direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×Dic9, C62.53D6, (C3×C9)⋊C42, C3.1Dic32, C18.3(C4×S3), C9⋊Dic3⋊1C4, C9⋊1(C4×Dic3), C3⋊1(C4×Dic9), (C2×C6).8D18, C6.18(C4×D9), (C2×C18).8D6, (C3×Dic9)⋊1C4, (C9×Dic3)⋊2C4, C2.2(S3×Dic9), C6.3(C2×Dic9), C2.2(Dic3×D9), C22.4(S3×D9), (C6×C18).2C22, (C2×Dic3).5D9, (C6×Dic9).1C2, (C2×Dic9).5S3, C6.24(S3×Dic3), C18.3(C2×Dic3), (C6×Dic3).14S3, (Dic3×C18).7C2, C6.3(C6.D6), C32.2(C4×Dic3), (C3×Dic3).5Dic3, C2.2(C18.D6), (C2×C6).14S32, (C3×C6).37(C4×S3), (C3×C18).7(C2×C4), (C2×C9⋊Dic3).1C2, (C3×C6).31(C2×Dic3), SmallGroup(432,87)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — Dic3×Dic9 |
Generators and relations for Dic3×Dic9
G = < a,b,c,d | a6=c18=1, b2=a3, d2=c9, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 444 in 106 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C9, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C18, C18, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, C4×Dic3, C3×C18, C2×Dic9, C2×Dic9, C2×C36, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×C18, C4×Dic9, Dic32, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, Dic3×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, D9, C4×S3, C2×Dic3, Dic9, D18, S32, C4×Dic3, C4×D9, C2×Dic9, S3×Dic3, C6.D6, S3×D9, C4×Dic9, Dic32, Dic3×D9, C18.D6, S3×Dic9, Dic3×Dic9
►On 144 points
Generators in S144
(1 70 7 58 13 64)(2 71 8 59 14 65)(3 72 9 60 15 66)(4 55 10 61 16 67)(5 56 11 62 17 68)(6 57 12 63 18 69)(19 100 31 94 25 106)(20 101 32 95 26 107)(21 102 33 96 27 108)(22 103 34 97 28 91)(23 104 35 98 29 92)(24 105 36 99 30 93)(37 89 43 77 49 83)(38 90 44 78 50 84)(39 73 45 79 51 85)(40 74 46 80 52 86)(41 75 47 81 53 87)(42 76 48 82 54 88)(109 132 121 144 115 138)(110 133 122 127 116 139)(111 134 123 128 117 140)(112 135 124 129 118 141)(113 136 125 130 119 142)(114 137 126 131 120 143)
(1 97 58 22)(2 98 59 23)(3 99 60 24)(4 100 61 25)(5 101 62 26)(6 102 63 27)(7 103 64 28)(8 104 65 29)(9 105 66 30)(10 106 67 31)(11 107 68 32)(12 108 69 33)(13 91 70 34)(14 92 71 35)(15 93 72 36)(16 94 55 19)(17 95 56 20)(18 96 57 21)(37 121 77 138)(38 122 78 139)(39 123 79 140)(40 124 80 141)(41 125 81 142)(42 126 82 143)(43 109 83 144)(44 110 84 127)(45 111 85 128)(46 112 86 129)(47 113 87 130)(48 114 88 131)(49 115 89 132)(50 116 90 133)(51 117 73 134)(52 118 74 135)(53 119 75 136)(54 120 76 137)
(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)
(1 123 10 114)(2 122 11 113)(3 121 12 112)(4 120 13 111)(5 119 14 110)(6 118 15 109)(7 117 16 126)(8 116 17 125)(9 115 18 124)(19 42 28 51)(20 41 29 50)(21 40 30 49)(22 39 31 48)(23 38 32 47)(24 37 33 46)(25 54 34 45)(26 53 35 44)(27 52 36 43)(55 143 64 134)(56 142 65 133)(57 141 66 132)(58 140 67 131)(59 139 68 130)(60 138 69 129)(61 137 70 128)(62 136 71 127)(63 135 72 144)(73 94 82 103)(74 93 83 102)(75 92 84 101)(76 91 85 100)(77 108 86 99)(78 107 87 98)(79 106 88 97)(80 105 89 96)(81 104 90 95)
