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## G = C16×3- 1+2order 432 = 24·33

### Direct product of C16 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C16×3- 1+2
 Chief series C1 — C2 — C4 — C12 — C24 — C3×C24 — C8×3- 1+2 — C16×3- 1+2
 Lower central C1 — C3 — C16×3- 1+2
 Upper central C1 — C48 — C16×3- 1+2

Generators and relations for C16×3- 1+2
G = < a,b,c | a16=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

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

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

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

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

176 conjugacy classes

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

176 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 type + + image C1 C2 C3 C3 C4 C6 C6 C8 C12 C12 C16 C24 C24 C48 C48 3- 1+2 C2×3- 1+2 C4×3- 1+2 C8×3- 1+2 C16×3- 1+2 kernel C16×3- 1+2 C8×3- 1+2 C144 C3×C48 C4×3- 1+2 C72 C3×C24 C2×3- 1+2 C36 C3×C12 3- 1+2 C18 C3×C6 C9 C32 C16 C8 C4 C2 C1 # reps 1 1 6 2 2 6 2 4 12 4 8 24 8 48 16 2 2 4 8 16

Matrix representation of C16×3- 1+2 in GL4(𝔽433) generated by

 151 0 0 0 0 354 0 0 0 0 354 0 0 0 0 354
,
 234 0 0 0 0 0 1 0 0 35 199 415 0 398 37 234
,
 234 0 0 0 0 1 0 0 0 0 234 0 0 74 398 198
G:=sub<GL(4,GF(433))| [151,0,0,0,0,354,0,0,0,0,354,0,0,0,0,354],[234,0,0,0,0,0,35,398,0,1,199,37,0,0,415,234],[234,0,0,0,0,1,0,74,0,0,234,398,0,0,0,198] >;

C16×3- 1+2 in GAP, Magma, Sage, TeX

C_{16}\times 3_-^{1+2}
% in TeX

G:=Group("C16xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,36);
// by ID

G=gap.SmallGroup(432,36);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-3,-2,-2,126,260,450,192,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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