metabelian, supersoluble, monomial
Aliases: D36.1S3, C36.30D6, C12.9D18, C6.12D36, C3⋊C8⋊1D9, C12.44S32, C4.8(S3×D9), (C3×C9)⋊1SD16, (C3×C18).1D4, C3⋊3(C72⋊C2), C9⋊1(D4.S3), (C3×D36).1C2, (C3×C6).29D12, (C3×C12).69D6, C12.D9⋊1C2, C18.1(C3⋊D4), (C3×C36).1C22, C6.1(C3⋊D12), C2.4(C3⋊D36), C32.2(C24⋊C2), C3.1(D12.S3), (C9×C3⋊C8)⋊2C2, (C3×C3⋊C8).4S3, SmallGroup(432,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D36.S3
G = < a,b,c,d | a36=b2=c3=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a27b, dcd-1=c-1 >
Subgroups: 504 in 72 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, SD16, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C3×C9, Dic9, C36, C36, D18, C3⋊Dic3, C3×C12, S3×C6, C24⋊C2, D4.S3, C3×D9, C3×C18, C72, Dic18, D36, C3×C3⋊C8, C3×D12, C32⋊4Q8, C9⋊Dic3, C3×C36, C6×D9, C72⋊C2, D12.S3, C9×C3⋊C8, C3×D36, C12.D9, D36.S3
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, D12, C3⋊D4, D18, S32, C24⋊C2, D4.S3, D36, C3⋊D12, S3×D9, C72⋊C2, D12.S3, C3⋊D36, D36.S3
►On 144 points
Generators in S144
(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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 108)(8 107)(9 106)(10 105)(11 104)(12 103)(13 102)(14 101)(15 100)(16 99)(17 98)(18 97)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 84)(32 83)(33 82)(34 81)(35 80)(36 79)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 144)(56 143)(57 142)(58 141)(59 140)(60 139)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)(73 97 85)(74 98 86)(75 99 87)(76 100 88)(77 101 89)(78 102 90)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)(109 121 133)(110 122 134)(111 123 135)(112 124 136)(113 125 137)(114 126 138)(115 127 139)(116 128 140)(117 129 141)(118 130 142)(119 131 143)(120 132 144)
(1 38 10 47 19 56 28 65)(2 39 11 48 20 57 29 66)(3 40 12 49 21 58 30 67)(4 41 13 50 22 59 31 68)(5 42 14 51 23 60 32 69)(6 43 15 52 24 61 33 70)(7 44 16 53 25 62 34 71)(8 45 17 54 26 63 35 72)(9 46 18 55 27 64 36 37)(73 111 82 120 91 129 100 138)(74 112 83 121 92 130 101 139)(75 113 84 122 93 131 102 140)(76 114 85 123 94 132 103 141)(77 115 86 124 95 133 104 142)(78 116 87 125 96 134 105 143)(79 117 88 126 97 135 106 144)(80 118 89 127 98 136 107 109)(81 119 90 128 99 137 108 110)
G:=sub<Sym(144)| (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,108)(8,107)(9,106)(10,105)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,144)(56,143)(57,142)(58,141)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127), (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(109,121,133)(110,122,134)(111,123,135)(112,124,136)(113,125,137)(114,126,138)(115,127,139)(116,128,140)(117,129,141)(118,130,142)(119,131,143)(120,132,144), (1,38,10,47,19,56,28,65)(2,39,11,48,20,57,29,66)(3,40,12,49,21,58,30,67)(4,41,13,50,22,59,31,68)(5,42,14,51,23,60,32,69)(6,43,15,52,24,61,33,70)(7,44,16,53,25,62,34,71)(8,45,17,54,26,63,35,72)(9,46,18,55,27,64,36,37)(73,111,82,120,91,129,100,138)(74,112,83,121,92,130,101,139)(75,113,84,122,93,131,102,140)(76,114,85,123,94,132,103,141)(77,115,86,124,95,133,104,142)(78,116,87,125,96,134,105,143)(79,117,88,126,97,135,106,144)(80,118,89,127,98,136,107,109)(81,119,90,128,99,137,108,110)>;
G:=Group( (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,108)(8,107)(9,106)(10,105)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,144)(56,143)(57,142)(58,141)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127), (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(109,121,133)(110,122,134)(111,123,135)(112,124,136)(113,125,137)(114,126,138)(115,127,139)(116,128,140)(117,129,141)(118,130,142)(119,131,143)(120,132,144), (1,38,10,47,19,56,28,65)(2,39,11,48,20,57,29,66)(3,40,12,49,21,58,30,67)(4,41,13,50,22,59,31,68)(5,42,14,51,23,60,32,69)(6,43,15,52,24,61,33,70)(7,44,16,53,25,62,34,71)(8,45,17,54,26,63,35,72)(9,46,18,55,27,64,36,37)(73,111,82,120,91,129,100,138)(74,112,83,121,92,130,101,139)(75,113,84,122,93,131,102,140)(76,114,85,123,94,132,103,141)(77,115,86,124,95,133,104,142)(78,116,87,125,96,134,105,143)(79,117,88,126,97,135,106,144)(80,118,89,127,98,136,107,109)(81,119,90,128,99,137,108,110) );