G:=sub<Sym(144)| (1,70,7,58,13,64)(2,71,8,59,14,65)(3,72,9,60,15,66)(4,55,10,61,16,67)(5,56,11,62,17,68)(6,57,12,63,18,69)(19,100,31,94,25,106)(20,101,32,95,26,107)(21,102,33,96,27,108)(22,103,34,97,28,91)(23,104,35,98,29,92)(24,105,36,99,30,93)(37,89,43,77,49,83)(38,90,44,78,50,84)(39,73,45,79,51,85)(40,74,46,80,52,86)(41,75,47,81,53,87)(42,76,48,82,54,88)(109,132,121,144,115,138)(110,133,122,127,116,139)(111,134,123,128,117,140)(112,135,124,129,118,141)(113,136,125,130,119,142)(114,137,126,131,120,143), (1,97,58,22)(2,98,59,23)(3,99,60,24)(4,100,61,25)(5,101,62,26)(6,102,63,27)(7,103,64,28)(8,104,65,29)(9,105,66,30)(10,106,67,31)(11,107,68,32)(12,108,69,33)(13,91,70,34)(14,92,71,35)(15,93,72,36)(16,94,55,19)(17,95,56,20)(18,96,57,21)(37,121,77,138)(38,122,78,139)(39,123,79,140)(40,124,80,141)(41,125,81,142)(42,126,82,143)(43,109,83,144)(44,110,84,127)(45,111,85,128)(46,112,86,129)(47,113,87,130)(48,114,88,131)(49,115,89,132)(50,116,90,133)(51,117,73,134)(52,118,74,135)(53,119,75,136)(54,120,76,137), (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), (1,123,10,114)(2,122,11,113)(3,121,12,112)(4,120,13,111)(5,119,14,110)(6,118,15,109)(7,117,16,126)(8,116,17,125)(9,115,18,124)(19,42,28,51)(20,41,29,50)(21,40,30,49)(22,39,31,48)(23,38,32,47)(24,37,33,46)(25,54,34,45)(26,53,35,44)(27,52,36,43)(55,143,64,134)(56,142,65,133)(57,141,66,132)(58,140,67,131)(59,139,68,130)(60,138,69,129)(61,137,70,128)(62,136,71,127)(63,135,72,144)(73,94,82,103)(74,93,83,102)(75,92,84,101)(76,91,85,100)(77,108,86,99)(78,107,87,98)(79,106,88,97)(80,105,89,96)(81,104,90,95)>;
G:=Group( (1,70,7,58,13,64)(2,71,8,59,14,65)(3,72,9,60,15,66)(4,55,10,61,16,67)(5,56,11,62,17,68)(6,57,12,63,18,69)(19,100,31,94,25,106)(20,101,32,95,26,107)(21,102,33,96,27,108)(22,103,34,97,28,91)(23,104,35,98,29,92)(24,105,36,99,30,93)(37,89,43,77,49,83)(38,90,44,78,50,84)(39,73,45,79,51,85)(40,74,46,80,52,86)(41,75,47,81,53,87)(42,76,48,82,54,88)(109,132,121,144,115,138)(110,133,122,127,116,139)(111,134,123,128,117,140)(112,135,124,129,118,141)(113,136,125,130,119,142)(114,137,126,131,120,143), (1,97,58,22)(2,98,59,23)(3,99,60,24)(4,100,61,25)(5,101,62,26)(6,102,63,27)(7,103,64,28)(8,104,65,29)(9,105,66,30)(10,106,67,31)(11,107,68,32)(12,108,69,33)(13,91,70,34)(14,92,71,35)(15,93,72,36)(16,94,55,19)(17,95,56,20)(18,96,57,21)(37,121,77,138)(38,122,78,139)(39,123,79,140)(40,124,80,141)(41,125,81,142)(42,126,82,143)(43,109,83,144)(44,110,84,127)(45,111,85,128)(46,112,86,129)(47,113,87,130)(48,114,88,131)(49,115,89,132)(50,116,90,133)(51,117,73,134)(52,118,74,135)(53,119,75,136)(54,120,76,137), (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), (1,123,10,114)(2,122,11,113)(3,121,12,112)(4,120,13,111)(5,119,14,110)(6,118,15,109)(7,117,16,126)(8,116,17,125)(9,115,18,124)(19,42,28,51)(20,41,29,50)(21,40,30,49)(22,39,31,48)(23,38,32,47)(24,37,33,46)(25,54,34,45)(26,53,35,44)(27,52,36,43)(55,143,64,134)(56,142,65,133)(57,141,66,132)(58,140,67,131)(59,139,68,130)(60,138,69,129)(61,137,70,128)(62,136,71,127)(63,135,72,144)(73,94,82,103)(74,93,83,102)(75,92,84,101)(76,91,85,100)(77,108,86,99)(78,107,87,98)(79,106,88,97)(80,105,89,96)(81,104,90,95) );
G=PermutationGroup([[(1,70,7,58,13,64),(2,71,8,59,14,65),(3,72,9,60,15,66),(4,55,10,61,16,67),(5,56,11,62,17,68),(6,57,12,63,18,69),(19,100,31,94,25,106),(20,101,32,95,26,107),(21,102,33,96,27,108),(22,103,34,97,28,91),(23,104,35,98,29,92),(24,105,36,99,30,93),(37,89,43,77,49,83),(38,90,44,78,50,84),(39,73,45,79,51,85),(40,74,46,80,52,86),(41,75,47,81,53,87),(42,76,48,82,54,88),(109,132,121,144,115,138),(110,133,122,127,116,139),(111,134,123,128,117,140),(112,135,124,129,118,141),(113,136,125,130,119,142),(114,137,126,131,120,143)], [(1,97,58,22),(2,98,59,23),(3,99,60,24),(4,100,61,25),(5,101,62,26),(6,102,63,27),(7,103,64,28),(8,104,65,29),(9,105,66,30),(10,106,67,31),(11,107,68,32),(12,108,69,33),(13,91,70,34),(14,92,71,35),(15,93,72,36),(16,94,55,19),(17,95,56,20),(18,96,57,21),(37,121,77,138),(38,122,78,139),(39,123,79,140),(40,124,80,141),(41,125,81,142),(42,126,82,143),(43,109,83,144),(44,110,84,127),(45,111,85,128),(46,112,86,129),(47,113,87,130),(48,114,88,131),(49,115,89,132),(50,116,90,133),(51,117,73,134),(52,118,74,135),(53,119,75,136),(54,120,76,137)], [(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)], [(1,123,10,114),(2,122,11,113),(3,121,12,112),(4,120,13,111),(5,119,14,110),(6,118,15,109),(7,117,16,126),(8,116,17,125),(9,115,18,124),(19,42,28,51),(20,41,29,50),(21,40,30,49),(22,39,31,48),(23,38,32,47),(24,37,33,46),(25,54,34,45),(26,53,35,44),(27,52,36,43),(55,143,64,134),(56,142,65,133),(57,141,66,132),(58,140,67,131),(59,139,68,130),(60,138,69,129),(61,137,70,128),(62,136,71,127),(63,135,72,144),(73,94,82,103),(74,93,83,102),(75,92,84,101),(76,91,85,100),(77,108,86,99),(78,107,87,98),(79,106,88,97),(80,105,89,96),(81,104,90,95)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6F | 6G | 6H | 6I | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18I | 18J | ··· | 18R | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 27 | 27 | 27 | 27 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | + | + | - | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | S3 | Dic3 | D6 | Dic3 | D6 | D9 | C4×S3 | C4×S3 | Dic9 | D18 | C4×D9 | S32 | S3×Dic3 | C6.D6 | S3×D9 | Dic3×D9 | C18.D6 | S3×Dic9 |
| kernel | Dic3×Dic9 | C6×Dic9 | Dic3×C18 | C2×C9⋊Dic3 | C3×Dic9 | C9×Dic3 | C9⋊Dic3 | C2×Dic9 | C6×Dic3 | Dic9 | C2×C18 | C3×Dic3 | C62 | C2×Dic3 | C18 | C3×C6 | Dic3 | C2×C6 | C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 4 | 4 | 6 | 3 | 12 | 1 | 2 | 1 | 3 | 3 | 3 | 3 |
Matrix representation of Dic3×Dic9 ►in GL6(𝔽37)
| 0 | 36 | 0 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 3 |
| 0 | 0 | 0 | 0 | 25 | 20 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 35 | 36 |
G:=sub<GL(6,GF(37))| [0,1,0,0,0,0,36,36,0,0,0,0,0,0,1,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,31,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,25,0,0,0,0,3,20],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,35,0,0,0,0,0,36] >;
Dic3×Dic9 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times {\rm Dic}_9 % in TeX
G:=Group("Dic3xDic9"); // GroupNames label
G:=SmallGroup(432,87);
// by ID
G=gap.SmallGroup(432,87);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,36,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^18=1,b^2=a^3,d^2=c^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations