direct product, abelian, monomial, 3-elementary
Aliases: C3×C66, SmallGroup(198,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C66 |
| C1 — C3×C66 |
| C1 — C3×C66 |
Generators and relations for C3×C66
G = < a,b | a3=b66=1, ab=ba >
Smallest permutation representation of C3×C66
►Regular action on 198 points
Generators in S198
(1 164 93)(2 165 94)(3 166 95)(4 167 96)(5 168 97)(6 169 98)(7 170 99)(8 171 100)(9 172 101)(10 173 102)(11 174 103)(12 175 104)(13 176 105)(14 177 106)(15 178 107)(16 179 108)(17 180 109)(18 181 110)(19 182 111)(20 183 112)(21 184 113)(22 185 114)(23 186 115)(24 187 116)(25 188 117)(26 189 118)(27 190 119)(28 191 120)(29 192 121)(30 193 122)(31 194 123)(32 195 124)(33 196 125)(34 197 126)(35 198 127)(36 133 128)(37 134 129)(38 135 130)(39 136 131)(40 137 132)(41 138 67)(42 139 68)(43 140 69)(44 141 70)(45 142 71)(46 143 72)(47 144 73)(48 145 74)(49 146 75)(50 147 76)(51 148 77)(52 149 78)(53 150 79)(54 151 80)(55 152 81)(56 153 82)(57 154 83)(58 155 84)(59 156 85)(60 157 86)(61 158 87)(62 159 88)(63 160 89)(64 161 90)(65 162 91)(66 163 92)
(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)(133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198)
G:=sub<Sym(198)| (1,164,93)(2,165,94)(3,166,95)(4,167,96)(5,168,97)(6,169,98)(7,170,99)(8,171,100)(9,172,101)(10,173,102)(11,174,103)(12,175,104)(13,176,105)(14,177,106)(15,178,107)(16,179,108)(17,180,109)(18,181,110)(19,182,111)(20,183,112)(21,184,113)(22,185,114)(23,186,115)(24,187,116)(25,188,117)(26,189,118)(27,190,119)(28,191,120)(29,192,121)(30,193,122)(31,194,123)(32,195,124)(33,196,125)(34,197,126)(35,198,127)(36,133,128)(37,134,129)(38,135,130)(39,136,131)(40,137,132)(41,138,67)(42,139,68)(43,140,69)(44,141,70)(45,142,71)(46,143,72)(47,144,73)(48,145,74)(49,146,75)(50,147,76)(51,148,77)(52,149,78)(53,150,79)(54,151,80)(55,152,81)(56,153,82)(57,154,83)(58,155,84)(59,156,85)(60,157,86)(61,158,87)(62,159,88)(63,160,89)(64,161,90)(65,162,91)(66,163,92), (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)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)>;
G:=Group( (1,164,93)(2,165,94)(3,166,95)(4,167,96)(5,168,97)(6,169,98)(7,170,99)(8,171,100)(9,172,101)(10,173,102)(11,174,103)(12,175,104)(13,176,105)(14,177,106)(15,178,107)(16,179,108)(17,180,109)(18,181,110)(19,182,111)(20,183,112)(21,184,113)(22,185,114)(23,186,115)(24,187,116)(25,188,117)(26,189,118)(27,190,119)(28,191,120)(29,192,121)(30,193,122)(31,194,123)(32,195,124)(33,196,125)(34,197,126)(35,198,127)(36,133,128)(37,134,129)(38,135,130)(39,136,131)(40,137,132)(41,138,67)(42,139,68)(43,140,69)(44,141,70)(45,142,71)(46,143,72)(47,144,73)(48,145,74)(49,146,75)(50,147,76)(51,148,77)(52,149,78)(53,150,79)(54,151,80)(55,152,81)(56,153,82)(57,154,83)(58,155,84)(59,156,85)(60,157,86)(61,158,87)(62,159,88)(63,160,89)(64,161,90)(65,162,91)(66,163,92), (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)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198) );
G=PermutationGroup([[(1,164,93),(2,165,94),(3,166,95),(4,167,96),(5,168,97),(6,169,98),(7,170,99),(8,171,100),(9,172,101),(10,173,102),(11,174,103),(12,175,104),(13,176,105),(14,177,106),(15,178,107),(16,179,108),(17,180,109),(18,181,110),(19,182,111),(20,183,112),(21,184,113),(22,185,114),(23,186,115),(24,187,116),(25,188,117),(26,189,118),(27,190,119),(28,191,120),(29,192,121),(30,193,122),(31,194,123),(32,195,124),(33,196,125),(34,197,126),(35,198,127),(36,133,128),(37,134,129),(38,135,130),(39,136,131),(40,137,132),(41,138,67),(42,139,68),(43,140,69),(44,141,70),(45,142,71),(46,143,72),(47,144,73),(48,145,74),(49,146,75),(50,147,76),(51,148,77),(52,149,78),(53,150,79),(54,151,80),(55,152,81),(56,153,82),(57,154,83),(58,155,84),(59,156,85),(60,157,86),(61,158,87),(62,159,88),(63,160,89),(64,161,90),(65,162,91),(66,163,92)], [(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),(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)]])
C3×C66 is a maximal subgroup of
C3⋊Dic33
198 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 11A | ··· | 11J | 22A | ··· | 22J | 33A | ··· | 33CB | 66A | ··· | 66CB |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
198 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C11 | C22 | C33 | C66 |
| kernel | C3×C66 | C3×C33 | C66 | C33 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 8 | 10 | 10 | 80 | 80 |
Matrix representation of C3×C66 ►in GL2(𝔽67) generated by
| 37 | 0 |
| 0 | 37 |
| 31 | 0 |
| 0 | 2 |
G:=sub<GL(2,GF(67))| [37,0,0,37],[31,0,0,2] >;
C3×C66 in GAP, Magma, Sage, TeX
C_3\times C_{66} % in TeX
G:=Group("C3xC66"); // GroupNames label
G:=SmallGroup(198,10);
// by ID
G=gap.SmallGroup(198,10);
# by ID
G:=PCGroup([4,-2,-3,-3,-11]);
// Polycyclic
G:=Group<a,b|a^3=b^66=1,a*b=b*a>;
// generators/relations
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