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G = Dic3×C19order 228 = 22·3·19

Direct product of C19 and Dic3

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic3×C19, C3⋊C76, C573C4, C6.C38, C38.2S3, C114.3C2, C2.(S3×C19), SmallGroup(228,3)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C19
C1C3C6C114 — Dic3×C19
C3 — Dic3×C19
C1C38

Generators and relations for Dic3×C19
 G = < a,b,c | a19=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C76

Smallest permutation representation of Dic3×C19
Regular action 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)
(1 204 108 84 186 60)(2 205 109 85 187 61)(3 206 110 86 188 62)(4 207 111 87 189 63)(5 208 112 88 190 64)(6 209 113 89 172 65)(7 191 114 90 173 66)(8 192 96 91 174 67)(9 193 97 92 175 68)(10 194 98 93 176 69)(11 195 99 94 177 70)(12 196 100 95 178 71)(13 197 101 77 179 72)(14 198 102 78 180 73)(15 199 103 79 181 74)(16 200 104 80 182 75)(17 201 105 81 183 76)(18 202 106 82 184 58)(19 203 107 83 185 59)(20 160 148 132 57 223)(21 161 149 133 39 224)(22 162 150 115 40 225)(23 163 151 116 41 226)(24 164 152 117 42 227)(25 165 134 118 43 228)(26 166 135 119 44 210)(27 167 136 120 45 211)(28 168 137 121 46 212)(29 169 138 122 47 213)(30 170 139 123 48 214)(31 171 140 124 49 215)(32 153 141 125 50 216)(33 154 142 126 51 217)(34 155 143 127 52 218)(35 156 144 128 53 219)(36 157 145 129 54 220)(37 158 146 130 55 221)(38 159 147 131 56 222)
(1 45 84 167)(2 46 85 168)(3 47 86 169)(4 48 87 170)(5 49 88 171)(6 50 89 153)(7 51 90 154)(8 52 91 155)(9 53 92 156)(10 54 93 157)(11 55 94 158)(12 56 95 159)(13 57 77 160)(14 39 78 161)(15 40 79 162)(16 41 80 163)(17 42 81 164)(18 43 82 165)(19 44 83 166)(20 197 132 179)(21 198 133 180)(22 199 115 181)(23 200 116 182)(24 201 117 183)(25 202 118 184)(26 203 119 185)(27 204 120 186)(28 205 121 187)(29 206 122 188)(30 207 123 189)(31 208 124 190)(32 209 125 172)(33 191 126 173)(34 192 127 174)(35 193 128 175)(36 194 129 176)(37 195 130 177)(38 196 131 178)(58 228 106 134)(59 210 107 135)(60 211 108 136)(61 212 109 137)(62 213 110 138)(63 214 111 139)(64 215 112 140)(65 216 113 141)(66 217 114 142)(67 218 96 143)(68 219 97 144)(69 220 98 145)(70 221 99 146)(71 222 100 147)(72 223 101 148)(73 224 102 149)(74 225 103 150)(75 226 104 151)(76 227 105 152)

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

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

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

Dic3×C19 is a maximal subgroup of   D57⋊C4  C3⋊D76  C57⋊Q8  S3×C76

114 conjugacy classes

class 1  2  3 4A4B 6 19A···19R38A···38R57A···57R76A···76AJ114A···114R
order12344619···1938···3857···5776···76114···114
size1123321···11···12···23···32···2

114 irreducible representations

dim1111112222
type+++-
imageC1C2C4C19C38C76S3Dic3S3×C19Dic3×C19
kernelDic3×C19C114C57Dic3C6C3C38C19C2C1
# reps112181836111818

Matrix representation of Dic3×C19 in GL2(𝔽229) generated by

270
027
,
0228
11
,
18913
5340
G:=sub<GL(2,GF(229))| [27,0,0,27],[0,1,228,1],[189,53,13,40] >;

Dic3×C19 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{19}
% in TeX

G:=Group("Dic3xC19");
// GroupNames label

G:=SmallGroup(228,3);
// by ID

G=gap.SmallGroup(228,3);
# by ID

G:=PCGroup([4,-2,-19,-2,-3,152,2435]);
// Polycyclic

G:=Group<a,b,c|a^19=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Dic3×C19 in TeX

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