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## G = C9×D16order 288 = 25·32

### Direct product of C9 and D16

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C9×D16, C1445C2, C161C18, C48.2C6, D81C18, C36.37D4, C18.15D8, C72.24C22, C3.(C3×D16), (C9×D8)⋊5C2, (C3×D16).C3, C4.1(D4×C9), C2.3(C9×D8), C8.2(C2×C18), (C3×D8).2C6, C6.15(C3×D8), C24.21(C2×C6), C12.36(C3×D4), SmallGroup(288,61)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C9×D16
 Chief series C1 — C2 — C4 — C12 — C24 — C72 — C9×D8 — C9×D16
 Lower central C1 — C2 — C4 — C8 — C9×D16
 Upper central C1 — C18 — C36 — C72 — C9×D16

Generators and relations for C9×D16
G = < a,b,c | a9=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C9×D16
On 144 points
Generators in S144
(1 130 19 83 51 102 121 72 42)(2 131 20 84 52 103 122 73 43)(3 132 21 85 53 104 123 74 44)(4 133 22 86 54 105 124 75 45)(5 134 23 87 55 106 125 76 46)(6 135 24 88 56 107 126 77 47)(7 136 25 89 57 108 127 78 48)(8 137 26 90 58 109 128 79 33)(9 138 27 91 59 110 113 80 34)(10 139 28 92 60 111 114 65 35)(11 140 29 93 61 112 115 66 36)(12 141 30 94 62 97 116 67 37)(13 142 31 95 63 98 117 68 38)(14 143 32 96 64 99 118 69 39)(15 144 17 81 49 100 119 70 40)(16 129 18 82 50 101 120 71 41)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 20)(18 19)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 34)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 52)(50 51)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 80)(81 84)(82 83)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 106)(98 105)(99 104)(100 103)(101 102)(107 112)(108 111)(109 110)(113 128)(114 127)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)(129 130)(131 144)(132 143)(133 142)(134 141)(135 140)(136 139)(137 138)

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

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

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

99 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 8A 8B 9A ··· 9F 12A 12B 16A 16B 16C 16D 18A ··· 18F 18G ··· 18R 24A 24B 24C 24D 36A ··· 36F 48A ··· 48H 72A ··· 72L 144A ··· 144X order 1 2 2 2 3 3 4 6 6 6 6 6 6 8 8 9 ··· 9 12 12 16 16 16 16 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 48 ··· 48 72 ··· 72 144 ··· 144 size 1 1 8 8 1 1 2 1 1 8 8 8 8 2 2 1 ··· 1 2 2 2 2 2 2 1 ··· 1 8 ··· 8 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 D4 D8 C3×D4 D16 C3×D8 D4×C9 C3×D16 C9×D8 C9×D16 kernel C9×D16 C144 C9×D8 C3×D16 C48 C3×D8 D16 C16 D8 C36 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 6 6 12 1 2 2 4 4 6 8 12 24

Matrix representation of C9×D16 in GL2(𝔽433) generated by

 296 0 0 296
,
 260 41 392 260
,
 260 41 41 173
G:=sub<GL(2,GF(433))| [296,0,0,296],[260,392,41,260],[260,41,41,173] >;

C9×D16 in GAP, Magma, Sage, TeX

C_9\times D_{16}
% in TeX

G:=Group("C9xD16");
// GroupNames label

G:=SmallGroup(288,61);
// by ID

G=gap.SmallGroup(288,61);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-2,197,142,2355,1186,528,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c|a^9=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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