metabelian, supersoluble, monomial
Aliases: C62.19D6, C3⋊Dic3⋊C12, He3⋊6(C4⋊C4), (C6×C12).1S3, (C6×C12).1C6, C6.15(S3×C12), C32⋊C12⋊3C4, C6.Dic6⋊C3, C62.5(C2×C6), (C2×He3).4Q8, C6.5(C3×Dic6), (C3×C6).5Dic6, (C2×He3).26D4, C2.1(He3⋊6D4), C2.1(He3⋊3Q8), C32⋊5(Dic3⋊C4), (C22×He3).17C22, C32⋊3(C3×C4⋊C4), (C2×C4×He3).1C2, (C3×C6).2(C3×Q8), (C2×C12).5(C3×S3), (C2×C6).39(S3×C6), (C3×C6).3(C2×C12), (C3×C6).15(C4×S3), (C3×C6).11(C3×D4), C6.14(C3×C3⋊D4), C2.4(C4×C32⋊C6), (C2×C3⋊Dic3).1C6, C3.2(C3×Dic3⋊C4), (C3×C6).19(C3⋊D4), (C2×C32⋊C12).6C2, (C2×C4).1(C32⋊C6), (C2×He3).19(C2×C4), C22.4(C2×C32⋊C6), SmallGroup(432,139)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.19D6
G = < a,b,c,d | a6=b6=1, c6=d2=a3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b3c5 >
Subgroups: 397 in 107 conjugacy classes, 40 normal (36 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, Dic3⋊C4, C3×C4⋊C4, C2×He3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C32⋊C12, C32⋊C12, C4×He3, C22×He3, C3×Dic3⋊C4, C6.Dic6, C2×C32⋊C12, C2×C4×He3, C62.19D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, S3×C6, Dic3⋊C4, C3×C4⋊C4, C32⋊C6, C3×Dic6, S3×C12, C3×C3⋊D4, C2×C32⋊C6, C3×Dic3⋊C4, He3⋊3Q8, C4×C32⋊C6, He3⋊6D4, C62.19D6
►On 144 points
Generators in S144
(1 61 84 3 67 78)(2 70 81 4 64 75)(5 69 76 7 63 82)(6 66 73 8 72 79)(9 71 74 11 65 80)(10 68 83 12 62 77)(13 89 49 15 95 55)(14 86 58 16 92 52)(17 138 121 19 144 127)(18 135 130 20 141 124)(21 105 120 23 99 114)(22 102 117 24 108 111)(25 136 123 27 142 129)(26 133 132 28 139 126)(29 85 53 31 91 59)(30 94 50 32 88 56)(33 137 128 35 143 122)(34 134 125 36 140 131)(37 107 118 39 101 112)(38 104 115 40 98 109)(41 90 54 43 96 60)(42 87 51 44 93 57)(45 97 116 47 103 110)(46 106 113 48 100 119)
(1 47 5 23 11 37)(2 48 6 24 12 38)(3 45 7 21 9 39)(4 46 8 22 10 40)(13 35 29 28 44 18)(14 36 30 25 41 19)(15 33 31 26 42 20)(16 34 32 27 43 17)(49 122 53 126 57 130)(50 123 54 127 58 131)(51 124 55 128 59 132)(52 125 56 129 60 121)(61 103 69 99 65 107)(62 104 70 100 66 108)(63 105 71 101 67 97)(64 106 72 102 68 98)(73 111 77 115 81 119)(74 112 78 116 82 120)(75 113 79 117 83 109)(76 114 80 118 84 110)(85 139 93 135 89 143)(86 140 94 136 90 144)(87 141 95 137 91 133)(88 142 96 138 92 134)
(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 32 3 30)(2 18 4 20)(5 16 7 14)(6 28 8 26)(9 41 11 43)(10 33 12 35)(13 40 15 38)(17 21 19 23)(22 31 24 29)(25 37 27 39)(34 45 36 47)(42 48 44 46)(49 117 55 111)(50 80 56 74)(51 115 57 109)(52 78 58 84)(53 113 59 119)(54 76 60 82)(61 96 67 90)(62 139 68 133)(63 94 69 88)(64 137 70 143)(65 92 71 86)(66 135 72 141)(73 122 79 128)(75 132 81 126)(77 130 83 124)(85 98 91 104)(87 108 93 102)(89 106 95 100)(97 136 103 142)(99 134 105 140)(101 144 107 138)(110 121 116 127)(112 131 118 125)(114 129 120 123)
