non-abelian, supersoluble, monomial
Aliases: C62.3D6, He3⋊5(C4⋊C4), He3⋊3C4⋊2C4, (C2×He3).3Q8, (C3×C6).2Dic6, (C2×He3).14D4, C6.4(D6⋊S3), C2.1(He3⋊2D4), C2.2(He3⋊2Q8), C6.9(C32⋊2Q8), C32⋊2(Dic3⋊C4), C6.15(C6.D6), C22.5(C32⋊D6), C3.2(C62.C22), (C22×He3).3C22, (C2×C6).49S32, (C3×C6).7(C4×S3), (C3×C6).9(C3⋊D4), (C2×C3⋊Dic3).3S3, C2.4(He3⋊(C2×C4)), (C2×C32⋊C12).1C2, (C2×He3).14(C2×C4), (C2×He3⋊3C4).4C2, SmallGroup(432,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.3D6
G = < a,b,c,d | a6=b6=1, c6=a3, d2=b3, ab=ba, cac-1=dad-1=a-1b4, cbc-1=b-1, bd=db, dcd-1=b3c5 >
Subgroups: 495 in 107 conjugacy classes, 33 normal (17 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, C62, C62, Dic3⋊C4, C4⋊Dic3, C2×He3, C6×Dic3, C2×C3⋊Dic3, C32⋊C12, He3⋊3C4, C22×He3, Dic3⋊Dic3, C2×C32⋊C12, C2×He3⋊3C4, C62.3D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, S32, Dic3⋊C4, C6.D6, D6⋊S3, C32⋊2Q8, C32⋊D6, C62.C22, He3⋊2Q8, He3⋊2D4, He3⋊(C2×C4), C62.3D6
►On 144 points
Generators in S144
(1 120 80 3 114 74)(2 83 111 4 77 117)(5 104 56 7 98 50)(6 59 107 8 53 101)(9 102 58 11 108 52)(10 49 105 12 55 99)(13 81 113 15 75 119)(14 116 84 16 110 78)(17 91 134 19 85 140)(18 137 94 20 143 88)(21 93 144 23 87 138)(22 135 96 24 141 90)(25 64 129 27 70 123)(26 132 67 28 126 61)(29 124 63 31 130 69)(30 66 127 32 72 121)(33 106 54 35 100 60)(34 57 97 36 51 103)(37 133 86 39 139 92)(38 89 136 40 95 142)(41 73 109 43 79 115)(42 112 76 44 118 82)(45 128 71 47 122 65)(46 62 131 48 68 125)
(1 22 14 20 42 39)(2 40 43 17 15 23)(3 24 16 18 44 37)(4 38 41 19 13 21)(5 46 35 27 11 30)(6 31 12 28 36 47)(7 48 33 25 9 32)(8 29 10 26 34 45)(49 132 57 128 53 124)(50 125 54 129 58 121)(51 122 59 130 55 126)(52 127 56 131 60 123)(61 103 65 107 69 99)(62 100 70 108 66 104)(63 105 67 97 71 101)(64 102 72 98 68 106)(73 85 81 93 77 89)(74 90 78 94 82 86)(75 87 83 95 79 91)(76 92 80 96 84 88)(109 140 113 144 117 136)(110 137 118 133 114 141)(111 142 115 134 119 138)(112 139 120 135 116 143)
(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 36 20 31)(2 32 17 33)(3 34 18 29)(4 30 19 35)(5 13 27 38)(6 39 28 14)(7 15 25 40)(8 37 26 16)(9 43 48 23)(10 24 45 44)(11 41 46 21)(12 22 47 42)(49 92 128 84)(50 77 129 85)(51 90 130 82)(52 75 131 95)(53 88 132 80)(54 73 121 93)(55 86 122 78)(56 83 123 91)(57 96 124 76)(58 81 125 89)(59 94 126 74)(60 79 127 87)(61 118 107 141)(62 134 108 111)(63 116 97 139)(64 144 98 109)(65 114 99 137)(66 142 100 119)(67 112 101 135)(68 140 102 117)(69 110 103 133)(70 138 104 115)(71 120 105 143)(72 136 106 113)
G:=sub<Sym(144)| (1,120,80,3,114,74)(2,83,111,4,77,117)(5,104,56,7,98,50)(6,59,107,8,53,101)(9,102,58,11,108,52)(10,49,105,12,55,99)(13,81,113,15,75,119)(14,116,84,16,110,78)(17,91,134,19,85,140)(18,137,94,20,143,88)(21,93,144,23,87,138)(22,135,96,24,141,90)(25,64,129,27,70,123)(26,132,67,28,126,61)(29,124,63,31,130,69)(30,66,127,32,72,121)(33,106,54,35,100,60)(34,57,97,36,51,103)(37,133,86,39,139,92)(38,89,136,40,95,142)(41,73,109,43,79,115)(42,112,76,44,118,82)(45,128,71,47,122,65)(46,62,131,48,68,125), (1,22,14,20,42,39)(2,40,43,17,15,23)(3,24,16,18,44,37)(4,38,41,19,13,21)(5,46,35,27,11,30)(6,31,12,28,36,47)(7,48,33,25,9,32)(8,29,10,26