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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 [×2], C3 [×4], C4, C22 [×2], C6 [×4], C6 [×8], C8, D4 [×2], C32, C12 [×4], C2×C6 [×8], C16, D8 [×2], C3×C6, C3×C6 [×2], C24 [×4], C3×D4 [×8], D16, C3×C12, C62 [×2], C48 [×4], C3×D8 [×8], C3×C24, D4×C32 [×2], C3×D16 [×4], C3×C48, C32×D8 [×2], C32×D16
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], D4, C32, C2×C6 [×4], D8, C3×C6 [×3], C3×D4 [×4], D16, C62, C3×D8 [×4], D4×C32, C3×D16 [×4], C32×D8, C32×D16

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

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

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

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

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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