G:=sub<Sym(144)| (1,61,84,3,67,78)(2,70,81,4,64,75)(5,69,76,7,63,82)(6,66,73,8,72,79)(9,71,74,11,65,80)(10,68,83,12,62,77)(13,89,49,15,95,55)(14,86,58,16,92,52)(17,138,121,19,144,127)(18,135,130,20,141,124)(21,105,120,23,99,114)(22,102,117,24,108,111)(25,136,123,27,142,129)(26,133,132,28,139,126)(29,85,53,31,91,59)(30,94,50,32,88,56)(33,137,128,35,143,122)(34,134,125,36,140,131)(37,107,118,39,101,112)(38,104,115,40,98,109)(41,90,54,43,96,60)(42,87,51,44,93,57)(45,97,116,47,103,110)(46,106,113,48,100,119), (1,47,5,23,11,37)(2,48,6,24,12,38)(3,45,7,21,9,39)(4,46,8,22,10,40)(13,35,29,28,44,18)(14,36,30,25,41,19)(15,33,31,26,42,20)(16,34,32,27,43,17)(49,122,53,126,57,130)(50,123,54,127,58,131)(51,124,55,128,59,132)(52,125,56,129,60,121)(61,103,69,99,65,107)(62,104,70,100,66,108)(63,105,71,101,67,97)(64,106,72,102,68,98)(73,111,77,115,81,119)(74,112,78,116,82,120)(75,113,79,117,83,109)(76,114,80,118,84,110)(85,139,93,135,89,143)(86,140,94,136,90,144)(87,141,95,137,91,133)(88,142,96,138,92,134), (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), (1,32,3,30)(2,18,4,20)(5,16,7,14)(6,28,8,26)(9,41,11,43)(10,33,12,35)(13,40,15,38)(17,21,19,23)(22,31,24,29)(25,37,27,39)(34,45,36,47)(42,48,44,46)(49,117,55,111)(50,80,56,74)(51,115,57,109)(52,78,58,84)(53,113,59,119)(54,76,60,82)(61,96,67,90)(62,139,68,133)(63,94,69,88)(64,137,70,143)(65,92,71,86)(66,135,72,141)(73,122,79,128)(75,132,81,126)(77,130,83,124)(85,98,91,104)(87,108,93,102)(89,106,95,100)(97,136,103,142)(99,134,105,140)(101,144,107,138)(110,121,116,127)(112,131,118,125)(114,129,120,123)>;
G:=Group( (1,61,84,3,67,78)(2,70,81,4,64,75)(5,69,76,7,63,82)(6,66,73,8,72,79)(9,71,74,11,65,80)(10,68,83,12,62,77)(13,89,49,15,95,55)(14,86,58,16,92,52)(17,138,121,19,144,127)(18,135,130,20,141,124)(21,105,120,23,99,114)(22,102,117,24,108,111)(25,136,123,27,142,129)(26,133,132,28,139,126)(29,85,53,31,91,59)(30,94,50,32,88,56)(33,137,128,35,143,122)(34,134,125,36,140,131)(37,107,118,39,101,112)(38,104,115,40,98,109)(41,90,54,43,96,60)(42,87,51,44,93,57)(45,97,116,47,103,110)(46,106,113,48,100,119), (1,47,5,23,11,37)(2,48,6,24,12,38)(3,45,7,21,9,39)(4,46,8,22,10,40)(13,35,29,28,44,18)(14,36,30,25,41,19)(15,33,31,26,42,20)(16,34,32,27,43,17)(49,122,53,126,57,130)(50,123,54,127,58,131)(51,124,55,128,59,132)(52,125,56,129,60,121)(61,103,69,99,65,107)(62,104,70,100,66,108)(63,105,71,101,67,97)(64,106,72,102,68,98)(73,111,77,115,81,119)(74,112,78,116,82,120)(75,113,79,117,83,109)(76,114,80,118,84,110)(85,139,93,135,89,143)(86,140,94,136,90,144)(87,141,95,137,91,133)(88,142,96,138,92,134), (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), (1,32,3,30)(2,18,4,20)(5,16,7,14)(6,28,8,26)(9,41,11,43)(10,33,12,35)(13,40,15,38)(17,21,19,23)(22,31,24,29)(25,37,27,39)(34,45,36,47)(42,48,44,46)(49,117,55,111)(50,80,56,74)(51,115,57,109)(52,78,58,84)(53,113,59,119)(54,76,60,82)(61,96,67,90)(62,139,68,133)(63,94,69,88)(64,137,70,143)(65,92,71,86)(66,135,72,141)(73,122,79,128)(75,132,81,126)(77,130,83,124)(85,98,91,104)(87,108,93,102)(89,106,95,100)(97,136,103,142)(99,134,105,140)(101,144,107,138)(110,121,116,127)(112,131,118,125)(114,129,120,123) );
