direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C16×3- 1+2, C144⋊C3, C9⋊2C48, C72.4C6, C32.C48, C18.2C24, C36.5C12, C48.2C32, (C3×C48).C3, (C3×C24).9C6, C3.2(C3×C48), (C3×C6).5C24, C6.3(C3×C24), C24.17(C3×C6), C12.12(C3×C12), (C3×C12).11C12, C2.(C8×3- 1+2), C4.2(C4×3- 1+2), C8.2(C2×3- 1+2), (C2×3- 1+2).2C8, (C4×3- 1+2).5C4, (C8×3- 1+2).4C2, SmallGroup(432,36)
Series: Derived ►Chief ►Lower central ►Upper central
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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