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G = C3×C69order 207 = 32·23

Abelian group of type [3,69]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C69, SmallGroup(207,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C69
C1C23C69 — C3×C69
C1 — C3×C69
C1 — C3×C69

Generators and relations for C3×C69
 G = < a,b | a3=b69=1, ab=ba >


Smallest permutation representation of C3×C69
Regular action on 207 points
Generators in S207
(1 192 115)(2 193 116)(3 194 117)(4 195 118)(5 196 119)(6 197 120)(7 198 121)(8 199 122)(9 200 123)(10 201 124)(11 202 125)(12 203 126)(13 204 127)(14 205 128)(15 206 129)(16 207 130)(17 139 131)(18 140 132)(19 141 133)(20 142 134)(21 143 135)(22 144 136)(23 145 137)(24 146 138)(25 147 70)(26 148 71)(27 149 72)(28 150 73)(29 151 74)(30 152 75)(31 153 76)(32 154 77)(33 155 78)(34 156 79)(35 157 80)(36 158 81)(37 159 82)(38 160 83)(39 161 84)(40 162 85)(41 163 86)(42 164 87)(43 165 88)(44 166 89)(45 167 90)(46 168 91)(47 169 92)(48 170 93)(49 171 94)(50 172 95)(51 173 96)(52 174 97)(53 175 98)(54 176 99)(55 177 100)(56 178 101)(57 179 102)(58 180 103)(59 181 104)(60 182 105)(61 183 106)(62 184 107)(63 185 108)(64 186 109)(65 187 110)(66 188 111)(67 189 112)(68 190 113)(69 191 114)
(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)(139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207)

G:=sub<Sym(207)| (1,192,115)(2,193,116)(3,194,117)(4,195,118)(5,196,119)(6,197,120)(7,198,121)(8,199,122)(9,200,123)(10,201,124)(11,202,125)(12,203,126)(13,204,127)(14,205,128)(15,206,129)(16,207,130)(17,139,131)(18,140,132)(19,141,133)(20,142,134)(21,143,135)(22,144,136)(23,145,137)(24,146,138)(25,147,70)(26,148,71)(27,149,72)(28,150,73)(29,151,74)(30,152,75)(31,153,76)(32,154,77)(33,155,78)(34,156,79)(35,157,80)(36,158,81)(37,159,82)(38,160,83)(39,161,84)(40,162,85)(41,163,86)(42,164,87)(43,165,88)(44,166,89)(45,167,90)(46,168,91)(47,169,92)(48,170,93)(49,171,94)(50,172,95)(51,173,96)(52,174,97)(53,175,98)(54,176,99)(55,177,100)(56,178,101)(57,179,102)(58,180,103)(59,181,104)(60,182,105)(61,183,106)(62,184,107)(63,185,108)(64,186,109)(65,187,110)(66,188,111)(67,189,112)(68,190,113)(69,191,114), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207)>;

G:=Group( (1,192,115)(2,193,116)(3,194,117)(4,195,118)(5,196,119)(6,197,120)(7,198,121)(8,199,122)(9,200,123)(10,201,124)(11,202,125)(12,203,126)(13,204,127)(14,205,128)(15,206,129)(16,207,130)(17,139,131)(18,140,132)(19,141,133)(20,142,134)(21,143,135)(22,144,136)(23,145,137)(24,146,138)(25,147,70)(26,148,71)(27,149,72)(28,150,73)(29,151,74)(30,152,75)(31,153,76)(32,154,77)(33,155,78)(34,156,79)(35,157,80)(36,158,81)(37,159,82)(38,160,83)(39,161,84)(40,162,85)(41,163,86)(42,164,87)(43,165,88)(44,166,89)(45,167,90)(46,168,91)(47,169,92)(48,170,93)(49,171,94)(50,172,95)(51,173,96)(52,174,97)(53,175,98)(54,176,99)(55,177,100)(56,178,101)(57,179,102)(58,180,103)(59,181,104)(60,182,105)(61,183,106)(62,184,107)(63,185,108)(64,186,109)(65,187,110)(66,188,111)(67,189,112)(68,190,113)(69,191,114), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207) );

G=PermutationGroup([(1,192,115),(2,193,116),(3,194,117),(4,195,118),(5,196,119),(6,197,120),(7,198,121),(8,199,122),(9,200,123),(10,201,124),(11,202,125),(12,203,126),(13,204,127),(14,205,128),(15,206,129),(16,207,130),(17,139,131),(18,140,132),(19,141,133),(20,142,134),(21,143,135),(22,144,136),(23,145,137),(24,146,138),(25,147,70),(26,148,71),(27,149,72),(28,150,73),(29,151,74),(30,152,75),(31,153,76),(32,154,77),(33,155,78),(34,156,79),(35,157,80),(36,158,81),(37,159,82),(38,160,83),(39,161,84),(40,162,85),(41,163,86),(42,164,87),(43,165,88),(44,166,89),(45,167,90),(46,168,91),(47,169,92),(48,170,93),(49,171,94),(50,172,95),(51,173,96),(52,174,97),(53,175,98),(54,176,99),(55,177,100),(56,178,101),(57,179,102),(58,180,103),(59,181,104),(60,182,105),(61,183,106),(62,184,107),(63,185,108),(64,186,109),(65,187,110),(66,188,111),(67,189,112),(68,190,113),(69,191,114)], [(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),(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207)])

C3×C69 is a maximal subgroup of   C3⋊D69

207 conjugacy classes

class 1 3A···3H23A···23V69A···69FT
order13···323···2369···69
size11···11···11···1

207 irreducible representations

dim1111
type+
imageC1C3C23C69
kernelC3×C69C69C32C3
# reps1822176

Matrix representation of C3×C69 in GL2(𝔽139) generated by

420
042
,
450
0136
G:=sub<GL(2,GF(139))| [42,0,0,42],[45,0,0,136] >;

C3×C69 in GAP, Magma, Sage, TeX

C_3\times C_{69}
% in TeX

G:=Group("C3xC69");
// GroupNames label

G:=SmallGroup(207,2);
// by ID

G=gap.SmallGroup(207,2);
# by ID

G:=PCGroup([3,-3,-3,-23]);
// Polycyclic

G:=Group<a,b|a^3=b^69=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C69 in TeX

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