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G = C3×C66order 198 = 2·32·11

Abelian group of type [3,66]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C66, SmallGroup(198,10)

Series: Derived Chief Lower central Upper central

C1 — C3×C66
C1C11C33C3×C33 — 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 85 163)(2 86 164)(3 87 165)(4 88 166)(5 89 167)(6 90 168)(7 91 169)(8 92 170)(9 93 171)(10 94 172)(11 95 173)(12 96 174)(13 97 175)(14 98 176)(15 99 177)(16 100 178)(17 101 179)(18 102 180)(19 103 181)(20 104 182)(21 105 183)(22 106 184)(23 107 185)(24 108 186)(25 109 187)(26 110 188)(27 111 189)(28 112 190)(29 113 191)(30 114 192)(31 115 193)(32 116 194)(33 117 195)(34 118 196)(35 119 197)(36 120 198)(37 121 133)(38 122 134)(39 123 135)(40 124 136)(41 125 137)(42 126 138)(43 127 139)(44 128 140)(45 129 141)(46 130 142)(47 131 143)(48 132 144)(49 67 145)(50 68 146)(51 69 147)(52 70 148)(53 71 149)(54 72 150)(55 73 151)(56 74 152)(57 75 153)(58 76 154)(59 77 155)(60 78 156)(61 79 157)(62 80 158)(63 81 159)(64 82 160)(65 83 161)(66 84 162)
(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)| (1,85,163)(2,86,164)(3,87,165)(4,88,166)(5,89,167)(6,90,168)(7,91,169)(8,92,170)(9,93,171)(10,94,172)(11,95,173)(12,96,174)(13,97,175)(14,98,176)(15,99,177)(16,100,178)(17,101,179)(18,102,180)(19,103,181)(20,104,182)(21,105,183)(22,106,184)(23,107,185)(24,108,186)(25,109,187)(26,110,188)(27,111,189)(28,112,190)(29,113,191)(30,114,192)(31,115,193)(32,116,194)(33,117,195)(34,118,196)(35,119,197)(36,120,198)(37,121,133)(38,122,134)(39,123,135)(40,124,136)(41,125,137)(42,126,138)(43,127,139)(44,128,140)(45,129,141)(46,130,142)(47,131,143)(48,132,144)(49,67,145)(50,68,146)(51,69,147)(52,70,148)(53,71,149)(54,72,150)(55,73,151)(56,74,152)(57,75,153)(58,76,154)(59,77,155)(60,78,156)(61,79,157)(62,80,158)(63,81,159)(64,82,160)(65,83,161)(66,84,162), (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( (1,85,163)(2,86,164)(3,87,165)(4,88,166)(5,89,167)(6,90,168)(7,91,169)(8,92,170)(9,93,171)(10,94,172)(11,95,173)(12,96,174)(13,97,175)(14,98,176)(15,99,177)(16,100,178)(17,101,179)(18,102,180)(19,103,181)(20,104,182)(21,105,183)(22,106,184)(23,107,185)(24,108,186)(25,109,187)(26,110,188)(27,111,189)(28,112,190)(29,113,191)(30,114,192)(31,115,193)(32,116,194)(33,117,195)(34,118,196)(35,119,197)(36,120,198)(37,121,133)(38,122,134)(39,123,135)(40,124,136)(41,125,137)(42,126,138)(43,127,139)(44,128,140)(45,129,141)(46,130,142)(47,131,143)(48,132,144)(49,67,145)(50,68,146)(51,69,147)(52,70,148)(53,71,149)(54,72,150)(55,73,151)(56,74,152)(57,75,153)(58,76,154)(59,77,155)(60,78,156)(61,79,157)(62,80,158)(63,81,159)(64,82,160)(65,83,161)(66,84,162), (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([(1,85,163),(2,86,164),(3,87,165),(4,88,166),(5,89,167),(6,90,168),(7,91,169),(8,92,170),(9,93,171),(10,94,172),(11,95,173),(12,96,174),(13,97,175),(14,98,176),(15,99,177),(16,100,178),(17,101,179),(18,102,180),(19,103,181),(20,104,182),(21,105,183),(22,106,184),(23,107,185),(24,108,186),(25,109,187),(26,110,188),(27,111,189),(28,112,190),(29,113,191),(30,114,192),(31,115,193),(32,116,194),(33,117,195),(34,118,196),(35,119,197),(36,120,198),(37,121,133),(38,122,134),(39,123,135),(40,124,136),(41,125,137),(42,126,138),(43,127,139),(44,128,140),(45,129,141),(46,130,142),(47,131,143),(48,132,144),(49,67,145),(50,68,146),(51,69,147),(52,70,148),(53,71,149),(54,72,150),(55,73,151),(56,74,152),(57,75,153),(58,76,154),(59,77,155),(60,78,156),(61,79,157),(62,80,158),(63,81,159),(64,82,160),(65,83,161),(66,84,162)], [(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···3H6A···6H11A···11J22A···22J33A···33CB66A···66CB
order123···36···611···1122···2233···3366···66
size111···11···11···11···11···11···1

198 irreducible representations

dim11111111
type++
imageC1C2C3C6C11C22C33C66
kernelC3×C66C3×C33C66C33C3×C6C32C6C3
# reps118810108080

Matrix representation of C3×C66 in GL2(𝔽67) generated by

370
037
,
310
02
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

Export

Subgroup lattice of C3×C66 in TeX

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