direct product, metabelian, supersoluble, monomial
Aliases: C32×D24, C33⋊10D8, C12.9C62, C24⋊3(C3×S3), C24⋊1(C3×C6), (C3×C24)⋊10C6, (C3×C24)⋊13S3, D12⋊1(C3×C6), C8⋊1(S3×C32), C3⋊1(C32×D8), C32⋊7(C3×D8), (C3×D12)⋊10C6, (C32×C24)⋊2C2, (C3×C6).85D12, C6.38(C3×D12), C12.113(S3×C6), C6.2(D4×C32), (C3×C12).232D6, (C32×C6).53D4, C2.4(C32×D12), (C32×D12)⋊16C2, (C32×C12).80C22, C4.9(S3×C3×C6), (C3×C6).45(C3×D4), (C3×C12).84(C2×C6), SmallGroup(432,467)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×D24
G = < a,b,c,d | a3=b3=c24=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 472 in 164 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, D8, C3×S3, C3×C6, C3×C6, C3×C6, C24, C24, C24, D12, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C62, D24, C3×D8, S3×C32, C32×C6, C3×C24, C3×C24, C3×C24, C3×D12, D4×C32, C32×C12, S3×C3×C6, C3×D24, C32×D8, C32×C24, C32×D12, C32×D24
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, D8, C3×S3, C3×C6, D12, C3×D4, S3×C6, C62, D24, C3×D8, S3×C32, C3×D12, D4×C32, S3×C3×C6, C3×D24, C32×D8, C32×D12, C32×D24
►On 144 points
Generators in S144
(1 52 100)(2 53 101)(3 54 102)(4 55 103)(5 56 104)(6 57 105)(7 58 106)(8 59 107)(9 60 108)(10 61 109)(11 62 110)(12 63 111)(13 64 112)(14 65 113)(15 66 114)(16 67 115)(17 68 116)(18 69 117)(19 70 118)(20 71 119)(21 72 120)(22 49 97)(23 50 98)(24 51 99)(25 134 74)(26 135 75)(27 136 76)(28 137 77)(29 138 78)(30 139 79)(31 140 80)(32 141 81)(33 142 82)(34 143 83)(35 144 84)(36 121 85)(37 122 86)(38 123 87)(39 124 88)(40 125 89)(41 126 90)(42 127 91)(43 128 92)(44 129 93)(45 130 94)(46 131 95)(47 132 96)(48 133 73)
(1 108 68)(2 109 69)(3 110 70)(4 111 71)(5 112 72)(6 113 49)(7 114 50)(8 115 51)(9 116 52)(10 117 53)(11 118 54)(12 119 55)(13 120 56)(14 97 57)(15 98 58)(16 99 59)(17 100 60)(18 101 61)(19 102 62)(20 103 63)(21 104 64)(22 105 65)(23 106 66)(24 107 67)(25 90 142)(26 91 143)(27 92 144)(28 93 121)(29 94 122)(30 95 123)(31 96 124)(32 73 125)(33 74 126)(34 75 127)(35 76 128)(36 77 129)(37 78 130)(38 79 131)(39 80 132)(40 81 133)(41 82 134)(42 83 135)(43 84 136)(44 85 137)(45 86 138)(46 87 139)(47 88 140)(48 89 141)
(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)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 126)(11 125)(12 124)(13 123)(14 122)(15 121)(16 144)(17 143)(18 142)(19 141)(20 140)(21 139)(22 138)(23 137)(24 136)(25 101)(26 100)(27 99)(28 98)(29 97)(30 120)(31 119)(32 118)(33 117)(34 116)(35 115)(36 114)(37 113)(38 112)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)
