non-abelian, supersoluble, monomial
Aliases: C62.2D6, He3⋊4(C4⋊C4), (C3×C6).5D12, C32⋊C12⋊1C4, C32⋊(C4⋊Dic3), (C2×He3).2Q8, (C3×C6).1Dic6, C3⋊Dic3⋊2Dic3, (C2×He3).13D4, C6.14(S3×Dic3), C2.1(He3⋊3D4), C2.1(He3⋊2Q8), C6.8(C32⋊2Q8), C6.33(C3⋊D12), C32⋊1(Dic3⋊C4), C22.4(C32⋊D6), C3.3(Dic3⋊Dic3), (C22×He3).2C22, (C2×C6).48S32, (C3×C6).6(C4×S3), (C3×C6).8(C3⋊D4), (C2×C3⋊Dic3).2S3, (C3×C6).4(C2×Dic3), C2.4(C6.S32), (C2×C32⋊C12).3C2, (C2×He3).13(C2×C4), (C2×He3⋊3C4).1C2, SmallGroup(432,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.D6
G = < a,b,c,d | a6=b6=1, c6=d2=a3, ab=ba, cac-1=dad-1=a-1b4, cbc-1=b-1, bd=db, dcd-1=b3c5 >
Subgroups: 495 in 107 conjugacy classes, 35 normal (31 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C62, C62, Dic3⋊C4, C4⋊Dic3, C2×He3, C6×Dic3, C2×C3⋊Dic3, C32⋊C12, C32⋊C12, He3⋊3C4, C22×He3, Dic3⋊Dic3, C62.C22, C2×C32⋊C12, C2×He3⋊3C4, C62.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, S32, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C32⋊2Q8, C32⋊D6, Dic3⋊Dic3, He3⋊2Q8, C6.S32, He3⋊3D4, C62.D6
►On 144 points
Generators in S144
(1 75 111 3 81 117)(2 114 78 4 120 84)(5 132 102 7 126 108)(6 105 123 8 99 129)(9 95 50 11 89 56)(10 53 86 12 59 92)(13 128 106 15 122 100)(14 97 131 16 103 125)(17 136 69 19 142 63)(18 72 139 20 66 133)(21 79 119 23 73 113)(22 110 82 24 116 76)(25 70 141 27 64 135)(26 144 61 28 138 67)(29 68 143 31 62 137)(30 134 71 32 140 65)(33 130 104 35 124 98)(34 107 121 36 101 127)(37 55 96 39 49 90)(38 87 58 40 93 52)(41 112 80 43 118 74)(42 83 115 44 77 109)(45 51 88 47 57 94)(46 91 54 48 85 60)
(1 6 42 36 21 14)(2 15 22 33 43 7)(3 8 44 34 23 16)(4 13 24 35 41 5)(9 26 46 32 38 17)(10 18 39 29 47 27)(11 28 48 30 40 19)(12 20 37 31 45 25)(49 68 57 64 53 72)(50 61 54 65 58 69)(51 70 59 66 55 62)(52 63 56 67 60 71)(73 103 81 99 77 107)(74 108 78 100 82 104)(75 105 83 101 79 97)(76 98 80 102 84 106)(85 134 93 142 89 138)(86 139 90 143 94 135)(87 136 95 144 91 140)(88 141 92 133 96 137)(109 121 113 125 117 129)(110 130 118 126 114 122)(111 123 115 127 119 131)(112 132 120 128 116 124)
(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 10 3 12)(2 30 4 32)(5 46 7 48)(6 18 8 20)(9 33 11 35)(13 38 15 40)(14 27 16 25)(17 22 19 24)(21 47 23 45)(26 43 28 41)(29 34 31 36)(37 42 39 44)(49 111 55 117)(50 132 56 126)(51 109 57 115)(52 130 58 124)(53 119 59 113)(54 128 60 122)(61 120 67 114)(62 129 68 123)(63 118 69 112)(64 127 70 121)(65 116 71 110)(66 125 72 131)(73 90 79 96)(74 140 80 134)(75 88 81 94)(76 138 82 144)(77 86 83 92)(78 136 84 142)(85 104 91 98)(87 102 93 108)(89 100 95 106)(97 137 103 143)(99 135 105 141)(101 133 107 139)
G:=sub<Sym(144)| (1,75,111,3,81,117)(2,114,78,4,120,84)(5,132,102,7,126,108)(6,105,123,8,99,129)(9,95,50,11,89,56)(10,53,86,12,59,92)(13,128,106,15,122,100)(14,97,131,16,103,125)(17,136,69,19,142,63)(18,72,139,20,66,133)(21,79,119,23,73,113)(22,110,82,24,116,76)(25,70,141,27,64,135)(26,144,61,28,138,67)(29,68,143,31,62,137)(30,134,71,32,140,65)(33,130,104,35,124,98)(34,107,121,36,101,127)(37,55,96,39,49,90)(38,87,58,40,93,52)(41,112,80,43,118,74)(42,83,115,44,77,109)(45,51,88,47,57,94)(46,91,54,48,85,60), (1,6,42,36,21,14)(2,15,22,33,43,7)(3,8,44,34,23,16)(4,13,24,35,41,5)(9,26,46,32,38,17)(10,18,39,29,47,27)(11,28,48,30,40,19)(12,20,37,31,45,25)(49,68,57,64,53,72)(50,61,54,65,58,69)(51,70,59,66,55,62)(52,63,56,67,60,71)(73,103,81,99,77,107)(74,108,78,100,82,104)(75,105,83,101,79,97)(76,98,80,102,84,106)(85,134,93,142,89,138)(86,139,90,143,94,135)(87,136,95,144,91,140)(88,141,92,133,96,137)(109,121,113,125,117,129)(110,130,118,126,114,122)(111,123,115,127,119,131)(112,132,120,128,116,124), (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), (1,10,3,12)(2,30,4,32)(5,46,7,48)(6,18,8,20)(9,33,11,35)(13,38,15,40)(14,27,16,25)(17,22,19,24)(21,47,23,45)(26,43,28,41)(29,34,31,36)(37,42,39,44)(49,111,55,117)(50,132,56,126)(51,109,57,115)(52,130,58,124)(53,119,59,113)(54,128,60,122)(61,120,67,114)(62,129,68,123)(63,118,69,112)(64,127,70,121)(65,116,71,110)(66,125,72,131)(73,90,79,96)(74,140,80,134)(75,88,81,94)(76,138,82,144)(77,86,83,92)(78,136,84,142)(85,104,91,98)(87,102,93,108)(89,100,95,106)(97,137,103,143)(99,135,105,141)(101,133,107,139)>;
G:=Group( (1,75,111,3,81,117)(2,114,78,4,120,84)(5,132,102,7,126,108)(6,105,123,8,99,129)(9,95,50,11,89,56)(10,53,86,12,59,92)(13,128,106,15,122,100)(14,97,131,16,103,125)(17,136,69,19,142,63)(18,72,139,20,66,133)(21,79,119,23,73,113)(22,110,82,24,116,76)(25,70,141,27,64,135)(26,144,61,28,138,67)(29,68,143,31,62,137)(30,134,71,32,140,65)(33,130,104,35,124,98)(34,107,121,36,101,127)(37,55,96,39,49,90)(38,87,58,40,93,52)(41,112,80,43,118,74)(42,83,115,44,77,109)(45,51,88,47,57,94)(46,91,54,48,85,60), (1,6,42,36,21,14)(2,15,22,33,43,7)(3,8,44,34,23,16)(4,13,24,35,41,5)(9,26,46,32,38,17)(10,18,39,29,47,27)(11,28,48,30,40,19)(12,20,37,31,45,25)(49,68,57,64,53,72)(50,61,54,65,58,69)(51,70,59,66,55,62)(52,63,56,67,60,71)(73,103,81,99,77,107)(74,108,78,100,82,104)(75,105,83,101,79,97)(76,98,80,102,84,106)(85,134,93,142,89,138)(86,139,90,143,94,135)(87,136,95,144,91,140)(88,141,92,133,96,137)(109,121,113,125,117,129)(110,130,118,126,114,122)(111,123,115,127,119,131)(112,132,120,128,116,124), (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), (1,10,3,12)(2,30,4,32)(5,46,7,48)(6,18,8,20)(9,33,11,35)(13,38,15,40)(14,27,16,25)(17,22,19,24)(21,47,23,45)(26,43,28,41)(29,34,31,36)(37,42,39,44)(49,111,55,117)(50,132,56,126)(51,109,57,115)(52,130,58,124)(53,119,59,113)(54,128,60,122)(61,120,67,114)(62,129,68,123)(63,118,69,112)(64,127,70,121)(65,116,71,110)(66,125,72,131)(73,90,79,96)(74,140,80,134)(75,88,81,94)(76,138,82,144)(77,86,83,92)(78,136,84,142)(85,104,91,98)(87,102,93,108)(89,100,95,106)(97,137,103,143)(99,135,105,141)(101,133,107,139) );
G=PermutationGroup([[(1,75,111,3,81,117),(2,114,78,4,120,84),(5,132,102,7,126,108),(6,105,123,8,99,129),(9,95,50,11,89,56),(10,53,86,12,59,92),(13,128,106,15,122,100),(14,97,131,16,103,125),(17,136,69,19,142,63),(18,72,139,20,66,133),(21,79,119,23,73,113),(22,110,82,24,116,76),(25,70,141,27,64,135),(26,144,61,28,138,67),(29,68,143,31,62,137),(30,134,71,32,140,65),(33,130,104,35,124,98),(34,107,121,36,101,127),(37,55,96,39,49,90),(38,87,58,40,93,52),(41,112,80,43,118,74),(42,83,115,44,77,109),(45,51,88,47,57,94),(