direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C9×C3⋊C16, C3⋊C144, C6.C72, C12.2C36, C24.5C18, C72.10S3, C32.2C48, C36.10Dic3, (C3×C9)⋊1C16, C8.2(S3×C9), C18.4(C3⋊C8), (C3×C72).1C2, (C3×C18).1C8, (C3×C6).7C24, (C3×C36).1C4, C24.26(C3×S3), (C3×C24).19C6, C4.2(C9×Dic3), (C3×C12).19C12, C12.22(C3×Dic3), C2.(C9×C3⋊C8), (C3×C3⋊C16).C3, C6.8(C3×C3⋊C8), C3.4(C3×C3⋊C16), SmallGroup(432,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C9×C3⋊C16 |
Generators and relations for C9×C3⋊C16
G = < a,b,c | a9=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×C3⋊C16
►On 144 points
Generators in S144
(1 124 36 69 49 90 108 139 30)(2 125 37 70 50 91 109 140 31)(3 126 38 71 51 92 110 141 32)(4 127 39 72 52 93 111 142 17)(5 128 40 73 53 94 112 143 18)(6 113 41 74 54 95 97 144 19)(7 114 42 75 55 96 98 129 20)(8 115 43 76 56 81 99 130 21)(9 116 44 77 57 82 100 131 22)(10 117 45 78 58 83 101 132 23)(11 118 46 79 59 84 102 133 24)(12 119 47 80 60 85 103 134 25)(13 120 48 65 61 86 104 135 26)(14 121 33 66 62 87 105 136 27)(15 122 34 67 63 88 106 137 28)(16 123 35 68 64 89 107 138 29)
(1 69 108)(2 109 70)(3 71 110)(4 111 72)(5 73 112)(6 97 74)(7 75 98)(8 99 76)(9 77 100)(10 101 78)(11 79 102)(12 103 80)(13 65 104)(14 105 66)(15 67 106)(16 107 68)(17 93 39)(18 40 94)(19 95 41)(20 42 96)(21 81 43)(22 44 82)(23 83 45)(24 46 84)(25 85 47)(26 48 86)(27 87 33)(28 34 88)(29 89 35)(30 36 90)(31 91 37)(32 38 92)(49 139 124)(50 125 140)(51 141 126)(52 127 142)(53 143 128)(54 113 144)(55 129 114)(56 115 130)(57 131 116)(58 117 132)(59 133 118)(60 119 134)(61 135 120)(62 121 136)(63 137 122)(64 123 138)
(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)
G:=sub<Sym(144)| (1,124,36,69,49,90,108,139,30)(2,125,37,70,50,91,109,140,31)(3,126,38,71,51,92,110,141,32)(4,127,39,72,52,93,111,142,17)(5,128,40,73,53,94,112,143,18)(6,113,41,74,54,95,97,144,19)(7,114,42,75,55,96,98,129,20)(8,115,43,76,56,81,99,130,21)(9,116,44,77,57,82,100,131,22)(10,117,45,78,58,83,101,132,23)(11,118,46,79,59,84,102,133,24)(12,119,47,80,60,85,103,134,25)(13,120,48,65,61,86,104,135,26)(14,121,33,66,62,87,105,136,27)(15,122,34,67,63,88,106,137,28)(16,123,35,68,64,89,107,138,29), (1,69,108)(2,109,70)(3,71,110)(4,111,72)(5,73,112)(6,97,74)(7,75,98)(8,99,76)(9,77,100)(10,101,78)(11,79,102)(12,103,80)(13,65,104)(14,105,66)(15,67,106)(16,107,68)(17,93,39)(18,40,94)(19,95,41)(20,42,96)(21,81,43)(22,44,82)(23,83,45)(24,46,84)(25,85,47)(26,48,86)(27,87,33)(28,34,88)(29,89,35)(30,36,90)(31,91,37)(32,38,92)(49,139,124)(50,125,140)(51,141,126)(52,127,142)(53,143,128)(54,113,144)(55,129,114)(56,115,130)(57,131,116)(58,117,132)(59,133,118)(60,119,134)(61,135,120)(62,121,136)(63,137,122)(64,123,138), (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)>;
