metabelian, supersoluble, monomial
Aliases: D36⋊2S3, D12⋊2D9, C36.9D6, C12.13D18, (C3×C9)⋊2D8, C12.1S32, C3⋊2(D4⋊D9), C9⋊2(D4⋊S3), (C9×D12)⋊1C2, (C3×D36)⋊5C2, (C3×C18).5D4, C4.15(S3×D9), (C3×D12).1S3, (C3×C12).73D6, C6.8(C9⋊D4), C36.S3⋊1C2, C18.7(C3⋊D4), (C3×C36).8C22, C2.4(D6⋊D9), C32.2(D4⋊S3), C3.2(C32⋊2D8), C6.12(D6⋊S3), (C3×C6).41(C3⋊D4), SmallGroup(432,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D36⋊S3
G = < a,b,c,d | a12=b2=c9=d2=1, bab=a-1, ac=ca, dad=a7, bc=cb, dbd=a3b, dcd=c-1 >
Subgroups: 424 in 74 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C9, C9, C32, C12, C12, D6, C2×C6, D8, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, D12, D12, C3×D4, C3×C9, C36, C36, D18, C2×C18, C3×C12, S3×C6, D4⋊S3, C3×D9, S3×C9, C3×C18, C9⋊C8, D36, D4×C9, C32⋊4C8, C3×D12, C3×D12, C3×C36, C6×D9, S3×C18, D4⋊D9, C32⋊2D8, C36.S3, C3×D36, C9×D12, D36⋊S3
Quotients: C1, C2, C22, S3, D4, D6, D8, D9, C3⋊D4, D18, S32, D4⋊S3, C9⋊D4, D6⋊S3, S3×D9, D4⋊D9, C32⋊2D8, D6⋊D9, 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)
(1 143)(2 142)(3 141)(4 140)(5 139)(6 138)(7 137)(8 136)(9 135)(10 134)(11 133)(12 144)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 120)(34 119)(35 118)(36 117)(49 127)(50 126)(51 125)(52 124)(53 123)(54 122)(55 121)(56 132)(57 131)(58 130)(59 129)(60 128)(61 99)(62 98)(63 97)(64 108)(65 107)(66 106)(67 105)(68 104)(69 103)(70 102)(71 101)(72 100)(73 85)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)
(1 130 91 9 126 87 5 122 95)(2 131 92 10 127 88 6 123 96)(3 132 93 11 128 89 7 124 85)(4 121 94 12 129 90 8 125 86)(13 117 62 21 113 70 17 109 66)(14 118 63 22 114 71 18 110 67)(15 119 64 23 115 72 19 111 68)(16 120 65 24 116 61 20 112 69)(25 99 47 29 103 39 33 107 43)(26 100 48 30 104 40 34 108 44)(27 101 37 31 105 41 35 97 45)(28 102 38 32 106 42 36 98 46)(49 82 138 53 74 142 57 78 134)(50 83 139 54 75 143 58 79 135)(51 84 140 55 76 144 59 80 136)(52 73 141 56 77 133 60 81 137)
(1 33)(2 28)(3 35)(4 30)(5 25)(6 32)(7 27)(8 34)(9 29)(10 36)(11 31)(12 26)(13 58)(14 53)(15 60)(16 55)(17 50)(18 57)(19 52)(20 59)(21 54)(22 49)(23 56)(24 51)(37 128)(38 123)(39 130)(40 125)(41 132)(42 127)(43 122)(44 129)(45 124)(46 131)(47 126)(48 121)(61 80)(62 75)(63 82)(64 77)(65 84)(66 79)(67 74)(68 81)(69 76)(70 83)(71 78)(72 73)(85 97)(86 104)(87 99)(88 106)(89 101)(90 108)(91 103)(92 98)(93 105)(94 100)(95 107)(96 102)(109 135)(110 142)(111 137)(112 144)(113 139)(114 134)(115 141)(116 136)(117 143)(118 138)(119 133)(120 140)