G=PermutationGroup([[(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)], [(1,78),(2,77),(3,76),(4,75),(5,74),(6,73),(7,108),(8,107),(9,106),(10,105),(11,104),(12,103),(13,102),(14,101),(15,100),(16,99),(17,98),(18,97),(19,96),(20,95),(21,94),(22,93),(23,92),(24,91),(25,90),(26,89),(27,88),(28,87),(29,86),(30,85),(31,84),(32,83),(33,82),(34,81),(35,80),(36,79),(37,126),(38,125),(39,124),(40,123),(41,122),(42,121),(43,120),(44,119),(45,118),(46,117),(47,116),(48,115),(49,114),(50,113),(51,112),(52,111),(53,110),(54,109),(55,144),(56,143),(57,142),(58,141),(59,140),(60,139),(61,138),(62,137),(63,136),(64,135),(65,134),(66,133),(67,132),(68,131),(69,130),(70,129),(71,128),(72,127)], [(1,13,25),(2,14,26),(3,15,27),(4,16,28),(5,17,29),(6,18,30),(7,19,31),(8,20,32),(9,21,33),(10,22,34),(11,23,35),(12,24,36),(37,61,49),(38,62,50),(39,63,51),(40,64,52),(41,65,53),(42,66,54),(43,67,55),(44,68,56),(45,69,57),(46,70,58),(47,71,59),(48,72,60),(73,97,85),(74,98,86),(75,99,87),(76,100,88),(77,101,89),(78,102,90),(79,103,91),(80,104,92),(81,105,93),(82,106,94),(83,107,95),(84,108,96),(109,121,133),(110,122,134),(111,123,135),(112,124,136),(113,125,137),(114,126,138),(115,127,139),(116,128,140),(117,129,141),(118,130,142),(119,131,143),(120,132,144)], [(1,38,10,47,19,56,28,65),(2,39,11,48,20,57,29,66),(3,40,12,49,21,58,30,67),(4,41,13,50,22,59,31,68),(5,42,14,51,23,60,32,69),(6,43,15,52,24,61,33,70),(7,44,16,53,25,62,34,71),(8,45,17,54,26,63,35,72),(9,46,18,55,27,64,36,37),(73,111,82,120,91,129,100,138),(74,112,83,121,92,130,101,139),(75,113,84,122,93,131,102,140),(76,114,85,123,94,132,103,141),(77,115,86,124,95,133,104,142),(78,116,87,125,96,134,105,143),(79,117,88,126,97,135,106,144),(80,118,89,127,98,136,107,109),(81,119,90,128,99,137,108,110)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36L | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 36 | 2 | 2 | 4 | 2 | 108 | 2 | 2 | 4 | 36 | 36 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 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 | S3 | S3 | D4 | D6 | D6 | SD16 | D9 | C3⋊D4 | D12 | D18 | C24⋊C2 | D36 | C72⋊C2 | S32 | D4.S3 | C3⋊D12 | S3×D9 | D12.S3 | C3⋊D36 | D36.S3 |
| kernel | D36.S3 | C9×C3⋊C8 | C3×D36 | C12.D9 | D36 | C3×C3⋊C8 | C3×C18 | C36 | C3×C12 | C3×C9 | C3⋊C8 | C18 | C3×C6 | C12 | C32 | C6 | C3 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 3 | 4 | 6 | 12 | 1 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of D36.S3 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 28 |
| 0 | 0 | 0 | 0 | 45 | 3 |
| 32 | 27 | 0 | 0 | 0 | 0 |
| 27 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 68 |
| 0 | 0 | 0 | 0 | 50 | 55 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 67 | 6 | 0 | 0 | 0 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 59 |
| 0 | 0 | 0 | 0 | 14 | 7 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,45,0,0,0,0,28,3],[32,27,0,0,0,0,27,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,18,50,0,0,0,0,68,55],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,66,14,0,0,0,0,59,7] >;
D36.S3 in GAP, Magma, Sage, TeX
D_{36}.S_3 % in TeX
G:=Group("D36.S3"); // GroupNames label
G:=SmallGroup(432,62);
// by ID
G=gap.SmallGroup(432,62);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,36,254,58,571,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^36=b^2=c^3=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^27*b,d*c*d^-1=c^-1>;
// generators/relations