direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C76, D6.C38, C228⋊6C2, C12⋊2C38, C38.14D6, Dic3⋊2C38, C114.19C22, C3⋊1(C2×C76), C57⋊6(C2×C4), C2.1(S3×C38), C6.2(C2×C38), (S3×C38).2C2, (Dic3×C19)⋊5C2, SmallGroup(456,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C76 |
Generators and relations for S3×C76
G = < a,b,c | a76=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C76
►On 228 points
Generators in S228
(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)
(1 220 145)(2 221 146)(3 222 147)(4 223 148)(5 224 149)(6 225 150)(7 226 151)(8 227 152)(9 228 77)(10 153 78)(11 154 79)(12 155 80)(13 156 81)(14 157 82)(15 158 83)(16 159 84)(17 160 85)(18 161 86)(19 162 87)(20 163 88)(21 164 89)(22 165 90)(23 166 91)(24 167 92)(25 168 93)(26 169 94)(27 170 95)(28 171 96)(29 172 97)(30 173 98)(31 174 99)(32 175 100)(33 176 101)(34 177 102)(35 178 103)(36 179 104)(37 180 105)(38 181 106)(39 182 107)(40 183 108)(41 184 109)(42 185 110)(43 186 111)(44 187 112)(45 188 113)(46 189 114)(47 190 115)(48 191 116)(49 192 117)(50 193 118)(51 194 119)(52 195 120)(53 196 121)(54 197 122)(55 198 123)(56 199 124)(57 200 125)(58 201 126)(59 202 127)(60 203 128)(61 204 129)(62 205 130)(63 206 131)(64 207 132)(65 208 133)(66 209 134)(67 210 135)(68 211 136)(69 212 137)(70 213 138)(71 214 139)(72 215 140)(73 216 141)(74 217 142)(75 218 143)(76 219 144)
(77 228)(78 153)(79 154)(80 155)(81 156)(82 157)(83 158)(84 159)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 169)(95 170)(96 171)(97 172)(98 173)(99 174)(100 175)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 183)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(115 190)(116 191)(117 192)(118 193)(119 194)(120 195)(121 196)(122 197)(123 198)(124 199)(125 200)(126 201)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 208)(134 209)(135 210)(136 211)(137 212)(138 213)(139 214)(140 215)(141 216)(142 217)(143 218)(144 219)(145 220)(146 221)(147 222)(148 223)(149 224)(150 225)(151 226)(152 227)
G:=sub<Sym(228)| (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), (1,220,145)(2,221,146)(3,222,147)(4,223,148)(5,224,149)(6,225,150)(7,226,151)(8,227,152)(9,228,77)(10,153,78)(11,154,79)(12,155,80)(13,156,81)(14,157,82)(15,158,83)(16,159,84)(17,160,85)(18,161,86)(19,162,87)(20,163,88)(21,164,89)(22,165,90)(23,166,91)(24,167,92)(25,168,93)(26,169,94)(27,170,95)(28,171,96)(29,172,97)(30,173,98)(31,174,99)(32,175,100)(33,176,101)(34,177,102)(35,178,103)(36,179,104)(37,180,105)(38,181,106)(39,182,107)(40,183,108)(41,184,109)(42,185,110)(43,186,111)(44,187,112)(45,188,113)(46,189,114)(47,190,115)(48,191,116)(49,192,117)(50,193,118)(51,194,119)(52,195,120)(53,196,121)(54,197,122)(55,198,123)(56,199,124)(57,200,125)(58,201,126)(59,202,127)(60,203,128)(61,204,129)(62,205,130)(63,206,131)(64,207,132)(65,208,133)(66,209,134)(67,210,135)(68,211,136)(69,212,137)(70,213,138)(71,214,139)(72,215,140)(73,216,141)(74,217,142)(75,218,143)(76,219,144), (77,228)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,171)(97,172)(98,173)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227)>;
