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G = C9×C3⋊C16order 432 = 24·33

Direct product of C9 and C3⋊C16

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

Aliases: C9×C3⋊C16, C3⋊C144, C6.C72, C12.2C36, C24.5C18, C72.10S3, C32.2C48, C36.10Dic3, (C3×C9)⋊1C16, C8.2(S3×C9), C18.4(C3⋊C8), (C3×C72).1C2, (C3×C18).1C8, (C3×C6).7C24, (C3×C36).1C4, C24.26(C3×S3), (C3×C24).19C6, C4.2(C9×Dic3), (C3×C12).19C12, C12.22(C3×Dic3), C2.(C9×C3⋊C8), (C3×C3⋊C16).C3, C6.8(C3×C3⋊C8), C3.4(C3×C3⋊C16), SmallGroup(432,29)

Series: Derived Chief Lower central Upper central

C1C3 — C9×C3⋊C16
C1C3C6C12C3×C12C3×C24C3×C72 — C9×C3⋊C16
C3 — C9×C3⋊C16
C1C72

Generators and relations for C9×C3⋊C16
 G = < a,b,c | a9=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

2C3
2C6
2C9
2C12
2C18
3C16
2C24
2C36
3C48
2C72
3C144

Smallest permutation representation of C9×C3⋊C16
On 144 points
Generators in S144
(1 101 144 55 30 122 37 95 70)(2 102 129 56 31 123 38 96 71)(3 103 130 57 32 124 39 81 72)(4 104 131 58 17 125 40 82 73)(5 105 132 59 18 126 41 83 74)(6 106 133 60 19 127 42 84 75)(7 107 134 61 20 128 43 85 76)(8 108 135 62 21 113 44 86 77)(9 109 136 63 22 114 45 87 78)(10 110 137 64 23 115 46 88 79)(11 111 138 49 24 116 47 89 80)(12 112 139 50 25 117 48 90 65)(13 97 140 51 26 118 33 91 66)(14 98 141 52 27 119 34 92 67)(15 99 142 53 28 120 35 93 68)(16 100 143 54 29 121 36 94 69)
(1 55 37)(2 38 56)(3 57 39)(4 40 58)(5 59 41)(6 42 60)(7 61 43)(8 44 62)(9 63 45)(10 46 64)(11 49 47)(12 48 50)(13 51 33)(14 34 52)(15 53 35)(16 36 54)(17 104 82)(18 83 105)(19 106 84)(20 85 107)(21 108 86)(22 87 109)(23 110 88)(24 89 111)(25 112 90)(26 91 97)(27 98 92)(28 93 99)(29 100 94)(30 95 101)(31 102 96)(32 81 103)(65 117 139)(66 140 118)(67 119 141)(68 142 120)(69 121 143)(70 144 122)(71 123 129)(72 130 124)(73 125 131)(74 132 126)(75 127 133)(76 134 128)(77 113 135)(78 136 114)(79 115 137)(80 138 116)
(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)

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

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

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

216 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E8A8B8C8D9A···9F9G···9L12A12B12C12D12E···12J16A···16H18A···18F18G···18L24A···24H24I···24T36A···36L36M···36X48A···48P72A···72X72Y···72AV144A···144AV
order1233333446666688889···99···91212121212···1216···1618···1818···1824···2424···2436···3636···3648···4872···7272···72144···144
size1111222111122211111···12···211112···23···31···12···21···12···21···12···23···31···12···23···3

216 irreducible representations

dim111111111111111222222222222
type+++-
imageC1C2C3C4C6C8C9C12C16C18C24C36C48C72C144S3Dic3C3×S3C3⋊C8C3×Dic3C3⋊C16S3×C9C3×C3⋊C8C9×Dic3C3×C3⋊C16C9×C3⋊C8C9×C3⋊C16
kernelC9×C3⋊C16C3×C72C3×C3⋊C16C3×C36C3×C24C3×C18C3⋊C16C3×C12C3×C9C24C3×C6C12C32C6C3C72C36C24C18C12C9C8C6C4C3C2C1
# reps112224648681216244811222464681224

Matrix representation of C9×C3⋊C16 in GL3(𝔽433) generated by

25600
02960
00296
,
100
023435
00198
,
19500
02236
0233431
G:=sub<GL(3,GF(433))| [256,0,0,0,296,0,0,0,296],[1,0,0,0,234,0,0,35,198],[195,0,0,0,2,233,0,236,431] >;

C9×C3⋊C16 in GAP, Magma, Sage, TeX

C_9\times C_3\rtimes C_{16}
% in TeX

G:=Group("C9xC3:C16");
// GroupNames label

G:=SmallGroup(432,29);
// by ID

G=gap.SmallGroup(432,29);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-2,-2,-3,42,92,142,102,14118]);
// Polycyclic

G:=Group<a,b,c|a^9=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×C3⋊C16 in TeX

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