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## G = C32×D16order 288 = 25·32

### Direct product of C32 and D16

direct product, metacyclic, nilpotent (class 4), monomial

Aliases: C32×D16, C483C6, C8.2C62, (C3×C48)⋊5C2, C161(C3×C6), (C3×D8)⋊5C6, D81(C3×C6), (C3×C6).44D8, C6.21(C3×D8), C24.27(C2×C6), C12.45(C3×D4), (C32×D8)⋊9C2, C4.1(D4×C32), C2.3(C32×D8), (C3×C12).142D4, (C3×C24).60C22, SmallGroup(288,329)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C32×D16
 Chief series C1 — C2 — C4 — C8 — C24 — C3×C24 — C32×D8 — C32×D16
 Lower central C1 — C2 — C4 — C8 — C32×D16
 Upper central C1 — C3×C6 — C3×C12 — C3×C24 — C32×D16

Generators and relations for C32×D16
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 216 in 84 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, D4, C32, C12, C2×C6, C16, D8, C3×C6, C3×C6, C24, C3×D4, D16, C3×C12, C62, C48, C3×D8, C3×C24, D4×C32, C3×D16, C3×C48, C32×D8, C32×D16
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, D8, C3×C6, C3×D4, D16, C62, C3×D8, D4×C32, C3×D16, C32×D8, C32×D16

Smallest permutation representation of C32×D16
On 144 points
Generators in S144
(1 143 77)(2 144 78)(3 129 79)(4 130 80)(5 131 65)(6 132 66)(7 133 67)(8 134 68)(9 135 69)(10 136 70)(11 137 71)(12 138 72)(13 139 73)(14 140 74)(15 141 75)(16 142 76)(17 45 81)(18 46 82)(19 47 83)(20 48 84)(21 33 85)(22 34 86)(23 35 87)(24 36 88)(25 37 89)(26 38 90)(27 39 91)(28 40 92)(29 41 93)(30 42 94)(31 43 95)(32 44 96)(49 121 101)(50 122 102)(51 123 103)(52 124 104)(53 125 105)(54 126 106)(55 127 107)(56 128 108)(57 113 109)(58 114 110)(59 115 111)(60 116 112)(61 117 97)(62 118 98)(63 119 99)(64 120 100)
(1 27 108)(2 28 109)(3 29 110)(4 30 111)(5 31 112)(6 32 97)(7 17 98)(8 18 99)(9 19 100)(10 20 101)(11 21 102)(12 22 103)(13 23 104)(14 24 105)(15 25 106)(16 26 107)(33 50 137)(34 51 138)(35 52 139)(36 53 140)(37 54 141)(38 55 142)(39 56 143)(40 57 144)(41 58 129)(42 59 130)(43 60 131)(44 61 132)(45 62 133)(46 63 134)(47 64 135)(48 49 136)(65 95 116)(66 96 117)(67 81 118)(68 82 119)(69 83 120)(70 84 121)(71 85 122)(72 86 123)(73 87 124)(74 88 125)(75 89 126)(76 90 127)(77 91 128)(78 92 113)(79 93 114)(80 94 115)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 21)(18 20)(22 32)(23 31)(24 30)(25 29)(26 28)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(46 48)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(65 73)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)(81 85)(82 84)(86 96)(87 95)(88 94)(89 93)(90 92)(97 103)(98 102)(99 101)(104 112)(105 111)(106 110)(107 109)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(142 144)

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

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

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

99 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4 6A ··· 6H 6I ··· 6X 8A 8B 12A ··· 12H 16A 16B 16C 16D 24A ··· 24P 48A ··· 48AF order 1 2 2 2 3 ··· 3 4 6 ··· 6 6 ··· 6 8 8 12 ··· 12 16 16 16 16 24 ··· 24 48 ··· 48 size 1 1 8 8 1 ··· 1 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 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 D16 C3×D8 C3×D16 kernel C32×D16 C3×C48 C32×D8 C3×D16 C48 C3×D8 C3×C12 C3×C6 C12 C32 C6 C3 # reps 1 1 2 8 8 16 1 2 8 4 16 32

Matrix representation of C32×D16 in GL3(𝔽97) generated by

 1 0 0 0 35 0 0 0 35
,
 61 0 0 0 61 0 0 0 61
,
 96 0 0 0 95 71 0 26 95
,
 96 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(97))| [1,0,0,0,35,0,0,0,35],[61,0,0,0,61,0,0,0,61],[96,0,0,0,95,26,0,71,95],[96,0,0,0,0,1,0,1,0] >;

C32×D16 in GAP, Magma, Sage, TeX

C_3^2\times D_{16}
% in TeX

G:=Group("C3^2xD16");
// GroupNames label

G:=SmallGroup(288,329);
// by ID

G=gap.SmallGroup(288,329);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,533,3784,1901,242,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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