52,30,143,97),(24,88,49,31,140,106),(25,133,107,35,93,50),(26,142,104,36,90,59),(27,139,101,33,87,56),(28,136,98,34,96,53),(41,115,63,47,81,132),(42,112,72,48,78,129),(43,109,69,45,75,126),(44,118,66,46,84,123)], [(1,42,40,3,44,38),(2,43,37,4,41,39),(5,48,12,7,46,10),(6,45,9,8,47,11),(13,26,31,15,28,29),(14,27,32,16,25,30),(17,36,24,19,34,22),(18,33,21,20,35,23),(49,51,53,55,57,59),(50,52,54,56,58,60),(61,63,65,67,69,71),(62,64,66,68,70,72),(73,83,81,79,77,75),(74,84,82,80,78,76),(85,95,93,91,89,87),(86,96,94,92,90,88),(97,99,101,103,105,107),(98,100,102,104,106,108),(109,119,117,115,113,111),(110,120,118,116,114,112),(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)], [(1,16,5,20),(2,15,6,19),(3,14,7,18),(4,13,8,17),(9,36,37,26),(10,35,38,25),(11,34,39,28),(12,33,40,27),(21,42,32,48),(22,41,29,47),(23,44,30,46),(24,43,31,45),(49,65,106,122),(50,64,107,121),(51,63,108,132),(52,62,97,131),(53,61,98,130),(54,72,99,129),(55,71,100,128),(56,70,101,127),(57,69,102,126),(58,68,103,125),(59,67,104,124),(60,66,105,123),(73,86,119,138),(74,85,120,137),(75,96,109,136),(76,95,110,135),(77,94,111,134),(78,93,112,133),(79,92,113,144),(80,91,114,143),(81,90,115,142),(82,89,116,141),(83,88,117,140),(84,87,118,139)]])
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