G:=Group( (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), (1,220,145)(2,221,146)(3,222,147)(4,223,148)(5,224,149)(6,225,150)(7,226,151)(8,227,152)(9,228,77)(10,153,78)(11,154,79)(12,155,80)(13,156,81)(14,157,82)(15,158,83)(16,159,84)(17,160,85)(18,161,86)(19,162,87)(20,163,88)(21,164,89)(22,165,90)(23,166,91)(24,167,92)(25,168,93)(26,169,94)(27,170,95)(28,171,96)(29,172,97)(30,173,98)(31,174,99)(32,175,100)(33,176,101)(34,177,102)(35,178,103)(36,179,104)(37,180,105)(38,181,106)(39,182,107)(40,183,108)(41,184,109)(42,185,110)(43,186,111)(44,187,112)(45,188,113)(46,189,114)(47,190,115)(48,191,116)(49,192,117)(50,193,118)(51,194,119)(52,195,120)(53,196,121)(54,197,122)(55,198,123)(56,199,124)(57,200,125)(58,201,126)(59,202,127)(60,203,128)(61,204,129)(62,205,130)(63,206,131)(64,207,132)(65,208,133)(66,209,134)(67,210,135)(68,211,136)(69,212,137)(70,213,138)(71,214,139)(72,215,140)(73,216,141)(74,217,142)(75,218,143)(76,219,144), (77,228)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,171)(97,172)(98,173)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,225)(151,226)(152,227) );
G=PermutationGroup([[(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)], [(1,220,145),(2,221,146),(3,222,147),(4,223,148),(5,224,149),(6,225,150),(7,226,151),(8,227,152),(9,228,77),(10,153,78),(11,154,79),(12,155,80),(13,156,81),(14,157,82),(15,158,83),(16,159,84),(17,160,85),(18,161,86),(19,162,87),(20,163,88),(21,164,89),(22,165,90),(23,166,91),(24,167,92),(25,168,93),(26,169,94),(27,170,95),(28,171,96),(29,172,97),(30,173,98),(31,174,99),(32,175,100),(33,176,101),(34,177,102),(35,178,103),(36,179,104),(37,180,105),(38,181,106),(39,182,107),(40,183,108),(41,184,109),(42,185,110),(43,186,111),(44,187,112),(45,188,113),(46,189,114),(47,190,115),(48,191,116),(49,192,117),(50,193,118),(51,194,119),(52,195,120),(53,196,121),(54,197,122),(55,198,123),(56,199,124),(57,200,125),(58,201,126),(59,202,127),(60,203,128),(61,204,129),(62,205,130),(63,206,131),(64,207,132),(65,208,133),(66,209,134),(67,210,135),(68,211,136),(69,212,137),(70,213,138),(71,214,139),(72,215,140),(73,216,141),(74,217,142),(75,218,143),(76,219,144)], [(77,228),(78,153),(79,154),(80,155),(81,156),(82,157),(83,158),(84,159),(85,160),(86,161),(87,162),(88,163),(89,164),(90,165),(91,166),(92,167),(93,168),(94,169),(95,170),(96,171),(97,172),(98,173),(99,174),(100,175),(101,176),(102,177),(103,178),(104,179),(105,180),(106,181),(107,182),(108,183),(109,184),(110,185),(111,186),(112,187),(113,188),(114,189),(115,190),(116,191),(117,192),(118,193),(119,194),(120,195),(121,196),(122,197),(123,198),(124,199),(125,200),(126,201),(127,202),(128,203),(129,204),(130,205),(131,206),(132,207),(133,208),(134,209),(135,210),(136,211),(137,212),(138,213),(139,214),(140,215),(141,216),(142,217),(143,218),(144,219),(145,220),(146,221),(147,222),(148,223),(149,224),(150,225),(151,226),(152,227)]])
228 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 19A | ··· | 19R | 38A | ··· | 38R | 38S | ··· | 38BB | 57A | ··· | 57R | 76A | ··· | 76AJ | 76AK | ··· | 76BT | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
228 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C19 | C38 | C38 | C38 | C76 | S3 | D6 | C4×S3 | S3×C19 | S3×C38 | S3×C76 |
| kernel | S3×C76 | Dic3×C19 | C228 | S3×C38 | S3×C19 | C4×S3 | Dic3 | C12 | D6 | S3 | C76 | C38 | C19 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 18 | 18 | 18 | 18 | 72 | 1 | 1 | 2 | 18 | 18 | 36 |
Matrix representation of S3×C76 ►in GL3(𝔽229) generated by
| 107 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 61 |
| 1 | 0 | 0 |
| 0 | 0 | 228 |
| 0 | 1 | 228 |
| 228 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(229))| [107,0,0,0,61,0,0,0,61],[1,0,0,0,0,1,0,228,228],[228,0,0,0,0,1,0,1,0] >;
S3×C76 in GAP, Magma, Sage, TeX
S_3\times C_{76} % in TeX
G:=Group("S3xC76"); // GroupNames label
G:=SmallGroup(456,30);
// by ID
G=gap.SmallGroup(456,30);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-3,386,7604]);
// Polycyclic
G:=Group<a,b,c|a^76=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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