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), (1,143)(2,142)(3,141)(4,140)(5,139)(6,138)(7,137)(8,136)(9,135)(10,134)(11,133)(12,144)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,120)(34,119)(35,118)(36,117)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,132)(57,131)(58,130)(59,129)(60,128)(61,99)(62,98)(63,97)(64,108)(65,107)(66,106)(67,105)(68,104)(69,103)(70,102)(71,101)(72,100)(73,85)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86), (1,130,91,9,126,87,5,122,95)(2,131,92,10,127,88,6,123,96)(3,132,93,11,128,89,7,124,85)(4,121,94,12,129,90,8,125,86)(13,117,62,21,113,70,17,109,66)(14,118,63,22,114,71,18,110,67)(15,119,64,23,115,72,19,111,68)(16,120,65,24,116,61,20,112,69)(25,99,47,29,103,39,33,107,43)(26,100,48,30,104,40,34,108,44)(27,101,37,31,105,41,35,97,45)(28,102,38,32,106,42,36,98,46)(49,82,138,53,74,142,57,78,134)(50,83,139,54,75,143,58,79,135)(51,84,140,55,76,144,59,80,136)(52,73,141,56,77,133,60,81,137), (1,33)(2,28)(3,35)(4,30)(5,25)(6,32)(7,27)(8,34)(9,29)(10,36)(11,31)(12,26)(13,58)(14,53)(15,60)(16,55)(17,50)(18,57)(19,52)(20,59)(21,54)(22,49)(23,56)(24,51)(37,128)(38,123)(39,130)(40,125)(41,132)(42,127)(43,122)(44,129)(45,124)(46,131)(47,126)(48,121)(61,80)(62,75)(63,82)(64,77)(65,84)(66,79)(67,74)(68,81)(69,76)(70,83)(71,78)(72,73)(85,97)(86,104)(87,99)(88,106)(89,101)(90,108)(91,103)(92,98)(93,105)(94,100)(95,107)(96,102)(109,135)(110,142)(111,137)(112,144)(113,139)(114,134)(115,141)(116,136)(117,143)(118,138)(119,133)(120,140)>;
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), (1,143)(2,142)(3,141)(4,140)(5,139)(6,138)(7,137)(8,136)(9,135)(10,134)(11,133)(12,144)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,120)(34,119)(35,118)(36,117)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,132)(57,131)(58,130)(59,129)(60,128)(61,99)(62,98)(63,97)(64,108)(65,107)(66,106)(67,105)(68,104)(69,103)(70,102)(71,101)(72,100)(73,85)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86), (1,130,91,9,126,87,5,122,95)(2,131,92,10,127,88,6,123,96)(3,132,93,11,128,89,7,124,85)(4,121,94,12,129,90,8,125,86)(13,117,62,21,113,70,17,109,66)(14,118,63,22,114,71,18,110,67)(15,119,64,23,115,72,19,111,68)(16,120,65,24,116,61,20,112,69)(25,99,47,29,103,39,33,107,43)(26,100,48,30,104,40,34,108,44)(27,101,37,31,105,41,35,97,45)(28,102,38,32,106,42,36,98,46)(49,82,138,53,74,142,57,78,134)(50,83,139,54,75,143,58,79,135)(51,84,140,55,76,144,59,80,136)(52,73,141,56,77,133,60,81,137), (1,33)(2,28)(3,35)(4,30)(5,25)(6,32)(7,27)(8,34)(9,29)(10,36)(11,31)(12,26)(13,58)(14,53)(15,60)(16,55)(17,50)(18,57)(19,52)(20,59)(21,54)(22,49)(23,56)(24,51)(37,128)(38,123)(39,130)(40,125)(41,132)(42,127)(43,122)(44,129)(45,124)(46,131)(47,126)(48,121)(61,80)(62,75)(63,82)(64,77)(65,84)(66,79)(67,74)(68,81)(69,76)(70,83)(71,78)(72,73)(85,97)(86,104)(87,99)(88,106)(89,101)(90,108)(91,103)(92,98)(93,105)(94,100)(95,107)(96,102)(109,135)(110,142)(111,137)(112,144)(113,139)(114,134)(115,141)(116,136)(117,143)(118,138)(119,133)(120,140) );
