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G = C9×C24⋊C2order 432 = 24·33

Direct product of C9 and C24⋊C2

direct product, metacyclic, supersoluble, monomial

Aliases: C9×C24⋊C2, C726S3, C242C18, C36.71D6, Dic61C18, D12.1C18, C18.19D12, C82(S3×C9), C6.1(D4×C9), (C3×C9)⋊7SD16, (C3×C72)⋊12C2, C4.8(S3×C18), C31(C9×SD16), C2.3(C9×D12), C24.24(C3×S3), (C3×C24).15C6, C12.8(C2×C18), (C9×Dic6)⋊7C2, (C9×D12).3C2, (C3×D12).8C6, (C3×C18).19D4, C6.31(C3×D12), C12.106(S3×C6), (C3×Dic6).8C6, (C3×C36).73C22, C32.2(C3×SD16), (C3×C24⋊C2).C3, C3.4(C3×C24⋊C2), (C3×C6).39(C3×D4), (C3×C12).77(C2×C6), SmallGroup(432,111)

Series: Derived Chief Lower central Upper central

C1C12 — C9×C24⋊C2
C1C3C6C3×C6C3×C12C3×C36C9×D12 — C9×C24⋊C2
C3C6C12 — C9×C24⋊C2
C1C18C36C72

Generators and relations for C9×C24⋊C2
 G = < a,b,c | a9=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Subgroups: 172 in 68 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, D4, Q8, C9, C9, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, SD16, C18, C18 [×2], C3×S3, C3×C6, C24 [×2], C24, Dic6, D12, C3×D4, C3×Q8, C3×C9, C36, C36 [×2], C2×C18, C3×Dic3, C3×C12, S3×C6, C24⋊C2, C3×SD16, S3×C9, C3×C18, C72, C72, D4×C9, Q8×C9, C3×C24, C3×Dic6, C3×D12, C9×Dic3, C3×C36, S3×C18, C9×SD16, C3×C24⋊C2, C3×C72, C9×Dic6, C9×D12, C9×C24⋊C2
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, C9, D6, C2×C6, SD16, C18 [×3], C3×S3, D12, C3×D4, C2×C18, S3×C6, C24⋊C2, C3×SD16, S3×C9, D4×C9, C3×D12, S3×C18, C9×SD16, C3×C24⋊C2, C9×D12, C9×C24⋊C2

Smallest permutation representation of C9×C24⋊C2
On 144 points
Generators in S144
(1 31 65 9 39 49 17 47 57)(2 32 66 10 40 50 18 48 58)(3 33 67 11 41 51 19 25 59)(4 34 68 12 42 52 20 26 60)(5 35 69 13 43 53 21 27 61)(6 36 70 14 44 54 22 28 62)(7 37 71 15 45 55 23 29 63)(8 38 72 16 46 56 24 30 64)(73 121 115 89 137 107 81 129 99)(74 122 116 90 138 108 82 130 100)(75 123 117 91 139 109 83 131 101)(76 124 118 92 140 110 84 132 102)(77 125 119 93 141 111 85 133 103)(78 126 120 94 142 112 86 134 104)(79 127 97 95 143 113 87 135 105)(80 128 98 96 144 114 88 136 106)
(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 89)(2 76)(3 87)(4 74)(5 85)(6 96)(7 83)(8 94)(9 81)(10 92)(11 79)(12 90)(13 77)(14 88)(15 75)(16 86)(17 73)(18 84)(19 95)(20 82)(21 93)(22 80)(23 91)(24 78)(25 143)(26 130)(27 141)(28 128)(29 139)(30 126)(31 137)(32 124)(33 135)(34 122)(35 133)(36 144)(37 131)(38 142)(39 129)(40 140)(41 127)(42 138)(43 125)(44 136)(45 123)(46 134)(47 121)(48 132)(49 99)(50 110)(51 97)(52 108)(53 119)(54 106)(55 117)(56 104)(57 115)(58 102)(59 113)(60 100)(61 111)(62 98)(63 109)(64 120)(65 107)(66 118)(67 105)(68 116)(69 103)(70 114)(71 101)(72 112)

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

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

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

135 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B6A6B6C6D6E6F6G8A8B9A···9F9G···9L12A···12H12I12J18A···18F18G···18L18M···18R24A···24P36A···36R36S···36X72A···72AJ
order12233333446666666889···99···912···12121218···1818···1818···1824···2436···3636···3672···72
size111211222212112221212221···12···22···212121···12···212···122···22···212···122···2

135 irreducible representations

dim111111111111222222222222222222
type++++++++
imageC1C2C2C2C3C6C6C6C9C18C18C18S3D4D6SD16C3×S3D12C3×D4S3×C6C24⋊C2C3×SD16S3×C9D4×C9C3×D12S3×C18C9×SD16C3×C24⋊C2C9×D12C9×C24⋊C2
kernelC9×C24⋊C2C3×C72C9×Dic6C9×D12C3×C24⋊C2C3×C24C3×Dic6C3×D12C24⋊C2C24Dic6D12C72C3×C18C36C3×C9C24C18C3×C6C12C9C32C8C6C6C4C3C3C2C1
# reps111122226666111222224466461281224

Matrix representation of C9×C24⋊C2 in GL4(𝔽73) generated by

32000
03200
00640
00064
,
3000
04900
00667
0066
,
0100
1000
001616
001657
G:=sub<GL(4,GF(73))| [32,0,0,0,0,32,0,0,0,0,64,0,0,0,0,64],[3,0,0,0,0,49,0,0,0,0,6,6,0,0,67,6],[0,1,0,0,1,0,0,0,0,0,16,16,0,0,16,57] >;

C9×C24⋊C2 in GAP, Magma, Sage, TeX

C_9\times C_{24}\rtimes C_2
% in TeX

G:=Group("C9xC24:C2");
// GroupNames label

G:=SmallGroup(432,111);
// by ID

G=gap.SmallGroup(432,111);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,197,92,142,2355,192,14118]);
// 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^11>;
// generators/relations

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