direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D76, C57⋊5D4, C76⋊5C6, C228⋊3C2, D38⋊4C6, C12⋊3D19, C6.15D38, C114.15C22, C4⋊(C3×D19), C19⋊4(C3×D4), (C6×D19)⋊4C2, C2.4(C6×D19), C38.11(C2×C6), SmallGroup(456,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D76
G = < a,b,c | a3=b76=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D76
►On 228 points
Generators in S228
(1 80 200)(2 81 201)(3 82 202)(4 83 203)(5 84 204)(6 85 205)(7 86 206)(8 87 207)(9 88 208)(10 89 209)(11 90 210)(12 91 211)(13 92 212)(14 93 213)(15 94 214)(16 95 215)(17 96 216)(18 97 217)(19 98 218)(20 99 219)(21 100 220)(22 101 221)(23 102 222)(24 103 223)(25 104 224)(26 105 225)(27 106 226)(28 107 227)(29 108 228)(30 109 153)(31 110 154)(32 111 155)(33 112 156)(34 113 157)(35 114 158)(36 115 159)(37 116 160)(38 117 161)(39 118 162)(40 119 163)(41 120 164)(42 121 165)(43 122 166)(44 123 167)(45 124 168)(46 125 169)(47 126 170)(48 127 171)(49 128 172)(50 129 173)(51 130 174)(52 131 175)(53 132 176)(54 133 177)(55 134 178)(56 135 179)(57 136 180)(58 137 181)(59 138 182)(60 139 183)(61 140 184)(62 141 185)(63 142 186)(64 143 187)(65 144 188)(66 145 189)(67 146 190)(68 147 191)(69 148 192)(70 149 193)(71 150 194)(72 151 195)(73 152 196)(74 77 197)(75 78 198)(76 79 199)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(102 152)(103 151)(104 150)(105 149)(106 148)(107 147)(108 146)(109 145)(110 144)(111 143)(112 142)(113 141)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(153 189)(154 188)(155 187)(156 186)(157 185)(158 184)(159 183)(160 182)(161 181)(162 180)(163 179)(164 178)(165 177)(166 176)(167 175)(168 174)(169 173)(170 172)(190 228)(191 227)(192 226)(193 225)(194 224)(195 223)(196 222)(197 221)(198 220)(199 219)(200 218)(201 217)(202 216)(203 215)(204 214)(205 213)(206 212)(207 211)(208 210)
G:=sub<Sym(228)| (1,80,200)(2,81,201)(3,82,202)(4,83,203)(5,84,204)(6,85,205)(7,86,206)(8,87,207)(9,88,208)(10,89,209)(11,90,210)(12,91,211)(13,92,212)(14,93,213)(15,94,214)(16,95,215)(17,96,216)(18,97,217)(19,98,218)(20,99,219)(21,100,220)(22,101,221)(23,102,222)(24,103,223)(25,104,224)(26,105,225)(27,106,226)(28,107,227)(29,108,228)(30,109,153)(31,110,154)(32,111,155)(33,112,156)(34,113,157)(35,114,158)(36,115,159)(37,116,160)(38,117,161)(39,118,162)(40,119,163)(41,120,164)(42,121,165)(43,122,166)(44,123,167)(45,124,168)(46,125,169)(47,126,170)(48,127,171)(49,128,172)(50,129,173)(51,130,174)(52,131,175)(53,132,176)(54,133,177)(55,134,178)(56,135,179)(57,136,180)(58,137,181)(59,138,182)(60,139,183)(61,140,184)(62,141,185)(63,142,186)(64,143,187)(65,144,188)(66,145,189)(67,146,190)(68,147,191)(69,148,192)(70,149,193)(71,150,194)(72,151,195)(73,152,196)(74,77,197)(75,78,198)(76,79,199), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(153,189)(154,188)(155,187)(156,186)(157,185)(158,184)(159,183)(160,182)(161,181)(162,180)(163,179)(164,178)(165,177)(166,176)(167,175)(168,174)(169,173)(170,172)(190,228)(191,227)(192,226)(193,225)(194,224)(195,223)(196,222)(197,221)(198,220)(199,219)(200,218)(201,217)(202,216)(203,215)(204,214)(205,213)(206,212)(207,211)(208,210)>;
