direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C9⋊C16, C9⋊3C48, C72.6C6, C36.8C12, C18.3C24, C24.10D9, C12.10Dic9, (C3×C9)⋊2C16, C6.4(C9⋊C8), C8.2(C3×D9), (C3×C72).4C2, (C3×C18).2C8, (C3×C36).5C4, C24.14(C3×S3), (C3×C24).20S3, C4.2(C3×Dic9), C32.2(C3⋊C16), C12.8(C3×Dic3), (C3×C12).21Dic3, C2.(C3×C9⋊C8), C6.1(C3×C3⋊C8), C3.1(C3×C3⋊C16), (C3×C6).7(C3⋊C8), SmallGroup(432,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C3×C9⋊C16 |
Generators and relations for C3×C9⋊C16
G = < a,b,c | a3=b9=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C9⋊C16
►On 144 points
Generators in S144
(1 56 92)(2 57 93)(3 58 94)(4 59 95)(5 60 96)(6 61 81)(7 62 82)(8 63 83)(9 64 84)(10 49 85)(11 50 86)(12 51 87)(13 52 88)(14 53 89)(15 54 90)(16 55 91)(17 126 131)(18 127 132)(19 128 133)(20 113 134)(21 114 135)(22 115 136)(23 116 137)(24 117 138)(25 118 139)(26 119 140)(27 120 141)(28 121 142)(29 122 143)(30 123 144)(31 124 129)(32 125 130)(33 70 97)(34 71 98)(35 72 99)(36 73 100)(37 74 101)(38 75 102)(39 76 103)(40 77 104)(41 78 105)(42 79 106)(43 80 107)(44 65 108)(45 66 109)(46 67 110)(47 68 111)(48 69 112)
(1 23 70 92 137 33 56 116 97)(2 98 117 57 34 138 93 71 24)(3 25 72 94 139 35 58 118 99)(4 100 119 59 36 140 95 73 26)(5 27 74 96 141 37 60 120 101)(6 102 121 61 38 142 81 75 28)(7 29 76 82 143 39 62 122 103)(8 104 123 63 40 144 83 77 30)(9 31 78 84 129 41 64 124 105)(10 106 125 49 42 130 85 79 32)(11 17 80 86 131 43 50 126 107)(12 108 127 51 44 132 87 65 18)(13 19 66 88 133 45 52 128 109)(14 110 113 53 46 134 89 67 20)(15 21 68 90 135 47 54 114 111)(16 112 115 55 48 136 91 69 22)
(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)| (1,56,92)(2,57,93)(3,58,94)(4,59,95)(5,60,96)(6,61,81)(7,62,82)(8,63,83)(9,64,84)(10,49,85)(11,50,86)(12,51,87)(13,52,88)(14,53,89)(15,54,90)(16,55,91)(17,126,131)(18,127,132)(19,128,133)(20,113,134)(21,114,135)(22,115,136)(23,116,137)(24,117,138)(25,118,139)(26,119,140)(27,120,141)(28,121,142)(29,122,143)(30,123,144)(31,124,129)(32,125,130)(33,70,97)(34,71,98)(35,72,99)(36,73,100)(37,74,101)(38,75,102)(39,76,103)(40,77,104)(41,78,105)(42,79,106)(43,80,107)(44,65,108)(45,66,109)(46,67,110)(47,68,111)(48,69,112), (1,23,70,92,137,33,56,116,97)(2,98,117,57,34,138,93,71,24)(3,25,72,94,139,35,58,118,99)(4,100,119,59,36,140,95,73,26)(5,27,74,96,141,37,60,120,101)(6,102,121,61,38,142,81,75,28)(7,29,76,82,143,39,62,122,103)(8,104,123,63,40,144,83,77,30)(9,31,78,84,129,41,64,124,105)(10,106,125,49,42,130,85,79,32)(11,17,80,86,131,43,50,126,107)(12,108,127,51,44,132,87,65,18)(13,19,66,88,133,45,52,128,109)(14,110,113,53,46,134,89,67,20)(15,21,68,90,135,47,54,114,111)(16,112,115,55,48,136,91,69,22), (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( (1,56,92)(2,57,93)(3,58,94)(4,59,95)(5,60,96)(6,61,81)(7,62,82)(8,63,83)(9,64,84)(10,49,85)(11,50,86)(12,51,87)(13,52,88)(14,53,89)(15,54,90)(16,55,91)(17,126,131)(18,127,132)(19,128,133)(20,113,134)(21,114,135)(22,115,136)(23,116,137)(24,117,138)(25,118,139)(26,119,140)(27,120,141)(28,121,142)(29,122,143)(30,123,144)(31,124,129)(32,125,130)(33,70,97)(34,71,98)(35,72,99)(36,73,100)(37,74,101)(38,75,102)(39,76,103)(40,77,104)(41,78,105)(42,79,106)(43,80,107)(44,65,108)(45,66,109)(46,67,110)(47,68,111)(48,69,112), (1,23,70,92,137,33,56,116,97)(2,98,117,57,34,138,93,71,24)(3,25,72,94,139,35,58,118,99)(4,100,119,59,36,140,95,73,26)(5,27,74,96,141,37,60,120,101)(6,102,121,61,38,142,81,75,28)(7,29,76,82,143,39,62,122,103)(8,104,123,63,40,144,83,77,30)(9,31,78,84,129,41,64,124,105)(10,106,125,49,42,130,85,79,32)(11,17,80,86,131,43,50,126,107)(12,108,127,51,44,132,87,65,18)(13,19,66,88,133,45,52,128,109)(14,110,113,53,46,134,89,67,20)(15,21,68,90,135,47,54,114,111)(16,112,115,55,48,136,91,69,22), (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([[(1,56,92),(2,57,93),(3,58,94),(4,59,95),(5,60,96),(6,61,81),(7,62,82),(8,63,83),(9,64,84),(10,49,85),(11,50,86),(12,51,87),(13,52,88),(14,53,89),(15,54,90),(16,55,91),(17,126,131),(18,127,132),(19,128,133),(20,113,134),(21,114,135),(22,115,136),(23,116,137),(24,117,138),(25,118,139),(26,119,140),(27,120,141),(28,121,142),(29,122,143),(30,123,144),(31,124,129),(32,125,130),(33,70,97),(34,71,98),(35,72,99),(36,73,100),(37,74,101),(38,75,102),(39,76,103),(40,77,104),(41,78,105),(42,79,106),(43,80,107),(44,65,108),(45,66,109),(46,67,110),(47,68,111),(48,69,112)], [(1,23,70,92,137,33,56,116,97),(2,98,117,57,34,138,93,71,24),(3,25,72,94,139,35,58,118,99),(4,100,119,59,36,140,95,73,26),(5,27,74,96,141,37,60,120,101),(6,102,121,61,38,142,81,75,28),(7,29,76,82,143,39,62,122,103),(8,104,123,63,40,144,83,77,30),(9,31,78,84,129,41,64,124,105),(10,106,125,49,42,130,85,79,32),(11,17,80,86,131,43,50,126,107),(12,108,127,51,44,132,87,65,18),(13,19,66,88,133,45,52,128,109),(14,110,113,53,46,134,89,67,20),(15,21,68,90,135,47,54,114,111),(16,112,115,55,48,136,91,69,22)], [(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)]])
144 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 16A | ··· | 16H | 18A | ··· | 18I | 24A | ··· | 24H | 24I | ··· | 24T | 36A | ··· | 36R | 48A | ··· | 48P | 72A | ··· | 72AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | S3 | Dic3 | D9 | C3×S3 | C3⋊C8 | Dic9 | C3×Dic3 | C3⋊C16 | C3×D9 | C9⋊C8 | C3×C3⋊C8 | C3×Dic9 | C9⋊C16 | C3×C3⋊C16 | C3×C9⋊C8 | C3×C9⋊C16 |
| kernel | C3×C9⋊C16 | C3×C72 | C9⋊C16 | C3×C36 | C72 | C3×C18 | C36 | C3×C9 | C18 | C9 | C3×C24 | C3×C12 | C24 | C24 | C3×C6 | C12 | C12 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 1 | 1 | 3 | 2 | 2 | 3 | 2 | 4 | 6 | 6 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C3×C9⋊C16 ►in GL3(𝔽433) generated by
| 234 | 0 | 0 |
| 0 | 198 | 0 |
| 0 | 0 | 198 |
| 1 | 0 | 0 |
| 0 | 417 | 0 |
| 0 | 0 | 27 |
| 238 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 432 | 0 |
G:=sub<GL(3,GF(433))| [234,0,0,0,198,0,0,0,198],[1,0,0,0,417,0,0,0,27],[238,0,0,0,0,432,0,1,0] >;
C3×C9⋊C16 in GAP, Magma, Sage, TeX
C_3\times C_9\rtimes C_{16} % in TeX
G:=Group("C3xC9:C16"); // GroupNames label
G:=SmallGroup(432,28);
// by ID
G=gap.SmallGroup(432,28);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,-2,-3,-3,42,58,80,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^3=b^9=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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