metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C57⋊2D4, C3⋊2D76, D38⋊2S3, Dic3⋊D19, C6.5D38, C38.5D6, D114⋊3C2, C114.5C22, (C6×D19)⋊2C2, C19⋊1(C3⋊D4), C2.5(S3×D19), (Dic3×C19)⋊3C2, SmallGroup(456,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D76
G = < a,b,c | a3=b76=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D76
►On 228 points
Generators in S228
(1 79 226)(2 227 80)(3 81 228)(4 153 82)(5 83 154)(6 155 84)(7 85 156)(8 157 86)(9 87 158)(10 159 88)(11 89 160)(12 161 90)(13 91 162)(14 163 92)(15 93 164)(16 165 94)(17 95 166)(18 167 96)(19 97 168)(20 169 98)(21 99 170)(22 171 100)(23 101 172)(24 173 102)(25 103 174)(26 175 104)(27 105 176)(28 177 106)(29 107 178)(30 179 108)(31 109 180)(32 181 110)(33 111 182)(34 183 112)(35 113 184)(36 185 114)(37 115 186)(38 187 116)(39 117 188)(40 189 118)(41 119 190)(42 191 120)(43 121 192)(44 193 122)(45 123 194)(46 195 124)(47 125 196)(48 197 126)(49 127 198)(50 199 128)(51 129 200)(52 201 130)(53 131 202)(54 203 132)(55 133 204)(56 205 134)(57 135 206)(58 207 136)(59 137 208)(60 209 138)(61 139 210)(62 211 140)(63 141 212)(64 213 142)(65 143 214)(66 215 144)(67 145 216)(68 217 146)(69 147 218)(70 219 148)(71 149 220)(72 221 150)(73 151 222)(74 223 152)(75 77 224)(76 225 78)
(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)(77 170)(78 169)(79 168)(80 167)(81 166)(82 165)(83 164)(84 163)(85 162)(86 161)(87 160)(88 159)(89 158)(90 157)(91 156)(92 155)(93 154)(94 153)(95 228)(96 227)(97 226)(98 225)(99 224)(100 223)(101 222)(102 221)(103 220)(104 219)(105 218)(106 217)(107 216)(108 215)(109 214)(110 213)(111 212)(112 211)(113 210)(114 209)(115 208)(116 207)(117 206)(118 205)(119 204)(120 203)(121 202)(122 201)(123 200)(124 199)(125 198)(126 197)(127 196)(128 195)(129 194)(130 193)(131 192)(132 191)(133 190)(134 189)(135 188)(136 187)(137 186)(138 185)(139 184)(140 183)(141 182)(142 181)(143 180)(144 179)(145 178)(146 177)(147 176)(148 175)(149 174)(150 173)(151 172)(152 171)
G:=sub<Sym(228)| (1,79,226)(2,227,80)(3,81,228)(4,153,82)(5,83,154)(6,155,84)(7,85,156)(8,157,86)(9,87,158)(10,159,88)(11,89,160)(12,161,90)(13,91,162)(14,163,92)(15,93,164)(16,165,94)(17,95,166)(18,167,96)(19,97,168)(20,169,98)(21,99,170)(22,171,100)(23,101,172)(24,173,102)(25,103,174)(26,175,104)(27,105,176)(28,177,106)(29,107,178)(30,179,108)(31,109,180)(32,181,110)(33,111,182)(34,183,112)(35,113,184)(36,185,114)(37,115,186)(38,187,116)(39,117,188)(40,189,118)(41,119,190)(42,191,120)(43,121,192)(44,193,122)(45,123,194)(46,195,124)(47,125,196)(48,197,126)(49,127,198)(50,199,128)(51,129,200)(52,201,130)(53,131,202)(54,203,132)(55,133,204)(56,205,134)(57,135,206)(58,207,136)(59,137,208)(60,209,138)(61,139,210)(62,211,140)(63,141,212)(64,213,142)(65,143,214)(66,215,144)(67,145,216)(68,217,146)(69,147,218)(70,219,148)(71,149,220)(72,221,150)(73,151,222)(74,223,152)(75,77,224)(76,225,78), (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)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,156)(92,155)(93,154)(94,153)(95,228)(96,227)(97,226)(98,225)(99,224)(100,223)(101,222)(102,221)(103,220)(104,219)(105,218)(106,217)(107,216)(108,215)(109,214)(110,213)(111,212)(112,211)(113,210)(114,209)(115,208)(116,207)(117,206)(118,205)(119,204)(120,203)(121,202)(122,201)(123,200)(124,199)(125,198)(126,197)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,182)(142,181)(143,180)(144,179)(145,178)(146,177)(147,176)(148,175)(149,174)(150,173)(151,172)(152,171)>;