G:=Group( (1,80,200)(2,81,201)(3,82,202)(4,83,203)(5,84,204)(6,85,205)(7,86,206)(8,87,207)(9,88,208)(10,89,209)(11,90,210)(12,91,211)(13,92,212)(14,93,213)(15,94,214)(16,95,215)(17,96,216)(18,97,217)(19,98,218)(20,99,219)(21,100,220)(22,101,221)(23,102,222)(24,103,223)(25,104,224)(26,105,225)(27,106,226)(28,107,227)(29,108,228)(30,109,153)(31,110,154)(32,111,155)(33,112,156)(34,113,157)(35,114,158)(36,115,159)(37,116,160)(38,117,161)(39,118,162)(40,119,163)(41,120,164)(42,121,165)(43,122,166)(44,123,167)(45,124,168)(46,125,169)(47,126,170)(48,127,171)(49,128,172)(50,129,173)(51,130,174)(52,131,175)(53,132,176)(54,133,177)(55,134,178)(56,135,179)(57,136,180)(58,137,181)(59,138,182)(60,139,183)(61,140,184)(62,141,185)(63,142,186)(64,143,187)(65,144,188)(66,145,189)(67,146,190)(68,147,191)(69,148,192)(70,149,193)(71,150,194)(72,151,195)(73,152,196)(74,77,197)(75,78,198)(76,79,199), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(153,189)(154,188)(155,187)(156,186)(157,185)(158,184)(159,183)(160,182)(161,181)(162,180)(163,179)(164,178)(165,177)(166,176)(167,175)(168,174)(169,173)(170,172)(190,228)(191,227)(192,226)(193,225)(194,224)(195,223)(196,222)(197,221)(198,220)(199,219)(200,218)(201,217)(202,216)(203,215)(204,214)(205,213)(206,212)(207,211)(208,210) );
G=PermutationGroup([[(1,80,200),(2,81,201),(3,82,202),(4,83,203),(5,84,204),(6,85,205),(7,86,206),(8,87,207),(9,88,208),(10,89,209),(11,90,210),(12,91,211),(13,92,212),(14,93,213),(15,94,214),(16,95,215),(17,96,216),(18,97,217),(19,98,218),(20,99,219),(21,100,220),(22,101,221),(23,102,222),(24,103,223),(25,104,224),(26,105,225),(27,106,226),(28,107,227),(29,108,228),(30,109,153),(31,110,154),(32,111,155),(33,112,156),(34,113,157),(35,114,158),(36,115,159),(37,116,160),(38,117,161),(39,118,162),(40,119,163),(41,120,164),(42,121,165),(43,122,166),(44,123,167),(45,124,168),(46,125,169),(47,126,170),(48,127,171),(49,128,172),(50,129,173),(51,130,174),(52,131,175),(53,132,176),(54,133,177),(55,134,178),(56,135,179),(57,136,180),(58,137,181),(59,138,182),(60,139,183),(61,140,184),(62,141,185),(63,142,186),(64,143,187),(65,144,188),(66,145,189),(67,146,190),(68,147,191),(69,148,192),(70,149,193),(71,150,194),(72,151,195),(73,152,196),(74,77,197),(75,78,198),(76,79,199)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,76),(21,75),(22,74),(23,73),(24,72),(25,71),(26,70),(27,69),(28,68),(29,67),(30,66),(31,65),(32,64),(33,63),(34,62),(35,61),(36,60),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(77,101),(78,100),(79,99),(80,98),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(102,152),(103,151),(104,150),(105,149),(106,148),(107,147),(108,146),(109,145),(110,144),(111,143),(112,142),(113,141),(114,140),(115,139),(116,138),(117,137),(118,136),(119,135),(120,134),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128),(153,189),(154,188),(155,187),(156,186),(157,185),(158,184),(159,183),(160,182),(161,181),(162,180),(163,179),(164,178),(165,177),(166,176),(167,175),(168,174),(169,173),(170,172),(190,228),(191,227),(192,226),(193,225),(194,224),(195,223),(196,222),(197,221),(198,220),(199,219),(200,218),(201,217),(202,216),(203,215),(204,214),(205,213),(206,212),(207,211),(208,210)]])
123 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 38 | 38 | 1 | 1 | 2 | 1 | 1 | 38 | 38 | 38 | 38 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | C3×D4 | D19 | D38 | C3×D19 | D76 | C6×D19 | C3×D76 |
| kernel | C3×D76 | C228 | C6×D19 | D76 | C76 | D38 | C57 | C19 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of C3×D76 ►in GL3(𝔽229) generated by
| 94 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 228 | 0 | 0 |
| 0 | 36 | 110 |
| 0 | 119 | 141 |
| 228 | 0 | 0 |
| 0 | 134 | 104 |
| 0 | 107 | 95 |
G:=sub<GL(3,GF(229))| [94,0,0,0,1,0,0,0,1],[228,0,0,0,36,119,0,110,141],[228,0,0,0,134,107,0,104,95] >;
C3×D76 in GAP, Magma, Sage, TeX
C_3\times D_{76} % in TeX
G:=Group("C3xD76"); // GroupNames label
G:=SmallGroup(456,26);
// by ID
G=gap.SmallGroup(456,26);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,141,66,10804]);
// Polycyclic
G:=Group<a,b,c|a^3=b^76=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export