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G = C3×Dic19order 228 = 22·3·19

Direct product of C3 and Dic19

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

Aliases: C3×Dic19, C572C4, C193C12, C38.3C6, C6.2D19, C114.2C2, C2.(C3×D19), SmallGroup(228,4)

Series: Derived Chief Lower central Upper central

C1C19 — C3×Dic19
C1C19C38C114 — C3×Dic19
C19 — C3×Dic19
C1C6

Generators and relations for C3×Dic19
 G = < a,b,c | a3=b38=1, c2=b19, ab=ba, ac=ca, cbc-1=b-1 >

19C4
19C12

Smallest permutation representation of C3×Dic19
Regular action on 228 points
Generators in S228
(1 98 60)(2 99 61)(3 100 62)(4 101 63)(5 102 64)(6 103 65)(7 104 66)(8 105 67)(9 106 68)(10 107 69)(11 108 70)(12 109 71)(13 110 72)(14 111 73)(15 112 74)(16 113 75)(17 114 76)(18 77 39)(19 78 40)(20 79 41)(21 80 42)(22 81 43)(23 82 44)(24 83 45)(25 84 46)(26 85 47)(27 86 48)(28 87 49)(29 88 50)(30 89 51)(31 90 52)(32 91 53)(33 92 54)(34 93 55)(35 94 56)(36 95 57)(37 96 58)(38 97 59)(115 191 172)(116 192 173)(117 193 174)(118 194 175)(119 195 176)(120 196 177)(121 197 178)(122 198 179)(123 199 180)(124 200 181)(125 201 182)(126 202 183)(127 203 184)(128 204 185)(129 205 186)(130 206 187)(131 207 188)(132 208 189)(133 209 190)(134 210 153)(135 211 154)(136 212 155)(137 213 156)(138 214 157)(139 215 158)(140 216 159)(141 217 160)(142 218 161)(143 219 162)(144 220 163)(145 221 164)(146 222 165)(147 223 166)(148 224 167)(149 225 168)(150 226 169)(151 227 170)(152 228 171)
(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)
(1 134 20 115)(2 133 21 152)(3 132 22 151)(4 131 23 150)(5 130 24 149)(6 129 25 148)(7 128 26 147)(8 127 27 146)(9 126 28 145)(10 125 29 144)(11 124 30 143)(12 123 31 142)(13 122 32 141)(14 121 33 140)(15 120 34 139)(16 119 35 138)(17 118 36 137)(18 117 37 136)(19 116 38 135)(39 174 58 155)(40 173 59 154)(41 172 60 153)(42 171 61 190)(43 170 62 189)(44 169 63 188)(45 168 64 187)(46 167 65 186)(47 166 66 185)(48 165 67 184)(49 164 68 183)(50 163 69 182)(51 162 70 181)(52 161 71 180)(53 160 72 179)(54 159 73 178)(55 158 74 177)(56 157 75 176)(57 156 76 175)(77 193 96 212)(78 192 97 211)(79 191 98 210)(80 228 99 209)(81 227 100 208)(82 226 101 207)(83 225 102 206)(84 224 103 205)(85 223 104 204)(86 222 105 203)(87 221 106 202)(88 220 107 201)(89 219 108 200)(90 218 109 199)(91 217 110 198)(92 216 111 197)(93 215 112 196)(94 214 113 195)(95 213 114 194)

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

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

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

C3×Dic19 is a maximal subgroup of   D57⋊C4  C19⋊D12  C57⋊Q8  C12×D19

66 conjugacy classes

class 1  2 3A3B4A4B6A6B12A12B12C12D19A···19I38A···38I57A···57R114A···114R
order123344661212121219···1938···3857···57114···114
size1111191911191919192···22···22···22···2

66 irreducible representations

dim1111112222
type+++-
imageC1C2C3C4C6C12D19Dic19C3×D19C3×Dic19
kernelC3×Dic19C114Dic19C57C38C19C6C3C2C1
# reps112224991818

Matrix representation of C3×Dic19 in GL2(𝔽37) generated by

100
010
,
121
95
,
3122
06
G:=sub<GL(2,GF(37))| [10,0,0,10],[1,9,21,5],[31,0,22,6] >;

C3×Dic19 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{19}
% in TeX

G:=Group("C3xDic19");
// GroupNames label

G:=SmallGroup(228,4);
// by ID

G=gap.SmallGroup(228,4);
# by ID

G:=PCGroup([4,-2,-3,-2,-19,24,3459]);
// Polycyclic

G:=Group<a,b,c|a^3=b^38=1,c^2=b^19,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic19 in TeX

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