metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3⋊C64, C6.C32, C96.3C2, C24.2C8, C48.3C4, C32.2S3, C12.2C16, C16.3Dic3, C2.(C3⋊C32), C8.4(C3⋊C8), C4.2(C3⋊C16), SmallGroup(192,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C3⋊C64 |
Generators and relations for C3⋊C64
G = < a,b | a3=b64=1, bab-1=a-1 >
Smallest permutation representation of C3⋊C64
►Regular action on 192 points
Generators in S192
(1 182 115)(2 116 183)(3 184 117)(4 118 185)(5 186 119)(6 120 187)(7 188 121)(8 122 189)(9 190 123)(10 124 191)(11 192 125)(12 126 129)(13 130 127)(14 128 131)(15 132 65)(16 66 133)(17 134 67)(18 68 135)(19 136 69)(20 70 137)(21 138 71)(22 72 139)(23 140 73)(24 74 141)(25 142 75)(26 76 143)(27 144 77)(28 78 145)(29 146 79)(30 80 147)(31 148 81)(32 82 149)(33 150 83)(34 84 151)(35 152 85)(36 86 153)(37 154 87)(38 88 155)(39 156 89)(40 90 157)(41 158 91)(42 92 159)(43 160 93)(44 94 161)(45 162 95)(46 96 163)(47 164 97)(48 98 165)(49 166 99)(50 100 167)(51 168 101)(52 102 169)(53 170 103)(54 104 171)(55 172 105)(56 106 173)(57 174 107)(58 108 175)(59 176 109)(60 110 177)(61 178 111)(62 112 179)(63 180 113)(64 114 181)
(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)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)
G:=sub<Sym(192)| (1,182,115)(2,116,183)(3,184,117)(4,118,185)(5,186,119)(6,120,187)(7,188,121)(8,122,189)(9,190,123)(10,124,191)(11,192,125)(12,126,129)(13,130,127)(14,128,131)(15,132,65)(16,66,133)(17,134,67)(18,68,135)(19,136,69)(20,70,137)(21,138,71)(22,72,139)(23,140,73)(24,74,141)(25,142,75)(26,76,143)(27,144,77)(28,78,145)(29,146,79)(30,80,147)(31,148,81)(32,82,149)(33,150,83)(34,84,151)(35,152,85)(36,86,153)(37,154,87)(38,88,155)(39,156,89)(40,90,157)(41,158,91)(42,92,159)(43,160,93)(44,94,161)(45,162,95)(46,96,163)(47,164,97)(48,98,165)(49,166,99)(50,100,167)(51,168,101)(52,102,169)(53,170,103)(54,104,171)(55,172,105)(56,106,173)(57,174,107)(58,108,175)(59,176,109)(60,110,177)(61,178,111)(62,112,179)(63,180,113)(64,114,181), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)>;
G:=Group( (1,182,115)(2,116,183)(3,184,117)(4,118,185)(5,186,119)(6,120,187)(7,188,121)(8,122,189)(9,190,123)(10,124,191)(11,192,125)(12,126,129)(13,130,127)(14,128,131)(15,132,65)(16,66,133)(17,134,67)(18,68,135)(19,136,69)(20,70,137)(21,138,71)(22,72,139)(23,140,73)(24,74,141)(25,142,75)(26,76,143)(27,144,77)(28,78,145)(29,146,79)(30,80,147)(31,148,81)(32,82,149)(33,150,83)(34,84,151)(35,152,85)(36,86,153)(37,154,87)(38,88,155)(39,156,89)(40,90,157)(41,158,91)(42,92,159)(43,160,93)(44,94,161)(45,162,95)(46,96,163)(47,164,97)(48,98,165)(49,166,99)(50,100,167)(51,168,101)(52,102,169)(53,170,103)(54,104,171)(55,172,105)(56,106,173)(57,174,107)(58,108,175)(59,176,109)(60,110,177)(61,178,111)(62,112,179)(63,180,113)(64,114,181), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192) );
G=PermutationGroup([[(1,182,115),(2,116,183),(3,184,117),(4,118,185),(5,186,119),(6,120,187),(7,188,121),(8,122,189),(9,190,123),(10,124,191),(11,192,125),(12,126,129),(13,130,127),(14,128,131),(15,132,65),(16,66,133),(17,134,67),(18,68,135),(19,136,69),(20,70,137),(21,138,71),(22,72,139),(23,140,73),(24,74,141),(25,142,75),(26,76,143),(27,144,77),(28,78,145),(29,146,79),(30,80,147),(31,148,81),(32,82,149),(33,150,83),(34,84,151),(35,152,85),(36,86,153),(37,154,87),(38,88,155),(39,156,89),(40,90,157),(41,158,91),(42,92,159),(43,160,93),(44,94,161),(45,162,95),(46,96,163),(47,164,97),(48,98,165),(49,166,99),(50,100,167),(51,168,101),(52,102,169),(53,170,103),(54,104,171),(55,172,105),(56,106,173),(57,174,107),(58,108,175),(59,176,109),(60,110,177),(61,178,111),(62,112,179),(63,180,113),(64,114,181)], [(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),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)]])
96 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 16A | ··· | 16H | 24A | 24B | 24C | 24D | 32A | ··· | 32P | 48A | ··· | 48H | 64A | ··· | 64AF | 96A | ··· | 96P |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 | 32 | ··· | 32 | 48 | ··· | 48 | 64 | ··· | 64 | 96 | ··· | 96 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | C64 | S3 | Dic3 | C3⋊C8 | C3⋊C16 | C3⋊C32 | C3⋊C64 |
| kernel | C3⋊C64 | C96 | C48 | C24 | C12 | C6 | C3 | C32 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 32 | 1 | 1 | 2 | 4 | 8 | 16 |
Matrix representation of C3⋊C64 ►in GL3(𝔽193) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 192 | 192 |
| 74 | 0 | 0 |
| 0 | 186 | 40 |
| 0 | 47 | 7 |
G:=sub<GL(3,GF(193))| [1,0,0,0,0,192,0,1,192],[74,0,0,0,186,47,0,40,7] >;
C3⋊C64 in GAP, Magma, Sage, TeX
C_3\rtimes C_{64} % in TeX
G:=Group("C3:C64"); // GroupNames label
G:=SmallGroup(192,1);
// by ID
G=gap.SmallGroup(192,1);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,14,36,58,80,102,6278]);
// Polycyclic
G:=Group<a,b|a^3=b^64=1,b*a*b^-1=a^-1>;
// generators/relations
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