Copied to
clipboard

G = C9×Q16order 144 = 24·32

Direct product of C9 and Q16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×Q16, C8.C18, C24.3C6, C72.3C2, C18.16D4, Q8.2C18, C36.19C22, C3.(C3×Q16), C2.5(D4×C9), (C3×Q16).C3, C4.3(C2×C18), C6.16(C3×D4), (Q8×C9).2C2, (C3×Q8).7C6, C12.19(C2×C6), SmallGroup(144,27)

Series: Derived Chief Lower central Upper central

C1C4 — C9×Q16
C1C2C6C12C36Q8×C9 — C9×Q16
C1C2C4 — C9×Q16
C1C18C36 — C9×Q16

Generators and relations for C9×Q16
 G = < a,b,c | a9=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C12
2C12
2C36
2C36

Smallest permutation representation of C9×Q16
Regular action 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)
(1 138 54 25 42 28 59 11)(2 139 46 26 43 29 60 12)(3 140 47 27 44 30 61 13)(4 141 48 19 45 31 62 14)(5 142 49 20 37 32 63 15)(6 143 50 21 38 33 55 16)(7 144 51 22 39 34 56 17)(8 136 52 23 40 35 57 18)(9 137 53 24 41 36 58 10)(64 112 85 100 93 121 76 129)(65 113 86 101 94 122 77 130)(66 114 87 102 95 123 78 131)(67 115 88 103 96 124 79 132)(68 116 89 104 97 125 80 133)(69 117 90 105 98 126 81 134)(70 109 82 106 99 118 73 135)(71 110 83 107 91 119 74 127)(72 111 84 108 92 120 75 128)
(1 78 42 87)(2 79 43 88)(3 80 44 89)(4 81 45 90)(5 73 37 82)(6 74 38 83)(7 75 39 84)(8 76 40 85)(9 77 41 86)(10 130 24 101)(11 131 25 102)(12 132 26 103)(13 133 27 104)(14 134 19 105)(15 135 20 106)(16 127 21 107)(17 128 22 108)(18 129 23 100)(28 114 138 123)(29 115 139 124)(30 116 140 125)(31 117 141 126)(32 109 142 118)(33 110 143 119)(34 111 144 120)(35 112 136 121)(36 113 137 122)(46 96 60 67)(47 97 61 68)(48 98 62 69)(49 99 63 70)(50 91 55 71)(51 92 56 72)(52 93 57 64)(53 94 58 65)(54 95 59 66)

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

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

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

C9×Q16 is a maximal subgroup of   C9⋊SD32  C9⋊Q32  Q16⋊D9  D725C2

63 conjugacy classes

class 1  2 3A3B4A4B4C6A6B8A8B9A···9F12A12B12C12D12E12F18A···18F24A24B24C24D36A···36F36G···36R72A···72L
order123344466889···912121212121218···182424242436···3636···3672···72
size111124411221···12244441···122222···24···42···2

63 irreducible representations

dim111111111222222
type++++-
imageC1C2C2C3C6C6C9C18C18D4Q16C3×D4C3×Q16D4×C9C9×Q16
kernelC9×Q16C72Q8×C9C3×Q16C24C3×Q8Q16C8Q8C18C9C6C3C2C1
# reps11222466121224612

Matrix representation of C9×Q16 in GL2(𝔽73) generated by

550
055
,
041
1641
,
517
5068
G:=sub<GL(2,GF(73))| [55,0,0,55],[0,16,41,41],[5,50,17,68] >;

C9×Q16 in GAP, Magma, Sage, TeX

C_9\times Q_{16}
% in TeX

G:=Group("C9xQ16");
// GroupNames label

G:=SmallGroup(144,27);
// by ID

G=gap.SmallGroup(144,27);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-2,432,169,439,122,2019,1017,165]);
// Polycyclic

G:=Group<a,b,c|a^9=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×Q16 in TeX

׿
×
𝔽