direct product, metacyclic, nilpotent (class 4), monomial
Aliases: C32×D16, C48⋊3C6, C8.2C62, (C3×C48)⋊5C2, C16⋊1(C3×C6), (C3×D8)⋊5C6, D8⋊1(C3×C6), (C3×C6).44D8, C6.21(C3×D8), C24.27(C2×C6), C12.45(C3×D4), (C32×D8)⋊9C2, C4.1(D4×C32), C2.3(C32×D8), (C3×C12).142D4, (C3×C24).60C22, SmallGroup(288,329)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×D16
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 216 in 84 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, D4, C32, C12, C2×C6, C16, D8, C3×C6, C3×C6, C24, C3×D4, D16, C3×C12, C62, C48, C3×D8, C3×C24, D4×C32, C3×D16, C3×C48, C32×D8, C32×D16
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, D8, C3×C6, C3×D4, D16, C62, C3×D8, D4×C32, C3×D16, C32×D8, C32×D16
►On 144 points
Generators in S144
(1 143 77)(2 144 78)(3 129 79)(4 130 80)(5 131 65)(6 132 66)(7 133 67)(8 134 68)(9 135 69)(10 136 70)(11 137 71)(12 138 72)(13 139 73)(14 140 74)(15 141 75)(16 142 76)(17 45 81)(18 46 82)(19 47 83)(20 48 84)(21 33 85)(22 34 86)(23 35 87)(24 36 88)(25 37 89)(26 38 90)(27 39 91)(28 40 92)(29 41 93)(30 42 94)(31 43 95)(32 44 96)(49 121 101)(50 122 102)(51 123 103)(52 124 104)(53 125 105)(54 126 106)(55 127 107)(56 128 108)(57 113 109)(58 114 110)(59 115 111)(60 116 112)(61 117 97)(62 118 98)(63 119 99)(64 120 100)
(1 27 108)(2 28 109)(3 29 110)(4 30 111)(5 31 112)(6 32 97)(7 17 98)(8 18 99)(9 19 100)(10 20 101)(11 21 102)(12 22 103)(13 23 104)(14 24 105)(15 25 106)(16 26 107)(33 50 137)(34 51 138)(35 52 139)(36 53 140)(37 54 141)(38 55 142)(39 56 143)(40 57 144)(41 58 129)(42 59 130)(43 60 131)(44 61 132)(45 62 133)(46 63 134)(47 64 135)(48 49 136)(65 95 116)(66 96 117)(67 81 118)(68 82 119)(69 83 120)(70 84 121)(71 85 122)(72 86 123)(73 87 124)(74 88 125)(75 89 126)(76 90 127)(77 91 128)(78 92 113)(79 93 114)(80 94 115)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 21)(18 20)(22 32)(23 31)(24 30)(25 29)(26 28)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(46 48)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(65 73)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)(81 85)(82 84)(86 96)(87 95)(88 94)(89 93)(90 92)(97 103)(98 102)(99 101)(104 112)(105 111)(106 110)(107 109)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(142 144)
G:=sub<Sym(144)| (1,143,77)(2,144,78)(3,129,79)(4,130,80)(5,131,65)(6,132,66)(7,133,67)(8,134,68)(9,135,69)(10,136,70)(11,137,71)(12,138,72)(13,139,73)(14,140,74)(15,141,75)(16,142,76)(17,45,81)(18,46,82)(19,47,83)(20,48,84)(21,33,85)(22,34,86)(23,35,87)(24,36,88)(25,37,89)(26,38,90)(27,39,91)(28,40,92)(29,41,93)(30,42,94)(31,43,95)(32,44,96)(49,121,101)(50,122,102)(51,123,103)(52,124,104)(53,125,105)(54,126,106)(55,127,107)(56,128,108)(57,113,109)(58,114,110)(59,115,111)(60,116,112)(61,117,97)(62,118,98)(63,119,99)(64,120,100), (1,27,108)(2,28,109)(3,29,110)(4,30,111)(5,31,112)(6,32,97)(7,17,98)(8,18,99)(9,19,100)(10,20,101)(11,21,102)(12,22,103)(13,23,104)(14,24,105)(15,25,106)(16,26,107)(33,50,137)(34,51,138)(35,52,139)(36,53,140)(37,54,141)(38,55,142)(39,56,143)(40,57,144)(41,58,129)(42,59,130)(43,60,131)(44,61,132)(45,62,133)(46,63,134)(47,64,135)(48,49,136)(65,95,116)(66,96,117)(67,81,118)(68,82,119)(69,83,120)(70,84,121)(71,85,122)(72,86,123)(73,87,124)(74,88,125)(75,89,126)(76,90,127)(77,91,128)(78,92,113)(79,93,114)(80,94,115), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,21)(18,20)(22,32)(23,31)(24,30)(25,29)(26,28)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,48)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(65,73)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78)(81,85)(82,84)(86,96)(87,95)(88,94)(89,93)(90,92)(97,103)(98,102)(99,101)(104,112)(105,111)(106,110)(107,109)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(142,144)>;
