metabelian, supersoluble, monomial
Aliases: D6.Dic9, C36.42D6, C12.42D18, Dic3.Dic9, C9⋊C8⋊5S3, C12.64S32, (C4×S3).2D9, C4.27(S3×D9), C9⋊3(C8⋊S3), (C3×C9)⋊3M4(2), (S3×C18).3C4, (S3×C12).1S3, (S3×C36).2C2, C18.18(C4×S3), C2.3(S3×Dic9), C6.2(C2×Dic9), (C3×C12).160D6, C3⋊1(C4.Dic9), (C9×Dic3).1C4, (S3×C6).2Dic3, C6.23(S3×Dic3), C36.S3⋊10C2, (C3×C36).41C22, (C3×Dic3).1Dic3, C3.3(D6.Dic3), C32.2(C4.Dic3), (C3×C9⋊C8)⋊10C2, (C3×C18).6(C2×C4), (C3×C6).30(C2×Dic3), SmallGroup(432,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.Dic9
G = < a,b,c,d | a6=b2=1, c18=a3, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c17 >
Subgroups: 244 in 68 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, M4(2), C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C4.Dic3, S3×C9, C3×C18, C9⋊C8, C9⋊C8, C2×C36, C3×C3⋊C8, C32⋊4C8, S3×C12, C9×Dic3, C3×C36, S3×C18, C4.Dic9, D6.Dic3, C3×C9⋊C8, C36.S3, S3×C36, D6.Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), D9, C4×S3, C2×Dic3, Dic9, D18, S32, C8⋊S3, C4.Dic3, C2×Dic9, S3×Dic3, S3×D9, C4.Dic9, D6.Dic3, S3×Dic9, D6.Dic9
►On 144 points
Generators in S144
(1 7 13 19 25 31)(2 8 14 20 26 32)(3 9 15 21 27 33)(4 10 16 22 28 34)(5 11 17 23 29 35)(6 12 18 24 30 36)(37 67 61 55 49 43)(38 68 62 56 50 44)(39 69 63 57 51 45)(40 70 64 58 52 46)(41 71 65 59 53 47)(42 72 66 60 54 48)(73 103 97 91 85 79)(74 104 98 92 86 80)(75 105 99 93 87 81)(76 106 100 94 88 82)(77 107 101 95 89 83)(78 108 102 96 90 84)(109 115 121 127 133 139)(110 116 122 128 134 140)(111 117 123 129 135 141)(112 118 124 130 136 142)(113 119 125 131 137 143)(114 120 126 132 138 144)
(1 106)(2 107)(3 108)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 109)(52 110)(53 111)(54 112)(55 113)(56 114)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)
(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 46 10 55 19 64 28 37)(2 63 11 72 20 45 29 54)(3 44 12 53 21 62 30 71)(4 61 13 70 22 43 31 52)(5 42 14 51 23 60 32 69)(6 59 15 68 24 41 33 50)(7 40 16 49 25 58 34 67)(8 57 17 66 26 39 35 48)(9 38 18 47 27 56 36 65)(73 137 82 110 91 119 100 128)(74 118 83 127 92 136 101 109)(75 135 84 144 93 117 102 126)(76 116 85 125 94 134 103 143)(77 133 86 142 95 115 104 124)(78 114 87 123 96 132 105 141)(79 131 88 140 97 113 106 122)(80 112 89 121 98 130 107 139)(81 129 90 138 99 111 108 120)
G:=sub<Sym(144)| (1,7,13,19,25,31)(2,8,14,20,26,32)(3,9,15,21,27,33)(4,10,16,22,28,34)(5,11,17,23,29,35)(6,12,18,24,30,36)(37,67,61,55,49,43)(38,68,62,56,50,44)(39,69,63,57,51,45)(40,70,64,58,52,46)(41,71,65,59,53,47)(42,72,66,60,54,48)(73,103,97,91,85,79)(74,104,98,92,86,80)(75,105,99,93,87,81)(76,106,100,94,88,82)(77,107,101,95,89,83)(78,108,102,96,90,84)(109,115,121,127,133,139)(110,116,122,128,134,140)(111,117,123,129,135,141)(112,118,124,130,136,142)(113,119,125,131,137,143)(114,120,126,132,138,144), (1,106)(2,107)(3,108)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130), (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), (1,46,10,55,19,64,28,37)(2,63,11,72,20,45,29,54)(3,44,12,53,21,62,30,71)(4,61,13,70,22,43,31,52)(5,42,14,51,23,60,32,69)(6,59,15,68,24,41,33,50)(7,40,16,49,25,58,34,67)(8,57,17,66,26,39,35,48)(9,38,18,47,27,56,36,65)(73,137,82,110,91,119,100,128)(74,118,83,127,92,136,101,109)(75,135,84,144,93,117,102,126)(76,116,85,125,94,134,103,143)(77,133,86,142,95,115,104,124)(78,114,87,123,96,132,105,141)(79,131,88,140,97,113,106,122)(80,112,89,121,98,130,107,139)(81,129,90,138,99,111,108,120)>;