G:=Group( (1,124,36,69,49,90,108,139,30)(2,125,37,70,50,91,109,140,31)(3,126,38,71,51,92,110,141,32)(4,127,39,72,52,93,111,142,17)(5,128,40,73,53,94,112,143,18)(6,113,41,74,54,95,97,144,19)(7,114,42,75,55,96,98,129,20)(8,115,43,76,56,81,99,130,21)(9,116,44,77,57,82,100,131,22)(10,117,45,78,58,83,101,132,23)(11,118,46,79,59,84,102,133,24)(12,119,47,80,60,85,103,134,25)(13,120,48,65,61,86,104,135,26)(14,121,33,66,62,87,105,136,27)(15,122,34,67,63,88,106,137,28)(16,123,35,68,64,89,107,138,29), (1,69,108)(2,109,70)(3,71,110)(4,111,72)(5,73,112)(6,97,74)(7,75,98)(8,99,76)(9,77,100)(10,101,78)(11,79,102)(12,103,80)(13,65,104)(14,105,66)(15,67,106)(16,107,68)(17,93,39)(18,40,94)(19,95,41)(20,42,96)(21,81,43)(22,44,82)(23,83,45)(24,46,84)(25,85,47)(26,48,86)(27,87,33)(28,34,88)(29,89,35)(30,36,90)(31,91,37)(32,38,92)(49,139,124)(50,125,140)(51,141,126)(52,127,142)(53,143,128)(54,113,144)(55,129,114)(56,115,130)(57,131,116)(58,117,132)(59,133,118)(60,119,134)(61,135,120)(62,121,136)(63,137,122)(64,123,138), (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) );
G=PermutationGroup([[(1,124,36,69,49,90,108,139,30),(2,125,37,70,50,91,109,140,31),(3,126,38,71,51,92,110,141,32),(4,127,39,72,52,93,111,142,17),(5,128,40,73,53,94,112,143,18),(6,113,41,74,54,95,97,144,19),(7,114,42,75,55,96,98,129,20),(8,115,43,76,56,81,99,130,21),(9,116,44,77,57,82,100,131,22),(10,117,45,78,58,83,101,132,23),(11,118,46,79,59,84,102,133,24),(12,119,47,80,60,85,103,134,25),(13,120,48,65,61,86,104,135,26),(14,121,33,66,62,87,105,136,27),(15,122,34,67,63,88,106,137,28),(16,123,35,68,64,89,107,138,29)], [(1,69,108),(2,109,70),(3,71,110),(4,111,72),(5,73,112),(6,97,74),(7,75,98),(8,99,76),(9,77,100),(10,101,78),(11,79,102),(12,103,80),(13,65,104),(14,105,66),(15,67,106),(16,107,68),(17,93,39),(18,40,94),(19,95,41),(20,42,96),(21,81,43),(22,44,82),(23,83,45),(24,46,84),(25,85,47),(26,48,86),(27,87,33),(28,34,88),(29,89,35),(30,36,90),(31,91,37),(32,38,92),(49,139,124),(50,125,140),(51,141,126),(52,127,142),(53,143,128),(54,113,144),(55,129,114),(56,115,130),(57,131,116),(58,117,132),(59,133,118),(60,119,134),(61,135,120),(62,121,136),(63,137,122),(64,123,138)], [(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)]])
216 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 16A | ··· | 16H | 18A | ··· | 18F | 18G | ··· | 18L | 24A | ··· | 24H | 24I | ··· | 24T | 36A | ··· | 36L | 36M | ··· | 36X | 48A | ··· | 48P | 72A | ··· | 72X | 72Y | ··· | 72AV | 144A | ··· | 144AV |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C9 | C12 | C16 | C18 | C24 | C36 | C48 | C72 | C144 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3⋊C16 | S3×C9 | C3×C3⋊C8 | C9×Dic3 | C3×C3⋊C16 | C9×C3⋊C8 | C9×C3⋊C16 |
| kernel | C9×C3⋊C16 | C3×C72 | C3×C3⋊C16 | C3×C36 | C3×C24 | C3×C18 | C3⋊C16 | C3×C12 | C3×C9 | C24 | C3×C6 | C12 | C32 | C6 | C3 | C72 | C36 | C24 | C18 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 4 | 8 | 6 | 8 | 12 | 16 | 24 | 48 | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of C9×C3⋊C16 ►in GL3(𝔽433) generated by
| 256 | 0 | 0 |
| 0 | 296 | 0 |
| 0 | 0 | 296 |
| 1 | 0 | 0 |
| 0 | 234 | 35 |
| 0 | 0 | 198 |
| 195 | 0 | 0 |
| 0 | 2 | 236 |
| 0 | 233 | 431 |
G:=sub<GL(3,GF(433))| [256,0,0,0,296,0,0,0,296],[1,0,0,0,234,0,0,35,198],[195,0,0,0,2,233,0,236,431] >;
C9×C3⋊C16 in GAP, Magma, Sage, TeX
C_9\times C_3\rtimes C_{16} % in TeX
G:=Group("C9xC3:C16"); // GroupNames label
G:=SmallGroup(432,29);
// by ID
G=gap.SmallGroup(432,29);
# by ID
G:=PCGroup([7,-2,-3,-2,-3,-2,-2,-3,42,92,142,102,14118]);
// Polycyclic
G:=Group<a,b,c|a^9=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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