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)], [(1,143),(2,142),(3,141),(4,140),(5,139),(6,138),(7,137),(8,136),(9,135),(10,134),(11,133),(12,144),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,116),(26,115),(27,114),(28,113),(29,112),(30,111),(31,110),(32,109),(33,120),(34,119),(35,118),(36,117),(49,127),(50,126),(51,125),(52,124),(53,123),(54,122),(55,121),(56,132),(57,131),(58,130),(59,129),(60,128),(61,99),(62,98),(63,97),(64,108),(65,107),(66,106),(67,105),(68,104),(69,103),(70,102),(71,101),(72,100),(73,85),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86)], [(1,130,91,9,126,87,5,122,95),(2,131,92,10,127,88,6,123,96),(3,132,93,11,128,89,7,124,85),(4,121,94,12,129,90,8,125,86),(13,117,62,21,113,70,17,109,66),(14,118,63,22,114,71,18,110,67),(15,119,64,23,115,72,19,111,68),(16,120,65,24,116,61,20,112,69),(25,99,47,29,103,39,33,107,43),(26,100,48,30,104,40,34,108,44),(27,101,37,31,105,41,35,97,45),(28,102,38,32,106,42,36,98,46),(49,82,138,53,74,142,57,78,134),(50,83,139,54,75,143,58,79,135),(51,84,140,55,76,144,59,80,136),(52,73,141,56,77,133,60,81,137)], [(1,33),(2,28),(3,35),(4,30),(5,25),(6,32),(7,27),(8,34),(9,29),(10,36),(11,31),(12,26),(13,58),(14,53),(15,60),(16,55),(17,50),(18,57),(19,52),(20,59),(21,54),(22,49),(23,56),(24,51),(37,128),(38,123),(39,130),(40,125),(41,132),(42,127),(43,122),(44,129),(45,124),(46,131),(47,126),(48,121),(61,80),(62,75),(63,82),(64,77),(65,84),(66,79),(67,74),(68,81),(69,76),(70,83),(71,78),(72,73),(85,97),(86,104),(87,99),(88,106),(89,101),(90,108),(91,103),(92,98),(93,105),(94,100),(95,107),(96,102),(109,135),(110,142),(111,137),(112,144),(113,139),(114,134),(115,141),(116,136),(117,143),(118,138),(119,133),(120,140)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L | 36A | ··· | 36I |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 12 | 36 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 12 | 12 | 36 | 36 | 54 | 54 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D8 | D9 | C3⋊D4 | C3⋊D4 | D18 | C9⋊D4 | S32 | D4⋊S3 | D4⋊S3 | D6⋊S3 | S3×D9 | D4⋊D9 | C32⋊2D8 | D6⋊D9 | D36⋊S3 |
| kernel | D36⋊S3 | C36.S3 | C3×D36 | C9×D12 | D36 | C3×D12 | C3×C18 | C36 | C3×C12 | C3×C9 | D12 | C18 | C3×C6 | C12 | C6 | C12 | C9 | C32 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 3 | 6 | 1 | 1 | 1 | 1 | 3 | 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 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 7 | 68 | 0 | 0 | 0 | 0 |
| 68 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 60 |
| 0 | 0 | 0 | 0 | 13 | 43 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 3 |
| 0 | 0 | 0 | 0 | 70 | 28 |
| 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 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[7,68,0,0,0,0,68,66,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,30,13,0,0,0,0,60,43],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,70,0,0,0,0,3,28],[32,27,0,0,0,0,27,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;
D36⋊S3 in GAP, Magma, Sage, TeX
D_{36}\rtimes S_3 % in TeX
G:=Group("D36:S3"); // GroupNames label
G:=SmallGroup(432,68);
// by ID
G=gap.SmallGroup(432,68);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,254,135,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations