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G = C12×D19order 456 = 23·3·19

Direct product of C12 and D19

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

Aliases: C12×D19, C766C6, C2284C2, D38.2C6, C6.14D38, Dic195C6, C114.14C22, C575(C2×C4), C194(C2×C12), C2.1(C6×D19), C38.10(C2×C6), (C6×D19).2C2, (C3×Dic19)⋊5C2, SmallGroup(456,25)

Series: Derived Chief Lower central Upper central

C1C19 — C12×D19
C1C19C38C114C6×D19 — C12×D19
C19 — C12×D19
C1C12

Generators and relations for C12×D19
 G = < a,b,c | a12=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

19C2
19C2
19C4
19C22
19C6
19C6
19C2×C4
19C12
19C2×C6
19C2×C12

Smallest permutation representation of C12×D19
On 228 points
Generators in S228
(1 217 68 120 83 182 20 200 46 142 101 160)(2 218 69 121 84 183 21 201 47 143 102 161)(3 219 70 122 85 184 22 202 48 144 103 162)(4 220 71 123 86 185 23 203 49 145 104 163)(5 221 72 124 87 186 24 204 50 146 105 164)(6 222 73 125 88 187 25 205 51 147 106 165)(7 223 74 126 89 188 26 206 52 148 107 166)(8 224 75 127 90 189 27 207 53 149 108 167)(9 225 76 128 91 190 28 208 54 150 109 168)(10 226 58 129 92 172 29 209 55 151 110 169)(11 227 59 130 93 173 30 191 56 152 111 170)(12 228 60 131 94 174 31 192 57 134 112 171)(13 210 61 132 95 175 32 193 39 135 113 153)(14 211 62 133 77 176 33 194 40 136 114 154)(15 212 63 115 78 177 34 195 41 137 96 155)(16 213 64 116 79 178 35 196 42 138 97 156)(17 214 65 117 80 179 36 197 43 139 98 157)(18 215 66 118 81 180 37 198 44 140 99 158)(19 216 67 119 82 181 38 199 45 141 100 159)
(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 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(53 57)(54 56)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(89 95)(90 94)(91 93)(96 105)(97 104)(98 103)(99 102)(100 101)(106 114)(107 113)(108 112)(109 111)(115 124)(116 123)(117 122)(118 121)(119 120)(125 133)(126 132)(127 131)(128 130)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(141 142)(150 152)(153 166)(154 165)(155 164)(156 163)(157 162)(158 161)(159 160)(167 171)(168 170)(173 190)(174 189)(175 188)(176 187)(177 186)(178 185)(179 184)(180 183)(181 182)(191 208)(192 207)(193 206)(194 205)(195 204)(196 203)(197 202)(198 201)(199 200)(210 223)(211 222)(212 221)(213 220)(214 219)(215 218)(216 217)(224 228)(225 227)

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

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

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

132 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F12A12B12C12D12E12F12G12H19A···19I38A···38I57A···57R76A···76R114A···114R228A···228AJ
order1222334444666666121212121212121219···1938···3857···5776···76114···114228···228
size1119191111191911191919191111191919192···22···22···22···22···22···2

132 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C3C4C6C6C6C12D19D38C3×D19C4×D19C6×D19C12×D19
kernelC12×D19C3×Dic19C228C6×D19C4×D19C3×D19Dic19C76D38D19C12C6C4C3C2C1
# reps11112422289918181836

Matrix representation of C12×D19 in GL2(𝔽37) generated by

230
023
,
3633
727
,
2730
3010
G:=sub<GL(2,GF(37))| [23,0,0,23],[36,7,33,27],[27,30,30,10] >;

C12×D19 in GAP, Magma, Sage, TeX

C_{12}\times D_{19}
% in TeX

G:=Group("C12xD19");
// GroupNames label

G:=SmallGroup(456,25);
// by ID

G=gap.SmallGroup(456,25);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-19,66,10804]);
// Polycyclic

G:=Group<a,b,c|a^12=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C12×D19 in TeX

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