G=PermutationGroup([[(1,61,84,3,67,78),(2,70,81,4,64,75),(5,69,76,7,63,82),(6,66,73,8,72,79),(9,71,74,11,65,80),(10,68,83,12,62,77),(13,89,49,15,95,55),(14,86,58,16,92,52),(17,138,121,19,144,127),(18,135,130,20,141,124),(21,105,120,23,99,114),(22,102,117,24,108,111),(25,136,123,27,142,129),(26,133,132,28,139,126),(29,85,53,31,91,59),(30,94,50,32,88,56),(33,137,128,35,143,122),(34,134,125,36,140,131),(37,107,118,39,101,112),(38,104,115,40,98,109),(41,90,54,43,96,60),(42,87,51,44,93,57),(45,97,116,47,103,110),(46,106,113,48,100,119)], [(1,47,5,23,11,37),(2,48,6,24,12,38),(3,45,7,21,9,39),(4,46,8,22,10,40),(13,35,29,28,44,18),(14,36,30,25,41,19),(15,33,31,26,42,20),(16,34,32,27,43,17),(49,122,53,126,57,130),(50,123,54,127,58,131),(51,124,55,128,59,132),(52,125,56,129,60,121),(61,103,69,99,65,107),(62,104,70,100,66,108),(63,105,71,101,67,97),(64,106,72,102,68,98),(73,111,77,115,81,119),(74,112,78,116,82,120),(75,113,79,117,83,109),(76,114,80,118,84,110),(85,139,93,135,89,143),(86,140,94,136,90,144),(87,141,95,137,91,133),(88,142,96,138,92,134)], [(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)], [(1,32,3,30),(2,18,4,20),(5,16,7,14),(6,28,8,26),(9,41,11,43),(10,33,12,35),(13,40,15,38),(17,21,19,23),(22,31,24,29),(25,37,27,39),(34,45,36,47),(42,48,44,46),(49,117,55,111),(50,80,56,74),(51,115,57,109),(52,78,58,84),(53,113,59,119),(54,76,60,82),(61,96,67,90),(62,139,68,133),(63,94,69,88),(64,137,70,143),(65,92,71,86),(66,135,72,141),(73,122,79,128),(75,132,81,126),(77,130,83,124),(85,98,91,104),(87,108,93,102),(89,106,95,100),(97,136,103,142),(99,134,105,140),(101,144,107,138),(110,121,116,127),(112,131,118,125),(114,129,120,123)]])
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 | D6 | C3×S3 | Dic6 | C4×S3 | C3⋊D4 | C3×D4 | C3×Q8 | S3×C6 | C3×Dic6 | S3×C12 | C3×C3⋊D4 | C32⋊C6 | C2×C32⋊C6 | He3⋊3Q8 | C4×C32⋊C6 | He3⋊6D4 |
| kernel | C62.19D6 | C2×C32⋊C12 | C2×C4×He3 | C6.Dic6 | C32⋊C12 | C2×C3⋊Dic3 | C6×C12 | C3⋊Dic3 | C6×C12 | C2×He3 | C2×He3 | C62 | C2×C12 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C62.19D6 ►in GL10(𝔽13)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 | 0 | 1 | 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 | 2 | 2 | 2 | 12 | 12 | 2 |
| 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 | 0 |
| 0 | 12 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 11 | 11 | 11 | 0 | 0 | 3 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 8 | 4 | 2 | 9 | 10 | 9 |
| 2 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 | 12 | 8 | 8 | 10 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 9 | 7 | 0 | 1 | 4 | 1 |
G:=sub<GL(10,GF(13))| [4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,2,0,1],[12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,9,0,0,8,0,11,0,0,0,0,0,9,0,0,8,11,0,0,0,0,0,0,9,0,0,11,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[7,3,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8,0,0,0,0,0,9,0,0,8,4,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,1,0,9,0,0,0,0,0,0,0,0,3,10,0,0,0,0,0,0,0,0,0,9],[2,9,0,0,0,0,0,0,0,0,11,11,0,0,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,0,0,6,0,10,2,0,9,0,0,0,0,0,5,10,0,9,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,2,0,8,7,0,1,0,0,0,0,0,6,8,0,8,4,0,0,0,0,0,0,10,0,0,1] >;
C62.19D6 in GAP, Magma, Sage, TeX
C_6^2._{19}D_6 % in TeX
G:=Group("C6^2.19D6"); // GroupNames label
G:=SmallGroup(432,139);
// by ID
G=gap.SmallGroup(432,139);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,365,92,4037,2035,14118]);
// 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*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;
// generators/relations