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G = C9×D24order 432 = 24·33

Direct product of C9 and D24

direct product, metacyclic, supersoluble, monomial

Aliases: C9×D24, C725S3, C241C18, D121C18, C36.72D6, C18.20D12, (C3×C9)⋊4D8, C81(S3×C9), C31(C9×D8), (C3×D24).C3, C6.2(D4×C9), (C3×C72)⋊10C2, (C9×D12)⋊7C2, C4.9(S3×C18), C3.4(C3×D24), C2.4(C9×D12), C24.11(C3×S3), C12.9(C2×C18), (C3×C24).12C6, (C3×D12).9C6, (C3×C18).20D4, C6.32(C3×D12), C12.107(S3×C6), C32.2(C3×D8), (C3×C36).74C22, (C3×C6).40(C3×D4), (C3×C12).78(C2×C6), SmallGroup(432,112)

Series: Derived Chief Lower central Upper central

C1C12 — C9×D24
C1C3C6C3×C6C3×C12C3×C36C9×D12 — C9×D24
C3C6C12 — C9×D24
C1C18C36C72

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 [×2], C3 [×2], C3, C4, C22 [×2], S3 [×2], C6 [×2], C6 [×3], C8, D4 [×2], C9, C9, C32, C12 [×2], C12, D6 [×2], C2×C6 [×2], D8, C18, C18 [×3], C3×S3 [×2], C3×C6, C24 [×2], C24, D12 [×2], C3×D4 [×2], C3×C9, C36, C36, C2×C18 [×2], C3×C12, S3×C6 [×2], D24, C3×D8, S3×C9 [×2], C3×C18, C72, C72, D4×C9 [×2], C3×C24, C3×D12 [×2], C3×C36, S3×C18 [×2], C9×D8, C3×D24, C3×C72, C9×D12 [×2], C9×D24
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, C9, D6, C2×C6, D8, C18 [×3], 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
(1 144 78 9 128 86 17 136 94)(2 121 79 10 129 87 18 137 95)(3 122 80 11 130 88 19 138 96)(4 123 81 12 131 89 20 139 73)(5 124 82 13 132 90 21 140 74)(6 125 83 14 133 91 22 141 75)(7 126 84 15 134 92 23 142 76)(8 127 85 16 135 93 24 143 77)(25 111 70 41 103 62 33 119 54)(26 112 71 42 104 63 34 120 55)(27 113 72 43 105 64 35 97 56)(28 114 49 44 106 65 36 98 57)(29 115 50 45 107 66 37 99 58)(30 116 51 46 108 67 38 100 59)(31 117 52 47 109 68 39 101 60)(32 118 53 48 110 69 40 102 61)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 96)(58 95)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(97 139)(98 138)(99 137)(100 136)(101 135)(102 134)(103 133)(104 132)(105 131)(106 130)(107 129)(108 128)(109 127)(110 126)(111 125)(112 124)(113 123)(114 122)(115 121)(116 144)(117 143)(118 142)(119 141)(120 140)

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

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

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

135 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C6D6E6F6G6H6I8A8B9A···9F9G···9L12A···12H18A···18F18G···18L18M···18X24A···24P36A···36R72A···72AJ
order1222333334666666666889···99···912···1218···1818···1818···1824···2436···3672···72
size1112121122221122212121212221···12···22···21···12···212···122···22···22···2

135 irreducible representations

dim111111111222222222222222222
type+++++++++
imageC1C2C2C3C6C6C9C18C18S3D4D6D8C3×S3D12C3×D4S3×C6D24C3×D8S3×C9D4×C9C3×D12S3×C18C9×D8C3×D24C9×D12C9×D24
kernelC9×D24C3×C72C9×D12C3×D24C3×C24C3×D12D24C24D12C72C3×C18C36C3×C9C24C18C3×C6C12C9C32C8C6C6C4C3C3C2C1
# reps1122246612111222224466461281224

Matrix representation of C9×D24 in GL2(𝔽73) generated by

160
016
,
520
066
,
066
520
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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