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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 [×2], C3 [×2], C3, C4, C4, C22, S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4, C9, C9, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C2×C8, D9 [×2], C18, C18, C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C2×C12, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C24, C3×D9 [×2], 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 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, C12 [×2], D6, C2×C6, C2×C8, D9, C3×S3, C24 [×2], 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 37 114 9 45 98 17 29 106)(2 38 115 10 46 99 18 30 107)(3 39 116 11 47 100 19 31 108)(4 40 117 12 48 101 20 32 109)(5 41 118 13 25 102 21 33 110)(6 42 119 14 26 103 22 34 111)(7 43 120 15 27 104 23 35 112)(8 44 97 16 28 105 24 36 113)(49 122 79 65 138 95 57 130 87)(50 123 80 66 139 96 58 131 88)(51 124 81 67 140 73 59 132 89)(52 125 82 68 141 74 60 133 90)(53 126 83 69 142 75 61 134 91)(54 127 84 70 143 76 62 135 92)(55 128 85 71 144 77 63 136 93)(56 129 86 72 121 78 64 137 94)
(1 72)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 73)(97 136)(98 137)(99 138)(100 139)(101 140)(102 141)(103 142)(104 143)(105 144)(106 121)(107 122)(108 123)(109 124)(110 125)(111 126)(112 127)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 134)(120 135)

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), (1,37,114,9,45,98,17,29,106)(2,38,115,10,46,99,18,30,107)(3,39,116,11,47,100,19,31,108)(4,40,117,12,48,101,20,32,109)(5,41,118,13,25,102,21,33,110)(6,42,119,14,26,103,22,34,111)(7,43,120,15,27,104,23,35,112)(8,44,97,16,28,105,24,36,113)(49,122,79,65,138,95,57,130,87)(50,123,80,66,139,96,58,131,88)(51,124,81,67,140,73,59,132,89)(52,125,82,68,141,74,60,133,90)(53,126,83,69,142,75,61,134,91)(54,127,84,70,143,76,62,135,92)(55,128,85,71,144,77,63,136,93)(56,129,86,72,121,78,64,137,94), (1,72)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,73)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)>;

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), (1,37,114,9,45,98,17,29,106)(2,38,115,10,46,99,18,30,107)(3,39,116,11,47,100,19,31,108)(4,40,117,12,48,101,20,32,109)(5,41,118,13,25,102,21,33,110)(6,42,119,14,26,103,22,34,111)(7,43,120,15,27,104,23,35,112)(8,44,97,16,28,105,24,36,113)(49,122,79,65,138,95,57,130,87)(50,123,80,66,139,96,58,131,88)(51,124,81,67,140,73,59,132,89)(52,125,82,68,141,74,60,133,90)(53,126,83,69,142,75,61,134,91)(54,127,84,70,143,76,62,135,92)(55,128,85,71,144,77,63,136,93)(56,129,86,72,121,78,64,137,94), (1,72)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,73)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135) );

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)], [(1,37,114,9,45,98,17,29,106),(2,38,115,10,46,99,18,30,107),(3,39,116,11,47,100,19,31,108),(4,40,117,12,48,101,20,32,109),(5,41,118,13,25,102,21,33,110),(6,42,119,14,26,103,22,34,111),(7,43,120,15,27,104,23,35,112),(8,44,97,16,28,105,24,36,113),(49,122,79,65,138,95,57,130,87),(50,123,80,66,139,96,58,131,88),(51,124,81,67,140,73,59,132,89),(52,125,82,68,141,74,60,133,90),(53,126,83,69,142,75,61,134,91),(54,127,84,70,143,76,62,135,92),(55,128,85,71,144,77,63,136,93),(56,129,86,72,121,78,64,137,94)], [(1,72),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,85),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,73),(97,136),(98,137),(99,138),(100,139),(101,140),(102,141),(103,142),(104,143),(105,144),(106,121),(107,122),(108,123),(109,124),(110,125),(111,126),(112,127),(113,128),(114,129),(115,130),(116,131),(117,132),(118,133),(119,134),(120,135)])

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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