G:=Group( (1,7,13,19,25,31)(2,8,14,20,26,32)(3,9,15,21,27,33)(4,10,16,22,28,34)(5,11,17,23,29,35)(6,12,18,24,30,36)(37,67,61,55,49,43)(38,68,62,56,50,44)(39,69,63,57,51,45)(40,70,64,58,52,46)(41,71,65,59,53,47)(42,72,66,60,54,48)(73,103,97,91,85,79)(74,104,98,92,86,80)(75,105,99,93,87,81)(76,106,100,94,88,82)(77,107,101,95,89,83)(78,108,102,96,90,84)(109,115,121,127,133,139)(110,116,122,128,134,140)(111,117,123,129,135,141)(112,118,124,130,136,142)(113,119,125,131,137,143)(114,120,126,132,138,144), (1,106)(2,107)(3,108)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130), (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), (1,46,10,55,19,64,28,37)(2,63,11,72,20,45,29,54)(3,44,12,53,21,62,30,71)(4,61,13,70,22,43,31,52)(5,42,14,51,23,60,32,69)(6,59,15,68,24,41,33,50)(7,40,16,49,25,58,34,67)(8,57,17,66,26,39,35,48)(9,38,18,47,27,56,36,65)(73,137,82,110,91,119,100,128)(74,118,83,127,92,136,101,109)(75,135,84,144,93,117,102,126)(76,116,85,125,94,134,103,143)(77,133,86,142,95,115,104,124)(78,114,87,123,96,132,105,141)(79,131,88,140,97,113,106,122)(80,112,89,121,98,130,107,139)(81,129,90,138,99,111,108,120) );
G=PermutationGroup([[(1,7,13,19,25,31),(2,8,14,20,26,32),(3,9,15,21,27,33),(4,10,16,22,28,34),(5,11,17,23,29,35),(6,12,18,24,30,36),(37,67,61,55,49,43),(38,68,62,56,50,44),(39,69,63,57,51,45),(40,70,64,58,52,46),(41,71,65,59,53,47),(42,72,66,60,54,48),(73,103,97,91,85,79),(74,104,98,92,86,80),(75,105,99,93,87,81),(76,106,100,94,88,82),(77,107,101,95,89,83),(78,108,102,96,90,84),(109,115,121,127,133,139),(110,116,122,128,134,140),(111,117,123,129,135,141),(112,118,124,130,136,142),(113,119,125,131,137,143),(114,120,126,132,138,144)], [(1,106),(2,107),(3,108),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,90),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,131),(38,132),(39,133),(40,134),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,141),(48,142),(49,143),(50,144),(51,109),(52,110),(53,111),(54,112),(55,113),(56,114),(57,115),(58,116),(59,117),(60,118),(61,119),(62,120),(63,121),(64,122),(65,123),(66,124),(67,125),(68,126),(69,127),(70,128),(71,129),(72,130)], [(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)], [(1,46,10,55,19,64,28,37),(2,63,11,72,20,45,29,54),(3,44,12,53,21,62,30,71),(4,61,13,70,22,43,31,52),(5,42,14,51,23,60,32,69),(6,59,15,68,24,41,33,50),(7,40,16,49,25,58,34,67),(8,57,17,66,26,39,35,48),(9,38,18,47,27,56,36,65),(73,137,82,110,91,119,100,128),(74,118,83,127,92,136,101,109),(75,135,84,144,93,117,102,126),(76,116,85,125,94,134,103,143),(77,133,86,142,95,115,104,124),(78,114,87,123,96,132,105,141),(79,131,88,140,97,113,106,122),(80,112,89,121,98,130,107,139),(81,129,90,138,99,111,108,120)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36L | 36M | ··· | 36R |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 6 | 2 | 2 | 4 | 1 | 1 | 6 | 2 | 2 | 4 | 6 | 6 | 18 | 18 | 54 | 54 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | - | + | - | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | S3 | D6 | Dic3 | D6 | Dic3 | M4(2) | D9 | C4×S3 | Dic9 | D18 | Dic9 | C8⋊S3 | C4.Dic3 | C4.Dic9 | S32 | S3×Dic3 | S3×D9 | D6.Dic3 | S3×Dic9 | D6.Dic9 |
| kernel | D6.Dic9 | C3×C9⋊C8 | C36.S3 | S3×C36 | C9×Dic3 | S3×C18 | C9⋊C8 | S3×C12 | C36 | C3×Dic3 | C3×C12 | S3×C6 | C3×C9 | C4×S3 | C18 | Dic3 | C12 | D6 | C9 | C32 | C3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 3 | 3 | 4 | 4 | 12 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of D6.Dic9 ►in GL4(𝔽73) generated by
| 1 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 50 | 5 | 0 | 0 |
| 55 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 28 | 26 |
| 0 | 0 | 22 | 70 |
| 65 | 16 | 0 | 0 |
| 57 | 8 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 35 | 27 |
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[50,55,0,0,5,23,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,28,22,0,0,26,70],[65,57,0,0,16,8,0,0,0,0,46,35,0,0,0,27] >;
D6.Dic9 in GAP, Magma, Sage, TeX
D_6.{\rm Dic}_9 % in TeX
G:=Group("D6.Dic9"); // GroupNames label
G:=SmallGroup(432,67);
// by ID
G=gap.SmallGroup(432,67);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=1,c^18=a^3,d^2=c^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^17>;
// generators/relations