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G = C3×C9⋊C16order 432 = 24·33

Direct product of C3 and C9⋊C16

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

Aliases: C3×C9⋊C16, C93C48, C72.6C6, C36.8C12, C18.3C24, C24.10D9, C12.10Dic9, (C3×C9)⋊2C16, C6.4(C9⋊C8), C8.2(C3×D9), (C3×C72).4C2, (C3×C18).2C8, (C3×C36).5C4, C24.14(C3×S3), (C3×C24).20S3, C4.2(C3×Dic9), C32.2(C3⋊C16), C12.8(C3×Dic3), (C3×C12).21Dic3, C2.(C3×C9⋊C8), C6.1(C3×C3⋊C8), C3.1(C3×C3⋊C16), (C3×C6).7(C3⋊C8), SmallGroup(432,28)

Series: Derived Chief Lower central Upper central

C1C9 — C3×C9⋊C16
C1C3C9C18C36C72C3×C72 — C3×C9⋊C16
C9 — C3×C9⋊C16
C1C24

Generators and relations for C3×C9⋊C16
 G = < a,b,c | a3=b9=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

2C3
2C6
2C9
2C12
2C18
9C16
2C24
2C36
3C3⋊C16
9C48
2C72
3C3×C3⋊C16

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

144 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E8A8B8C8D9A···9I12A12B12C12D12E···12J16A···16H18A···18I24A···24H24I···24T36A···36R48A···48P72A···72AJ
order1233333446666688889···91212121212···1216···1618···1824···2424···2436···3648···4872···72
size1111222111122211112···211112···29···92···21···12···22···29···92···2

144 irreducible representations

dim11111111112222222222222222
type+++-+-
imageC1C2C3C4C6C8C12C16C24C48S3Dic3D9C3×S3C3⋊C8Dic9C3×Dic3C3⋊C16C3×D9C9⋊C8C3×C3⋊C8C3×Dic9C9⋊C16C3×C3⋊C16C3×C9⋊C8C3×C9⋊C16
kernelC3×C9⋊C16C3×C72C9⋊C16C3×C36C72C3×C18C36C3×C9C18C9C3×C24C3×C12C24C24C3×C6C12C12C32C8C6C6C4C3C3C2C1
# reps112224488161132232466461281224

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

23400
01980
00198
,
100
04170
0027
,
23800
001
04320
G:=sub<GL(3,GF(433))| [234,0,0,0,198,0,0,0,198],[1,0,0,0,417,0,0,0,27],[238,0,0,0,0,432,0,1,0] >;

C3×C9⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC9:C16");
// GroupNames label

G:=SmallGroup(432,28);
// by ID

G=gap.SmallGroup(432,28);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,-3,-3,42,58,80,10085,292,14118]);
// Polycyclic

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

Export

Subgroup lattice of C3×C9⋊C16 in TeX

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