direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D9×C24, C72⋊11C6, D18.4C12, C12.75D18, Dic9.4C12, C9⋊C8⋊13C6, (C3×C72)⋊6C2, C9⋊4(C2×C24), C6.5(S3×C12), C3.1(S3×C24), (C6×D9).4C4, (C4×D9).6C6, C6.21(C4×D9), C2.1(C12×D9), C4.12(C6×D9), C24.17(C3×S3), C12.84(S3×C6), C36.35(C2×C6), (C3×C24).22S3, (C12×D9).6C2, C32.4(S3×C8), C18.15(C2×C12), (C3×C12).212D6, (C3×Dic9).4C4, (C3×C36).65C22, (C3×C9)⋊5(C2×C8), (C3×C9⋊C8)⋊13C2, (C3×C6).63(C4×S3), (C3×C18).18(C2×C4), SmallGroup(432,105)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C24 |
Generators and relations for D9×C24
G = < a,b,c | a24=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 222 in 74 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C24, C3×D9, C3×C18, C9⋊C8, C72, C72, C4×D9, C3×C3⋊C8, C3×C24, S3×C12, C3×Dic9, C3×C36, C6×D9, C8×D9, S3×C24, C3×C9⋊C8, C3×C72, C12×D9, D9×C24
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C2×C8, D9, C3×S3, C24, C4×S3, C2×C12, D18, S3×C6, S3×C8, C2×C24, C3×D9, C4×D9, S3×C12, C6×D9, C8×D9, S3×C24, C12×D9, D9×C24
►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)
(1 104 130 9 112 138 17 120 122)(2 105 131 10 113 139 18 97 123)(3 106 132 11 114 140 19 98 124)(4 107 133 12 115 141 20 99 125)(5 108 134 13 116 142 21 100 126)(6 109 135 14 117 143 22 101 127)(7 110 136 15 118 144 23 102 128)(8 111 137 16 119 121 24 103 129)(25 86 66 41 78 58 33 94 50)(26 87 67 42 79 59 34 95 51)(27 88 68 43 80 60 35 96 52)(28 89 69 44 81 61 36 73 53)(29 90 70 45 82 62 37 74 54)(30 91 71 46 83 63 38 75 55)(31 92 72 47 84 64 39 76 56)(32 93 49 48 85 65 40 77 57)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 108)(26 109)(27 110)(28 111)(29 112)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 120)(38 97)(39 98)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 141)(58 142)(59 143)(60 144)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 131)(72 132)
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), (1,104,130,9,112,138,17,120,122)(2,105,131,10,113,139,18,97,123)(3,106,132,11,114,140,19,98,124)(4,107,133,12,115,141,20,99,125)(5,108,134,13,116,142,21,100,126)(6,109,135,14,117,143,22,101,127)(7,110,136,15,118,144,23,102,128)(8,111,137,16,119,121,24,103,129)(25,86,66,41,78,58,33,94,50)(26,87,67,42,79,59,34,95,51)(27,88,68,43,80,60,35,96,52)(28,89,69,44,81,61,36,73,53)(29,90,70,45,82,62,37,74,54)(30,91,71,46,83,63,38,75,55)(31,92,72,47,84,64,39,76,56)(32,93,49,48,85,65,40,77,57), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,73)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,108)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,141)(58,142)(59,143)(60,144)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)>;
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), (1,104,130,9,112,138,17,120,122)(2,105,131,10,113,139,18,97,123)(3,106,132,11,114,140,19,98,124)(4,107,133,12,115,141,20,99,125)(5,108,134,13,116,142,21,100,126)(6,109,135,14,117,143,22,101,127)(7,110,136,15,118,144,23,102,128)(8,111,137,16,119,121,24,103,129)(25,86,66,41,78,58,33,94,50)(26,87,67,42,79,59,34,95,51)(27,88,68,43,80,60,35,96,52)(28,89,69,44,81,61,36,73,53)(29,90,70,45,82,62,37,74,54)(30,91,71,46,83,63,38,75,55)(31,92,72,47,84,64,39,76,56)(32,93,49,48,85,65,40,77,57), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,73)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,108)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,141)(58,142)(59,143)(60,144)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132) );
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)], [(1,104,130,9,112,138,17,120,122),(2,105,131,10,113,139,18,97,123),(3,106,132,11,114,140,19,98,124),(4,107,133,12,115,141,20,99,125),(5,108,134,13,116,142,21,100,126),(6,109,135,14,117,143,22,101,127),(7,110,136,15,118,144,23,102,128),(8,111,137,16,119,121,24,103,129),(25,86,66,41,78,58,33,94,50),(26,87,67,42,79,59,34,95,51),(27,88,68,43,80,60,35,96,52),(28,89,69,44,81,61,36,73,53),(29,90,70,45,82,62,37,74,54),(30,91,71,46,83,63,38,75,55),(31,92,72,47,84,64,39,76,56),(32,93,49,48,85,65,40,77,57)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,73),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,81),(25,108),(26,109),(27,110),(28,111),(29,112),(30,113),(31,114),(32,115),(33,116),(34,117),(35,118),(36,119),(37,120),(38,97),(39,98),(40,99),(41,100),(42,101),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(49,133),(50,134),(51,135),(52,136),(53,137),(54,138),(55,139),(56,140),(57,141),(58,142),(59,143),(60,144),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,127),(68,128),(69,129),(70,130),(71,131),(72,132)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 18A | ··· | 18I | 24A | ··· | 24H | 24I | ··· | 24T | 24U | ··· | 24AB | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | D6 | D9 | C3×S3 | C4×S3 | D18 | S3×C6 | S3×C8 | C3×D9 | C4×D9 | S3×C12 | C6×D9 | C8×D9 | S3×C24 | C12×D9 | D9×C24 |
| kernel | D9×C24 | C3×C9⋊C8 | C3×C72 | C12×D9 | C8×D9 | C3×Dic9 | C6×D9 | C9⋊C8 | C72 | C4×D9 | C3×D9 | Dic9 | D18 | D9 | C3×C24 | C3×C12 | C24 | C24 | C3×C6 | C12 | C12 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 | 1 | 1 | 3 | 2 | 2 | 3 | 2 | 4 | 6 | 6 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of D9×C24 ►in GL2(𝔽73) generated by
| 21 | 0 |
| 0 | 21 |
| 2 | 0 |
| 11 | 37 |
| 61 | 68 |
| 14 | 12 |
G:=sub<GL(2,GF(73))| [21,0,0,21],[2,11,0,37],[61,14,68,12] >;
D9×C24 in GAP, Magma, Sage, TeX
D_9\times C_{24} % in TeX
G:=Group("D9xC24"); // GroupNames label
G:=SmallGroup(432,105);
// by ID
G=gap.SmallGroup(432,105);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,92,80,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^24=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations