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G = C9⋊C16order 144 = 24·32

The semidirect product of C9 and C16 acting via C16/C8=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C9⋊C16, C18.C8, C8.2D9, C36.2C4, C72.2C2, C24.6S3, C4.2Dic9, C12.4Dic3, C2.(C9⋊C8), C3.(C3⋊C16), C6.1(C3⋊C8), SmallGroup(144,1)

Series: Derived Chief Lower central Upper central

C1C9 — C9⋊C16
C1C3C9C18C36C72 — C9⋊C16
C9 — C9⋊C16
C1C8

Generators and relations for C9⋊C16
 G = < a,b | a9=b16=1, bab-1=a-1 >

9C16
3C3⋊C16

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

C9⋊C16 is a maximal subgroup of
C16×D9  C16⋊D9  C36.C8  C9⋊D16  D8.D9  C9⋊SD32  C9⋊Q32  C27⋊C16  C9⋊C48  C72.S3
C9⋊C16 is a maximal quotient of
C9⋊C32  C27⋊C16  C72.S3

48 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D9A9B9C12A12B16A···16H18A18B18C24A24B24C24D36A···36F72A···72L
order1234468888999121216···161818182424242436···3672···72
size1121121111222229···922222222···22···2

48 irreducible representations

dim1111122222222
type+++-+-
imageC1C2C4C8C16S3Dic3D9C3⋊C8Dic9C3⋊C16C9⋊C8C9⋊C16
kernelC9⋊C16C72C36C18C9C24C12C8C6C4C3C2C1
# reps11248113234612

Matrix representation of C9⋊C16 in GL2(𝔽17) generated by

614
78
,
72
010
G:=sub<GL(2,GF(17))| [6,7,14,8],[7,0,2,10] >;

C9⋊C16 in GAP, Magma, Sage, TeX

C_9\rtimes C_{16}
% in TeX

G:=Group("C9:C16");
// GroupNames label

G:=SmallGroup(144,1);
// by ID

G=gap.SmallGroup(144,1);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,12,31,50,2404,208,3461]);
// Polycyclic

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

Export

Subgroup lattice of C9⋊C16 in TeX

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