G:=Group( (1,79,226)(2,227,80)(3,81,228)(4,153,82)(5,83,154)(6,155,84)(7,85,156)(8,157,86)(9,87,158)(10,159,88)(11,89,160)(12,161,90)(13,91,162)(14,163,92)(15,93,164)(16,165,94)(17,95,166)(18,167,96)(19,97,168)(20,169,98)(21,99,170)(22,171,100)(23,101,172)(24,173,102)(25,103,174)(26,175,104)(27,105,176)(28,177,106)(29,107,178)(30,179,108)(31,109,180)(32,181,110)(33,111,182)(34,183,112)(35,113,184)(36,185,114)(37,115,186)(38,187,116)(39,117,188)(40,189,118)(41,119,190)(42,191,120)(43,121,192)(44,193,122)(45,123,194)(46,195,124)(47,125,196)(48,197,126)(49,127,198)(50,199,128)(51,129,200)(52,201,130)(53,131,202)(54,203,132)(55,133,204)(56,205,134)(57,135,206)(58,207,136)(59,137,208)(60,209,138)(61,139,210)(62,211,140)(63,141,212)(64,213,142)(65,143,214)(66,215,144)(67,145,216)(68,217,146)(69,147,218)(70,219,148)(71,149,220)(72,221,150)(73,151,222)(74,223,152)(75,77,224)(76,225,78), (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)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,156)(92,155)(93,154)(94,153)(95,228)(96,227)(97,226)(98,225)(99,224)(100,223)(101,222)(102,221)(103,220)(104,219)(105,218)(106,217)(107,216)(108,215)(109,214)(110,213)(111,212)(112,211)(113,210)(114,209)(115,208)(116,207)(117,206)(118,205)(119,204)(120,203)(121,202)(122,201)(123,200)(124,199)(125,198)(126,197)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,182)(142,181)(143,180)(144,179)(145,178)(146,177)(147,176)(148,175)(149,174)(150,173)(151,172)(152,171) );
G=PermutationGroup([[(1,79,226),(2,227,80),(3,81,228),(4,153,82),(5,83,154),(6,155,84),(7,85,156),(8,157,86),(9,87,158),(10,159,88),(11,89,160),(12,161,90),(13,91,162),(14,163,92),(15,93,164),(16,165,94),(17,95,166),(18,167,96),(19,97,168),(20,169,98),(21,99,170),(22,171,100),(23,101,172),(24,173,102),(25,103,174),(26,175,104),(27,105,176),(28,177,106),(29,107,178),(30,179,108),(31,109,180),(32,181,110),(33,111,182),(34,183,112),(35,113,184),(36,185,114),(37,115,186),(38,187,116),(39,117,188),(40,189,118),(41,119,190),(42,191,120),(43,121,192),(44,193,122),(45,123,194),(46,195,124),(47,125,196),(48,197,126),(49,127,198),(50,199,128),(51,129,200),(52,201,130),(53,131,202),(54,203,132),(55,133,204),(56,205,134),(57,135,206),(58,207,136),(59,137,208),(60,209,138),(61,139,210),(62,211,140),(63,141,212),(64,213,142),(65,143,214),(66,215,144),(67,145,216),(68,217,146),(69,147,218),(70,219,148),(71,149,220),(72,221,150),(73,151,222),(74,223,152),(75,77,224),(76,225,78)], [(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),(77,170),(78,169),(79,168),(80,167),(81,166),(82,165),(83,164),(84,163),(85,162),(86,161),(87,160),(88,159),(89,158),(90,157),(91,156),(92,155),(93,154),(94,153),(95,228),(96,227),(97,226),(98,225),(99,224),(100,223),(101,222),(102,221),(103,220),(104,219),(105,218),(106,217),(107,216),(108,215),(109,214),(110,213),(111,212),(112,211),(113,210),(114,209),(115,208),(116,207),(117,206),(118,205),(119,204),(120,203),(121,202),(122,201),(123,200),(124,199),(125,198),(126,197),(127,196),(128,195),(129,194),(130,193),(131,192),(132,191),(133,190),(134,189),(135,188),(136,187),(137,186),(138,185),(139,184),(140,183),(141,182),(142,181),(143,180),(144,179),(145,178),(146,177),(147,176),(148,175),(149,174),(150,173),(151,172),(152,171)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57I | 76A | ··· | 76R | 114A | ··· | 114I |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 |
| size | 1 | 1 | 38 | 114 | 2 | 6 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 | D19 | D38 | D76 | S3×D19 | C3⋊D76 |
| kernel | C3⋊D76 | Dic3×C19 | C6×D19 | D114 | D38 | C57 | C38 | C19 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 9 | 9 |
Matrix representation of C3⋊D76 ►in GL4(𝔽229) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 228 | 1 |
| 0 | 0 | 228 | 0 |
| 51 | 125 | 0 | 0 |
| 63 | 123 | 0 | 0 |
| 0 | 0 | 126 | 166 |
| 0 | 0 | 63 | 103 |
| 106 | 125 | 0 | 0 |
| 75 | 123 | 0 | 0 |
| 0 | 0 | 0 | 228 |
| 0 | 0 | 228 | 0 |
G:=sub<GL(4,GF(229))| [1,0,0,0,0,1,0,0,0,0,228,228,0,0,1,0],[51,63,0,0,125,123,0,0,0,0,126,63,0,0,166,103],[106,75,0,0,125,123,0,0,0,0,0,228,0,0,228,0] >;
C3⋊D76 in GAP, Magma, Sage, TeX
C_3\rtimes D_{76} % in TeX
G:=Group("C3:D76"); // GroupNames label
G:=SmallGroup(456,16);
// by ID
G=gap.SmallGroup(456,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,61,26,168,10804]);
// Polycyclic
G:=Group<a,b,c|a^3=b^76=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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