Copied to
clipboard

G = D9×C24order 432 = 24·33

Direct product of C24 and D9

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: D9×C24, C7211C6, D18.4C12, C12.75D18, Dic9.4C12, C9⋊C813C6, (C3×C72)⋊6C2, C94(C2×C24), C6.5(S3×C12), C3.1(S3×C24), (C6×D9).4C4, (C4×D9).6C6, C6.21(C4×D9), C2.1(C12×D9), C4.12(C6×D9), C24.17(C3×S3), C12.84(S3×C6), C36.35(C2×C6), (C3×C24).22S3, (C12×D9).6C2, C32.4(S3×C8), C18.15(C2×C12), (C3×C12).212D6, (C3×Dic9).4C4, (C3×C36).65C22, (C3×C9)⋊5(C2×C8), (C3×C9⋊C8)⋊13C2, (C3×C6).63(C4×S3), (C3×C18).18(C2×C4), SmallGroup(432,105)

Series: Derived Chief Lower central Upper central

C1C9 — D9×C24
C1C3C9C18C36C3×C36C12×D9 — D9×C24
C9 — D9×C24
C1C24

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 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I8A8B8C8D8E8F8G8H9A···9I12A12B12C12D12E···12J12K12L12M12N18A···18I24A···24H24I···24T24U···24AB36A···36R72A···72AJ
order1222333334444666666666888888889···91212121212···121212121218···1824···2424···2424···2436···3672···72
size1199112221199112229999111199992···211112···299992···21···12···29···92···22···2

144 irreducible representations

dim111111111111112222222222222222
type++++++++
imageC1C2C2C2C3C4C4C6C6C6C8C12C12C24S3D6D9C3×S3C4×S3D18S3×C6S3×C8C3×D9C4×D9S3×C12C6×D9C8×D9S3×C24C12×D9D9×C24
kernelD9×C24C3×C9⋊C8C3×C72C12×D9C8×D9C3×Dic9C6×D9C9⋊C8C72C4×D9C3×D9Dic9D18D9C3×C24C3×C12C24C24C3×C6C12C12C32C8C6C6C4C3C3C2C1
# reps1111222222844161132232466461281224

Matrix representation of D9×C24 in GL2(𝔽73) generated by

210
021
,
20
1137
,
6168
1412
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

׿
×
𝔽