G:=Group( (1,143,77)(2,144,78)(3,129,79)(4,130,80)(5,131,65)(6,132,66)(7,133,67)(8,134,68)(9,135,69)(10,136,70)(11,137,71)(12,138,72)(13,139,73)(14,140,74)(15,141,75)(16,142,76)(17,45,81)(18,46,82)(19,47,83)(20,48,84)(21,33,85)(22,34,86)(23,35,87)(24,36,88)(25,37,89)(26,38,90)(27,39,91)(28,40,92)(29,41,93)(30,42,94)(31,43,95)(32,44,96)(49,121,101)(50,122,102)(51,123,103)(52,124,104)(53,125,105)(54,126,106)(55,127,107)(56,128,108)(57,113,109)(58,114,110)(59,115,111)(60,116,112)(61,117,97)(62,118,98)(63,119,99)(64,120,100), (1,27,108)(2,28,109)(3,29,110)(4,30,111)(5,31,112)(6,32,97)(7,17,98)(8,18,99)(9,19,100)(10,20,101)(11,21,102)(12,22,103)(13,23,104)(14,24,105)(15,25,106)(16,26,107)(33,50,137)(34,51,138)(35,52,139)(36,53,140)(37,54,141)(38,55,142)(39,56,143)(40,57,144)(41,58,129)(42,59,130)(43,60,131)(44,61,132)(45,62,133)(46,63,134)(47,64,135)(48,49,136)(65,95,116)(66,96,117)(67,81,118)(68,82,119)(69,83,120)(70,84,121)(71,85,122)(72,86,123)(73,87,124)(74,88,125)(75,89,126)(76,90,127)(77,91,128)(78,92,113)(79,93,114)(80,94,115), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,21)(18,20)(22,32)(23,31)(24,30)(25,29)(26,28)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,48)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(65,73)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78)(81,85)(82,84)(86,96)(87,95)(88,94)(89,93)(90,92)(97,103)(98,102)(99,101)(104,112)(105,111)(106,110)(107,109)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(142,144) );
G=PermutationGroup([[(1,143,77),(2,144,78),(3,129,79),(4,130,80),(5,131,65),(6,132,66),(7,133,67),(8,134,68),(9,135,69),(10,136,70),(11,137,71),(12,138,72),(13,139,73),(14,140,74),(15,141,75),(16,142,76),(17,45,81),(18,46,82),(19,47,83),(20,48,84),(21,33,85),(22,34,86),(23,35,87),(24,36,88),(25,37,89),(26,38,90),(27,39,91),(28,40,92),(29,41,93),(30,42,94),(31,43,95),(32,44,96),(49,121,101),(50,122,102),(51,123,103),(52,124,104),(53,125,105),(54,126,106),(55,127,107),(56,128,108),(57,113,109),(58,114,110),(59,115,111),(60,116,112),(61,117,97),(62,118,98),(63,119,99),(64,120,100)], [(1,27,108),(2,28,109),(3,29,110),(4,30,111),(5,31,112),(6,32,97),(7,17,98),(8,18,99),(9,19,100),(10,20,101),(11,21,102),(12,22,103),(13,23,104),(14,24,105),(15,25,106),(16,26,107),(33,50,137),(34,51,138),(35,52,139),(36,53,140),(37,54,141),(38,55,142),(39,56,143),(40,57,144),(41,58,129),(42,59,130),(43,60,131),(44,61,132),(45,62,133),(46,63,134),(47,64,135),(48,49,136),(65,95,116),(66,96,117),(67,81,118),(68,82,119),(69,83,120),(70,84,121),(71,85,122),(72,86,123),(73,87,124),(74,88,125),(75,89,126),(76,90,127),(77,91,128),(78,92,113),(79,93,114),(80,94,115)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,21),(18,20),(22,32),(23,31),(24,30),(25,29),(26,28),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(46,48),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(65,73),(66,72),(67,71),(68,70),(74,80),(75,79),(76,78),(81,85),(82,84),(86,96),(87,95),(88,94),(89,93),(90,92),(97,103),(98,102),(99,101),(104,112),(105,111),(106,110),(107,109),(113,127),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121),(129,141),(130,140),(131,139),(132,138),(133,137),(134,136),(142,144)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4 | 6A | ··· | 6H | 6I | ··· | 6X | 8A | 8B | 12A | ··· | 12H | 16A | 16B | 16C | 16D | 24A | ··· | 24P | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 8 | 8 | 1 | ··· | 1 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D8 | C3×D4 | D16 | C3×D8 | C3×D16 |
| kernel | C32×D16 | C3×C48 | C32×D8 | C3×D16 | C48 | C3×D8 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 8 | 8 | 16 | 1 | 2 | 8 | 4 | 16 | 32 |
Matrix representation of C32×D16 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 35 |
| 61 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 61 |
| 96 | 0 | 0 |
| 0 | 95 | 71 |
| 0 | 26 | 95 |
| 96 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(97))| [1,0,0,0,35,0,0,0,35],[61,0,0,0,61,0,0,0,61],[96,0,0,0,95,26,0,71,95],[96,0,0,0,0,1,0,1,0] >;
C32×D16 in GAP, Magma, Sage, TeX
C_3^2\times D_{16} % in TeX
G:=Group("C3^2xD16"); // GroupNames label
G:=SmallGroup(288,329);
// by ID
G=gap.SmallGroup(288,329);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,533,3784,1901,242,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations