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## G = C9×D24order 432 = 24·33

### Direct product of C9 and D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C9×D24
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — C9×D12 — C9×D24
 Lower central C3 — C6 — C12 — C9×D24
 Upper central C1 — C18 — C36 — C72

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

Smallest permutation representation of 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

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