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## G = C32×D24order 432 = 24·33

### Direct product of C32 and D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C32×D24
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C32×C12 — C32×D12 — C32×D24
 Lower central C3 — C6 — C12 — C32×D24
 Upper central C1 — C3×C6 — C3×C12 — C3×C24

Generators and relations for C32×D24
G = < a,b,c,d | a3=b3=c24=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 472 in 164 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, D8, C3×S3, C3×C6, C3×C6, C3×C6, C24, C24, C24, D12, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C62, D24, C3×D8, S3×C32, C32×C6, C3×C24, C3×C24, C3×C24, C3×D12, D4×C32, C32×C12, S3×C3×C6, C3×D24, C32×D8, C32×C24, C32×D12, C32×D24
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, D8, C3×S3, C3×C6, D12, C3×D4, S3×C6, C62, D24, C3×D8, S3×C32, C3×D12, D4×C32, S3×C3×C6, C3×D24, C32×D8, C32×D12, C32×D24

Smallest permutation representation of C32×D24
On 144 points
Generators in S144
(1 52 100)(2 53 101)(3 54 102)(4 55 103)(5 56 104)(6 57 105)(7 58 106)(8 59 107)(9 60 108)(10 61 109)(11 62 110)(12 63 111)(13 64 112)(14 65 113)(15 66 114)(16 67 115)(17 68 116)(18 69 117)(19 70 118)(20 71 119)(21 72 120)(22 49 97)(23 50 98)(24 51 99)(25 134 74)(26 135 75)(27 136 76)(28 137 77)(29 138 78)(30 139 79)(31 140 80)(32 141 81)(33 142 82)(34 143 83)(35 144 84)(36 121 85)(37 122 86)(38 123 87)(39 124 88)(40 125 89)(41 126 90)(42 127 91)(43 128 92)(44 129 93)(45 130 94)(46 131 95)(47 132 96)(48 133 73)
(1 108 68)(2 109 69)(3 110 70)(4 111 71)(5 112 72)(6 113 49)(7 114 50)(8 115 51)(9 116 52)(10 117 53)(11 118 54)(12 119 55)(13 120 56)(14 97 57)(15 98 58)(16 99 59)(17 100 60)(18 101 61)(19 102 62)(20 103 63)(21 104 64)(22 105 65)(23 106 66)(24 107 67)(25 90 142)(26 91 143)(27 92 144)(28 93 121)(29 94 122)(30 95 123)(31 96 124)(32 73 125)(33 74 126)(34 75 127)(35 76 128)(36 77 129)(37 78 130)(38 79 131)(39 80 132)(40 81 133)(41 82 134)(42 83 135)(43 84 136)(44 85 137)(45 86 138)(46 87 139)(47 88 140)(48 89 141)
(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)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 126)(11 125)(12 124)(13 123)(14 122)(15 121)(16 144)(17 143)(18 142)(19 141)(20 140)(21 139)(22 138)(23 137)(24 136)(25 101)(26 100)(27 99)(28 98)(29 97)(30 120)(31 119)(32 118)(33 117)(34 116)(35 115)(36 114)(37 113)(38 112)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)

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

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

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

135 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4 6A ··· 6H 6I ··· 6Q 6R ··· 6AG 8A 8B 12A ··· 12Z 24A ··· 24AZ order 1 2 2 2 3 ··· 3 3 ··· 3 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 12 ··· 12 24 ··· 24 size 1 1 12 12 1 ··· 1 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 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 D8 C3×S3 D12 C3×D4 S3×C6 D24 C3×D8 C3×D12 C3×D24 kernel C32×D24 C32×C24 C32×D12 C3×D24 C3×C24 C3×D12 C3×C24 C32×C6 C3×C12 C33 C24 C3×C6 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 2 8 8 16 1 1 1 2 8 2 8 8 4 16 16 32

Matrix representation of C32×D24 in GL3(𝔽73) generated by

 1 0 0 0 8 0 0 0 8
,
 64 0 0 0 8 0 0 0 8
,
 72 0 0 0 66 0 0 0 52
,
 72 0 0 0 0 52 0 66 0
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,8],[64,0,0,0,8,0,0,0,8],[72,0,0,0,66,0,0,0,52],[72,0,0,0,0,66,0,52,0] >;

C32×D24 in GAP, Magma, Sage, TeX

C_3^2\times D_{24}
% in TeX

G:=Group("C3^2xD24");
// GroupNames label

G:=SmallGroup(432,467);
// by ID

G=gap.SmallGroup(432,467);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,533,764,3784,102,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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