,34,45)(49,132,57,128,53,124)(50,125,54,129,58,121)(51,122,59,130,55,126)(52,127,56,131,60,123)(61,103,65,107,69,99)(62,100,70,108,66,104)(63,105,67,97,71,101)(64,102,72,98,68,106)(73,85,81,93,77,89)(74,90,78,94,82,86)(75,87,83,95,79,91)(76,92,80,96,84,88)(109,140,113,144,117,136)(110,137,118,133,114,141)(111,142,115,134,119,138)(112,139,120,135,116,143), (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), (1,36,20,31)(2,32,17,33)(3,34,18,29)(4,30,19,35)(5,13,27,38)(6,39,28,14)(7,15,25,40)(8,37,26,16)(9,43,48,23)(10,24,45,44)(11,41,46,21)(12,22,47,42)(49,92,128,84)(50,77,129,85)(51,90,130,82)(52,75,131,95)(53,88,132,80)(54,73,121,93)(55,86,122,78)(56,83,123,91)(57,96,124,76)(58,81,125,89)(59,94,126,74)(60,79,127,87)(61,118,107,141)(62,134,108,111)(63,116,97,139)(64,144,98,109)(65,114,99,137)(66,142,100,119)(67,112,101,135)(68,140,102,117)(69,110,103,133)(70,138,104,115)(71,120,105,143)(72,136,106,113)>;
G:=Group( (1,120,80,3,114,74)(2,83,111,4,77,117)(5,104,56,7,98,50)(6,59,107,8,53,101)(9,102,58,11,108,52)(10,49,105,12,55,99)(13,81,113,15,75,119)(14,116,84,16,110,78)(17,91,134,19,85,140)(18,137,94,20,143,88)(21,93,144,23,87,138)(22,135,96,24,141,90)(25,64,129,27,70,123)(26,132,67,28,126,61)(29,124,63,31,130,69)(30,66,127,32,72,121)(33,106,54,35,100,60)(34,57,97,36,51,103)(37,133,86,39,139,92)(38,89,136,40,95,142)(41,73,109,43,79,115)(42,112,76,44,118,82)(45,128,71,47,122,65)(46,62,131,48,68,125), (1,22,14,20,42,39)(2,40,43,17,15,23)(3,24,16,18,44,37)(4,38,41,19,13,21)(5,46,35,27,11,30)(6,31,12,28,36,47)(7,48,33,25,9,32)(8,29,10,26,34,45)(49,132,57,128,53,124)(50,125,54,129,58,121)(51,122,59,130,55,126)(52,127,56,131,60,123)(61,103,65,107,69,99)(62,100,70,108,66,104)(63,105,67,97,71,101)(64,102,72,98,68,106)(73,85,81,93,77,89)(74,90,78,94,82,86)(75,87,83,95,79,91)(76,92,80,96,84,88)(109,140,113,144,117,136)(110,137,118,133,114,141)(111,142,115,134,119,138)(112,139,120,135,116,143), (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), (1,36,20,31)(2,32,17,33)(3,34,18,29)(4,30,19,35)(5,13,27,38)(6,39,28,14)(7,15,25,40)(8,37,26,16)(9,43,48,23)(10,24,45,44)(11,41,46,21)(12,22,47,42)(49,92,128,84)(50,77,129,85)(51,90,130,82)(52,75,131,95)(53,88,132,80)(54,73,121,93)(55,86,122,78)(56,83,123,91)(57,96,124,76)(58,81,125,89)(59,94,126,74)(60,79,127,87)(61,118,107,141)(62,134,108,111)(63,116,97,139)(64,144,98,109)(65,114,99,137)(66,142,100,119)(67,112,101,135)(68,140,102,117)(69,110,103,133)(70,138,104,115)(71,120,105,143)(72,136,106,113) );
G=PermutationGroup([[(1,120,80,3,114,74),(2,83,111,4,77,117),(5,104,56,7,98,50),(6,59,107,8,53,101),(9,102,58,11,108,52),(10,49,105,12,55,99),(13,81,113,15,75,119),(14,116,84,16,110,78),(17,91,134,19,85,140),(18,137,94,20,143,88),(21,93,144,23,87,138),(22,135,96,24,141,90),(25,64,129,27,70,123),(26,132,67,28,126,61),(29,124,63,31,130,69),(30,66,127,32,72,121),(33,106,54,35,100,60),(34,57,97,36,51,103),(37,133,86,39,139,92),(38,89,136,40,95,142),(41,73,109,43,79,115),(42,112,76,44,118,82),(45,128,71,47,122,65),(46,62,131,48,68,125)], [(1,22,14,20,42,39),(2,40,43,17,15,23),(3,24,16,18,44,37),(4,38,41,19,13,21),(5,46,35,27,11,30),(6,31,12,28,36,47),(7,48,33,25,9,32),(8,29,10,26,34,45),(49,132,57,128,53,124),(50,125,54,129,58,121),(51,122,59,130,55,126),(52,127,56,131,60,123),(61,103,65,107,69,99),(62,100,70,108,66,104),(63,105,67,97,71,101),(64,102,72,98,68,106),(73,85,81,93,77,89),(74,90,78,94,82,86),(75,87,83,95,79,91),(76,92,80,96,84,88),(109,140,113,144,117,136),(110,137,118,133,114,141),(111,142,115,134,119,138),(112,139,120,135,116,143)], [(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)], [(1,36,20,31),(2,32,17,33),(3,34,18,29),(4,30,19,35),(5,13,27,38),(6,39,28,14),(7,15,25,40),(8,37,26,16),(9,43,48,23),(10,24,45,44),(11,41,46,21),(12,22,47,42),(49,92,128,84),(50,77,129,85),(51,90,130,82),(52,75,131,95),(53,88,132,80),(54,73,121,93),(55,86,122,78),(56,83,123,91),(57,96,124,76),(58,81,125,89),(59,94,126,74),(60,79,127,87),(61,118,107,141),(62,134,108,111),(63,116,97,139),(64,144,98,109),(65,114,99,137),(66,142,100,119),(67,112,101,135),(68,140,102,117),(69,110,103,133),(70,138,104,115),(71,120,105,143),(72,136,106,113)]])
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 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | - | + | - | + | + | - | - | + | - | + | - | |||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | Dic6 | C4×S3 | C3⋊D4 | S32 | C6.D6 | D6⋊S3 | C32⋊2Q8 | C32⋊D6 | He3⋊2Q8 | He3⋊2D4 | He3⋊(C2×C4) |
| kernel | C62.3D6 | C2×C32⋊C12 | C2×He3⋊3C4 | He3⋊3C4 | C2×C3⋊Dic3 | C2×He3 | C2×He3 | C62 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
Matrix representation of C62.3D6 ►in GL10(𝔽13)
| 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 4 | 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 |
| 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 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 1 | 0 |
| 7 | 10 | 12 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 10 | 7 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 11 | 2 | 2 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 9 | 11 | 0 | 0 |
| 9 | 11 | 8 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 9 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 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 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
G:=sub<GL(10,GF(13))| [10,0,3,0,0,0,0,0,0,0,0,10,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,1,2,1,1,1,1,0,0,0,0,12,1,0,0,0,0],[12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,12,12,0,12,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[7,3,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,12,7,6,10,0,0,0,0,0,0,6,6,3,3,0,0,0,0,0,0,0,0,0,0,9,11,0,9,11,0,0,0,0,0,2,4,4,0,2,4,0,0,0,0,0,0,0,0,2,9,0,0,0,0,0,0,0,0,11,11,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,4,0,0],[9,2,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,0,0,8,9,4,11,0,0,0,0,0,0,4,5,2,9,0,0,0,0,0,0,0,0,0,0,12,0,0,12,12,0,0,0,0,0,0,12,0,12,12,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0] >;
C62.3D6 in GAP, Magma, Sage, TeX
C_6^2._3D_6
% in TeX
G:=Group("C6^2.3D6"); // GroupNames label
G:=SmallGroup(432,96);
// by ID
G=gap.SmallGroup(432,96);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,36,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,d^2=b^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