46,91,54,48,85,60)], [(1,6,42,36,21,14),(2,15,22,33,43,7),(3,8,44,34,23,16),(4,13,24,35,41,5),(9,26,46,32,38,17),(10,18,39,29,47,27),(11,28,48,30,40,19),(12,20,37,31,45,25),(49,68,57,64,53,72),(50,61,54,65,58,69),(51,70,59,66,55,62),(52,63,56,67,60,71),(73,103,81,99,77,107),(74,108,78,100,82,104),(75,105,83,101,79,97),(76,98,80,102,84,106),(85,134,93,142,89,138),(86,139,90,143,94,135),(87,136,95,144,91,140),(88,141,92,133,96,137),(109,121,113,125,117,129),(110,130,118,126,114,122),(111,123,115,127,119,131),(112,132,120,128,116,124)], [(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)], [(1,10,3,12),(2,30,4,32),(5,46,7,48),(6,18,8,20),(9,33,11,35),(13,38,15,40),(14,27,16,25),(17,22,19,24),(21,47,23,45),(26,43,28,41),(29,34,31,36),(37,42,39,44),(49,111,55,117),(50,132,56,126),(51,109,57,115),(52,130,58,124),(53,119,59,113),(54,128,60,122),(61,120,67,114),(62,129,68,123),(63,118,69,112),(64,127,70,121),(65,116,71,110),(66,125,72,131),(73,90,79,96),(74,140,80,134),(75,88,81,94),(76,138,82,144),(77,86,83,92),(78,136,84,142),(85,104,91,98),(87,102,93,108),(89,100,95,106),(97,137,103,143),(99,135,105,141),(101,133,107,139)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 6A | 6B | 6C | 6D | ··· | 6I | 6J | 6K | 6L | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 18 | ··· | 18 | 2 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 18 | ··· | 18 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | - | - | + | - | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 | S32 | S3×Dic3 | C3⋊D12 | C32⋊2Q8 | C32⋊D6 | He3⋊2Q8 | C6.S32 | He3⋊3D4 |
| kernel | C62.D6 | C2×C32⋊C12 | C2×He3⋊3C4 | C32⋊C12 | C2×C3⋊Dic3 | C2×He3 | C2×He3 | C3⋊Dic3 | C62 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
Matrix representation of C62.D6 ►in GL10(𝔽13)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 12 | 0 |
| 6 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 8 | 12 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 3 | 1 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 5 | 0 | 0 | 0 |
| 8 | 10 | 5 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 0 | 3 | 10 | 0 | 0 | 10 | 3 |
| 0 | 0 | 0 | 0 | 3 | 10 | 10 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 10 | 7 | 0 | 0 |
G:=sub<GL(10,GF(13))| [0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,1,2,1,1,1,1,0,0,0,0,12,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,12,0,0,0,0,0,1,1,0,1,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0],[6,12,12,11,0,0,0,0,0,0,0,8,0,3,0,0,0,0,0,0,6,12,7,1,0,0,0,0,0,0,0,8,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,8,8,8,0,8,8,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,0,8,0,0],[8,0,0,0,0,0,0,0,0,0,10,5,0,0,0,0,0,0,0,0,5,0,5,0,0,0,0,0,0,0,3,8,3,8,0,0,0,0,0,0,0,0,0,0,3,6,0,3,3,6,0,0,0,0,7,10,7,10,10,0,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,3,3,0,0] >;
C62.D6 in GAP, Magma, Sage, TeX
C_6^2.D_6
% in TeX
G:=Group("C6^2.D6"); // GroupNames label
G:=SmallGroup(432,95);
// by ID
G=gap.SmallGroup(432,95);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=a^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c^5>;
// generators/relations