G:=sub<Sym(144)| (1,52,100)(2,53,101)(3,54,102)(4,55,103)(5,56,104)(6,57,105)(7,58,106)(8,59,107)(9,60,108)(10,61,109)(11,62,110)(12,63,111)(13,64,112)(14,65,113)(15,66,114)(16,67,115)(17,68,116)(18,69,117)(19,70,118)(20,71,119)(21,72,120)(22,49,97)(23,50,98)(24,51,99)(25,134,74)(26,135,75)(27,136,76)(28,137,77)(29,138,78)(30,139,79)(31,140,80)(32,141,81)(33,142,82)(34,143,83)(35,144,84)(36,121,85)(37,122,86)(38,123,87)(39,124,88)(40,125,89)(41,126,90)(42,127,91)(43,128,92)(44,129,93)(45,130,94)(46,131,95)(47,132,96)(48,133,73), (1,108,68)(2,109,69)(3,110,70)(4,111,71)(5,112,72)(6,113,49)(7,114,50)(8,115,51)(9,116,52)(10,117,53)(11,118,54)(12,119,55)(13,120,56)(14,97,57)(15,98,58)(16,99,59)(17,100,60)(18,101,61)(19,102,62)(20,103,63)(21,104,64)(22,105,65)(23,106,66)(24,107,67)(25,90,142)(26,91,143)(27,92,144)(28,93,121)(29,94,122)(30,95,123)(31,96,124)(32,73,125)(33,74,126)(34,75,127)(35,76,128)(36,77,129)(37,78,130)(38,79,131)(39,80,132)(40,81,133)(41,82,134)(42,83,135)(43,84,136)(44,85,137)(45,86,138)(46,87,139)(47,88,140)(48,89,141), (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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,144)(17,143)(18,142)(19,141)(20,140)(21,139)(22,138)(23,137)(24,136)(25,101)(26,100)(27,99)(28,98)(29,97)(30,120)(31,119)(32,118)(33,117)(34,116)(35,115)(36,114)(37,113)(38,112)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)>;
G:=Group( (1,52,100)(2,53,101)(3,54,102)(4,55,103)(5,56,104)(6,57,105)(7,58,106)(8,59,107)(9,60,108)(10,61,109)(11,62,110)(12,63,111)(13,64,112)(14,65,113)(15,66,114)(16,67,115)(17,68,116)(18,69,117)(19,70,118)(20,71,119)(21,72,120)(22,49,97)(23,50,98)(24,51,99)(25,134,74)(26,135,75)(27,136,76)(28,137,77)(29,138,78)(30,139,79)(31,140,80)(32,141,81)(33,142,82)(34,143,83)(35,144,84)(36,121,85)(37,122,86)(38,123,87)(39,124,88)(40,125,89)(41,126,90)(42,127,91)(43,128,92)(44,129,93)(45,130,94)(46,131,95)(47,132,96)(48,133,73), (1,108,68)(2,109,69)(3,110,70)(4,111,71)(5,112,72)(6,113,49)(7,114,50)(8,115,51)(9,116,52)(10,117,53)(11,118,54)(12,119,55)(13,120,56)(14,97,57)(15,98,58)(16,99,59)(17,100,60)(18,101,61)(19,102,62)(20,103,63)(21,104,64)(22,105,65)(23,106,66)(24,107,67)(25,90,142)(26,91,143)(27,92,144)(28,93,121)(29,94,122)(30,95,123)(31,96,124)(32,73,125)(33,74,126)(34,75,127)(35,76,128)(36,77,129)(37,78,130)(38,79,131)(39,80,132)(40,81,133)(41,82,134)(42,83,135)(43,84,136)(44,85,137)(45,86,138)(46,87,139)(47,88,140)(48,89,141), (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), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,123)(14,122)(15,121)(16,144)(17,143)(18,142)(19,141)(20,140)(21,139)(22,138)(23,137)(24,136)(25,101)(26,100)(27,99)(28,98)(29,97)(30,120)(31,119)(32,118)(33,117)(34,116)(35,115)(36,114)(37,113)(38,112)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79) );
G=PermutationGroup([[(1,52,100),(2,53,101),(3,54,102),(4,55,103),(5,56,104),(6,57,105),(7,58,106),(8,59,107),(9,60,108),(10,61,109),(11,62,110),(12,63,111),(13,64,112),(14,65,113),(15,66,114),(16,67,115),(17,68,116),(18,69,117),(19,70,118),(20,71,119),(21,72,120),(22,49,97),(23,50,98),(24,51,99),(25,134,74),(26,135,75),(27,136,76),(28,137,77),(29,138,78),(30,139,79),(31,140,80),(32,141,81),(33,142,82),(34,143,83),(35,144,84),(36,121,85),(37,122,86),(38,123,87),(39,124,88),(40,125,89),(41,126,90),(42,127,91),(43,128,92),(44,129,93),(45,130,94),(46,131,95),(47,132,96),(48,133,73)], [(1,108,68),(2,109,69),(3,110,70),(4,111,71),(5,112,72),(6,113,49),(7,114,50),(8,115,51),(9,116,52),(10,117,53),(11,118,54),(12,119,55),(13,120,56),(14,97,57),(15,98,58),(16,99,59),(17,100,60),(18,101,61),(19,102,62),(20,103,63),(21,104,64),(22,105,65),(23,106,66),(24,107,67),(25,90,142),(26,91,143),(27,92,144),(28,93,121),(29,94,122),(30,95,123),(31,96,124),(32,73,125),(33,74,126),(34,75,127),(35,76,128),(36,77,129),(37,78,130),(38,79,131),(39,80,132),(40,81,133),(41,82,134),(42,83,135),(43,84,136),(44,85,137),(45,86,138),(46,87,139),(47,88,140),(48,89,141)], [(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)], [(1,135),(2,134),(3,133),(4,132),(5,131),(6,130),(7,129),(8,128),(9,127),(10,126),(11,125),(12,124),(13,123),(14,122),(15,121),(16,144),(17,143),(18,142),(19,141),(20,140),(21,139),(22,138),(23,137),(24,136),(25,101),(26,100),(27,99),(28,98),(29,97),(30,120),(31,119),(32,118),(33,117),(34,116),(35,115),(36,114),(37,113),(38,112),(39,111),(40,110),(41,109),(42,108),(43,107),(44,106),(45,105),(46,104),(47,103),(48,102),(49,78),(50,77),(51,76),(52,75),(53,74),(54,73),(55,96),(56,95),(57,94),(58,93),(59,92),(60,91),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Q | 4 | 6A | ··· | 6H | 6I | ··· | 6Q | 6R | ··· | 6AG | 8A | 8B | 12A | ··· | 12Z | 24A | ··· | 24AZ |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | D8 | C3×S3 | D12 | C3×D4 | S3×C6 | D24 | C3×D8 | C3×D12 | C3×D24 |
| kernel | C32×D24 | C32×C24 | C32×D12 | C3×D24 | C3×C24 | C3×D12 | C3×C24 | C32×C6 | C3×C12 | C33 | C24 | C3×C6 | C3×C6 | C12 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 8 | 8 | 16 | 1 | 1 | 1 | 2 | 8 | 2 | 8 | 8 | 4 | 16 | 16 | 32 |
Matrix representation of C32×D24 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 72 | 0 | 0 |
| 0 | 66 | 0 |
| 0 | 0 | 52 |
| 72 | 0 | 0 |
| 0 | 0 | 52 |
| 0 | 66 | 0 |
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,8],[64,0,0,0,8,0,0,0,8],[72,0,0,0,66,0,0,0,52],[72,0,0,0,0,66,0,52,0] >;
C32×D24 in GAP, Magma, Sage, TeX
C_3^2\times D_{24} % in TeX
G:=Group("C3^2xD24"); // GroupNames label
G:=SmallGroup(432,467);
// by ID
G=gap.SmallGroup(432,467);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,533,764,3784,102,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^24=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