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## G = C3×C78order 234 = 2·32·13

### Abelian group of type [3,78]

Aliases: C3×C78, SmallGroup(234,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C78
 Chief series C1 — C13 — C39 — C3×C39 — C3×C78
 Lower central C1 — C3×C78
 Upper central C1 — C3×C78

Generators and relations for C3×C78
G = < a,b | a3=b78=1, ab=ba >

Smallest permutation representation of C3×C78
Regular action on 234 points
Generators in S234
(1 93 179)(2 94 180)(3 95 181)(4 96 182)(5 97 183)(6 98 184)(7 99 185)(8 100 186)(9 101 187)(10 102 188)(11 103 189)(12 104 190)(13 105 191)(14 106 192)(15 107 193)(16 108 194)(17 109 195)(18 110 196)(19 111 197)(20 112 198)(21 113 199)(22 114 200)(23 115 201)(24 116 202)(25 117 203)(26 118 204)(27 119 205)(28 120 206)(29 121 207)(30 122 208)(31 123 209)(32 124 210)(33 125 211)(34 126 212)(35 127 213)(36 128 214)(37 129 215)(38 130 216)(39 131 217)(40 132 218)(41 133 219)(42 134 220)(43 135 221)(44 136 222)(45 137 223)(46 138 224)(47 139 225)(48 140 226)(49 141 227)(50 142 228)(51 143 229)(52 144 230)(53 145 231)(54 146 232)(55 147 233)(56 148 234)(57 149 157)(58 150 158)(59 151 159)(60 152 160)(61 153 161)(62 154 162)(63 155 163)(64 156 164)(65 79 165)(66 80 166)(67 81 167)(68 82 168)(69 83 169)(70 84 170)(71 85 171)(72 86 172)(73 87 173)(74 88 174)(75 89 175)(76 90 176)(77 91 177)(78 92 178)
(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)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)

G:=sub<Sym(234)| (1,93,179)(2,94,180)(3,95,181)(4,96,182)(5,97,183)(6,98,184)(7,99,185)(8,100,186)(9,101,187)(10,102,188)(11,103,189)(12,104,190)(13,105,191)(14,106,192)(15,107,193)(16,108,194)(17,109,195)(18,110,196)(19,111,197)(20,112,198)(21,113,199)(22,114,200)(23,115,201)(24,116,202)(25,117,203)(26,118,204)(27,119,205)(28,120,206)(29,121,207)(30,122,208)(31,123,209)(32,124,210)(33,125,211)(34,126,212)(35,127,213)(36,128,214)(37,129,215)(38,130,216)(39,131,217)(40,132,218)(41,133,219)(42,134,220)(43,135,221)(44,136,222)(45,137,223)(46,138,224)(47,139,225)(48,140,226)(49,141,227)(50,142,228)(51,143,229)(52,144,230)(53,145,231)(54,146,232)(55,147,233)(56,148,234)(57,149,157)(58,150,158)(59,151,159)(60,152,160)(61,153,161)(62,154,162)(63,155,163)(64,156,164)(65,79,165)(66,80,166)(67,81,167)(68,82,168)(69,83,169)(70,84,170)(71,85,171)(72,86,172)(73,87,173)(74,88,174)(75,89,175)(76,90,176)(77,91,177)(78,92,178), (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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)>;

G:=Group( (1,93,179)(2,94,180)(3,95,181)(4,96,182)(5,97,183)(6,98,184)(7,99,185)(8,100,186)(9,101,187)(10,102,188)(11,103,189)(12,104,190)(13,105,191)(14,106,192)(15,107,193)(16,108,194)(17,109,195)(18,110,196)(19,111,197)(20,112,198)(21,113,199)(22,114,200)(23,115,201)(24,116,202)(25,117,203)(26,118,204)(27,119,205)(28,120,206)(29,121,207)(30,122,208)(31,123,209)(32,124,210)(33,125,211)(34,126,212)(35,127,213)(36,128,214)(37,129,215)(38,130,216)(39,131,217)(40,132,218)(41,133,219)(42,134,220)(43,135,221)(44,136,222)(45,137,223)(46,138,224)(47,139,225)(48,140,226)(49,141,227)(50,142,228)(51,143,229)(52,144,230)(53,145,231)(54,146,232)(55,147,233)(56,148,234)(57,149,157)(58,150,158)(59,151,159)(60,152,160)(61,153,161)(62,154,162)(63,155,163)(64,156,164)(65,79,165)(66,80,166)(67,81,167)(68,82,168)(69,83,169)(70,84,170)(71,85,171)(72,86,172)(73,87,173)(74,88,174)(75,89,175)(76,90,176)(77,91,177)(78,92,178), (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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234) );

G=PermutationGroup([(1,93,179),(2,94,180),(3,95,181),(4,96,182),(5,97,183),(6,98,184),(7,99,185),(8,100,186),(9,101,187),(10,102,188),(11,103,189),(12,104,190),(13,105,191),(14,106,192),(15,107,193),(16,108,194),(17,109,195),(18,110,196),(19,111,197),(20,112,198),(21,113,199),(22,114,200),(23,115,201),(24,116,202),(25,117,203),(26,118,204),(27,119,205),(28,120,206),(29,121,207),(30,122,208),(31,123,209),(32,124,210),(33,125,211),(34,126,212),(35,127,213),(36,128,214),(37,129,215),(38,130,216),(39,131,217),(40,132,218),(41,133,219),(42,134,220),(43,135,221),(44,136,222),(45,137,223),(46,138,224),(47,139,225),(48,140,226),(49,141,227),(50,142,228),(51,143,229),(52,144,230),(53,145,231),(54,146,232),(55,147,233),(56,148,234),(57,149,157),(58,150,158),(59,151,159),(60,152,160),(61,153,161),(62,154,162),(63,155,163),(64,156,164),(65,79,165),(66,80,166),(67,81,167),(68,82,168),(69,83,169),(70,84,170),(71,85,171),(72,86,172),(73,87,173),(74,88,174),(75,89,175),(76,90,176),(77,91,177),(78,92,178)], [(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),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)])

C3×C78 is a maximal subgroup of   C3⋊Dic39

234 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 13A ··· 13L 26A ··· 26L 39A ··· 39CR 78A ··· 78CR order 1 2 3 ··· 3 6 ··· 6 13 ··· 13 26 ··· 26 39 ··· 39 78 ··· 78 size 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

234 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C13 C26 C39 C78 kernel C3×C78 C3×C39 C78 C39 C3×C6 C32 C6 C3 # reps 1 1 8 8 12 12 96 96

Matrix representation of C3×C78 in GL2(𝔽79) generated by

 1 0 0 55
,
 70 0 0 11
G:=sub<GL(2,GF(79))| [1,0,0,55],[70,0,0,11] >;

C3×C78 in GAP, Magma, Sage, TeX

C_3\times C_{78}
% in TeX

G:=Group("C3xC78");
// GroupNames label

G:=SmallGroup(234,16);
// by ID

G=gap.SmallGroup(234,16);
# by ID

G:=PCGroup([4,-2,-3,-3,-13]);
// Polycyclic

G:=Group<a,b|a^3=b^78=1,a*b=b*a>;
// generators/relations

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