direct product, metacyclic, supersoluble, monomial
Aliases: C9×D24, C72⋊5S3, C24⋊1C18, D12⋊1C18, C36.72D6, C18.20D12, (C3×C9)⋊4D8, C8⋊1(S3×C9), C3⋊1(C9×D8), (C3×D24).C3, C6.2(D4×C9), (C3×C72)⋊10C2, (C9×D12)⋊7C2, C4.9(S3×C18), C3.4(C3×D24), C2.4(C9×D12), C24.11(C3×S3), C12.9(C2×C18), (C3×C24).12C6, (C3×D12).9C6, (C3×C18).20D4, C6.32(C3×D12), C12.107(S3×C6), C32.2(C3×D8), (C3×C36).74C22, (C3×C6).40(C3×D4), (C3×C12).78(C2×C6), SmallGroup(432,112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×D24
G = < a,b,c | a9=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 220 in 74 conjugacy classes, 33 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C9, C9, C32, C12, C12, D6, C2×C6, D8, C18, C18, C3×S3, C3×C6, C24, C24, D12, C3×D4, C3×C9, C36, C36, C2×C18, C3×C12, S3×C6, D24, C3×D8, S3×C9, C3×C18, C72, C72, D4×C9, C3×C24, C3×D12, C3×C36, S3×C18, C9×D8, C3×D24, C3×C72, C9×D12, C9×D24
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2×C6, D8, C18, C3×S3, D12, C3×D4, C2×C18, S3×C6, D24, C3×D8, S3×C9, D4×C9, C3×D12, S3×C18, C9×D8, C3×D24, C9×D12, C9×D24
►On 144 points
Generators in S144
(1 121 52 9 129 60 17 137 68)(2 122 53 10 130 61 18 138 69)(3 123 54 11 131 62 19 139 70)(4 124 55 12 132 63 20 140 71)(5 125 56 13 133 64 21 141 72)(6 126 57 14 134 65 22 142 49)(7 127 58 15 135 66 23 143 50)(8 128 59 16 136 67 24 144 51)(25 119 75 41 111 91 33 103 83)(26 120 76 42 112 92 34 104 84)(27 97 77 43 113 93 35 105 85)(28 98 78 44 114 94 36 106 86)(29 99 79 45 115 95 37 107 87)(30 100 80 46 116 96 38 108 88)(31 101 81 47 117 73 39 109 89)(32 102 82 48 118 74 40 110 90)
(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)
(1 100)(2 99)(3 98)(4 97)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 49)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 61)(38 60)(39 59)(40 58)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(73 128)(74 127)(75 126)(76 125)(77 124)(78 123)(79 122)(80 121)(81 144)(82 143)(83 142)(84 141)(85 140)(86 139)(87 138)(88 137)(89 136)(90 135)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)
G:=sub<Sym(144)| (1,121,52,9,129,60,17,137,68)(2,122,53,10,130,61,18,138,69)(3,123,54,11,131,62,19,139,70)(4,124,55,12,132,63,20,140,71)(5,125,56,13,133,64,21,141,72)(6,126,57,14,134,65,22,142,49)(7,127,58,15,135,66,23,143,50)(8,128,59,16,136,67,24,144,51)(25,119,75,41,111,91,33,103,83)(26,120,76,42,112,92,34,104,84)(27,97,77,43,113,93,35,105,85)(28,98,78,44,114,94,36,106,86)(29,99,79,45,115,95,37,107,87)(30,100,80,46,116,96,38,108,88)(31,101,81,47,117,73,39,109,89)(32,102,82,48,118,74,40,110,90), (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), (1,100)(2,99)(3,98)(4,97)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,49)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,144)(82,143)(83,142)(84,141)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)>;
G:=Group( (1,121,52,9,129,60,17,137,68)(2,122,53,10,130,61,18,138,69)(3,123,54,11,131,62,19,139,70)(4,124,55,12,132,63,20,140,71)(5,125,56,13,133,64,21,141,72)(6,126,57,14,134,65,22,142,49)(7,127,58,15,135,66,23,143,50)(8,128,59,16,136,67,24,144,51)(25,119,75,41,111,91,33,103,83)(26,120,76,42,112,92,34,104,84)(27,97,77,43,113,93,35,105,85)(28,98,78,44,114,94,36,106,86)(29,99,79,45,115,95,37,107,87)(30,100,80,46,116,96,38,108,88)(31,101,81,47,117,73,39,109,89)(32,102,82,48,118,74,40,110,90), (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), (1,100)(2,99)(3,98)(4,97)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,49)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,144)(82,143)(83,142)(84,141)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129) );
G=PermutationGroup([[(1,121,52,9,129,60,17,137,68),(2,122,53,10,130,61,18,138,69),(3,123,54,11,131,62,19,139,70),(4,124,55,12,132,63,20,140,71),(5,125,56,13,133,64,21,141,72),(6,126,57,14,134,65,22,142,49),(7,127,58,15,135,66,23,143,50),(8,128,59,16,136,67,24,144,51),(25,119,75,41,111,91,33,103,83),(26,120,76,42,112,92,34,104,84),(27,97,77,43,113,93,35,105,85),(28,98,78,44,114,94,36,106,86),(29,99,79,45,115,95,37,107,87),(30,100,80,46,116,96,38,108,88),(31,101,81,47,117,73,39,109,89),(32,102,82,48,118,74,40,110,90)], [(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)], [(1,100),(2,99),(3,98),(4,97),(5,120),(6,119),(7,118),(8,117),(9,116),(10,115),(11,114),(12,113),(13,112),(14,111),(15,110),(16,109),(17,108),(18,107),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,49),(26,72),(27,71),(28,70),(29,69),(30,68),(31,67),(32,66),(33,65),(34,64),(35,63),(36,62),(37,61),(38,60),(39,59),(40,58),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(73,128),(74,127),(75,126),(76,125),(77,124),(78,123),(79,122),(80,121),(81,144),(82,143),(83,142),(84,141),(85,140),(86,139),(87,138),(88,137),(89,136),(90,135),(91,134),(92,133),(93,132),(94,131),(95,130),(96,129)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12H | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18X | 24A | ··· | 24P | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 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 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | S3 | D4 | D6 | D8 | C3×S3 | D12 | C3×D4 | S3×C6 | D24 | C3×D8 | S3×C9 | D4×C9 | C3×D12 | S3×C18 | C9×D8 | C3×D24 | C9×D12 | C9×D24 |
| kernel | C9×D24 | C3×C72 | C9×D12 | C3×D24 | C3×C24 | C3×D12 | D24 | C24 | D12 | C72 | C3×C18 | C36 | C3×C9 | C24 | C18 | C3×C6 | C12 | C9 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C9×D24 ►in GL2(𝔽73) generated by
| 16 | 0 |
| 0 | 16 |
| 52 | 0 |
| 0 | 66 |
| 0 | 66 |
| 52 | 0 |
G:=sub<GL(2,GF(73))| [16,0,0,16],[52,0,0,66],[0,52,66,0] >;
C9×D24 in GAP, Magma, Sage, TeX
C_9\times D_{24} % in TeX
G:=Group("C9xD24"); // GroupNames label
G:=SmallGroup(432,112);
// by ID
G=gap.SmallGroup(432,112);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,197,596,142,2355,192,14118]);
// Polycyclic
G:=Group<a,b,c|a^9=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations