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## G = D9×C24order 432 = 24·33

### Direct product of C24 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — D9×C24
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C12×D9 — D9×C24
 Lower central C9 — D9×C24
 Upper central C1 — 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

Smallest permutation representation of 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

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