metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C57⋊3D4, C19⋊2D12, D6⋊2D19, Dic19⋊S3, C6.6D38, C38.6D6, D114⋊4C2, C114.6C22, (S3×C38)⋊2C2, C3⋊1(C19⋊D4), C2.6(S3×D19), (C3×Dic19)⋊3C2, SmallGroup(456,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C19⋊D12
G = < a,b,c | a19=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C19⋊D12
►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)
(1 158 101 178 20 139 62 210 43 129 94 194)(2 157 102 177 21 138 63 228 44 128 95 193)(3 156 103 176 22 137 64 227 45 127 77 192)(4 155 104 175 23 136 65 226 46 126 78 191)(5 154 105 174 24 135 66 225 47 125 79 209)(6 153 106 173 25 134 67 224 48 124 80 208)(7 171 107 172 26 152 68 223 49 123 81 207)(8 170 108 190 27 151 69 222 50 122 82 206)(9 169 109 189 28 150 70 221 51 121 83 205)(10 168 110 188 29 149 71 220 52 120 84 204)(11 167 111 187 30 148 72 219 53 119 85 203)(12 166 112 186 31 147 73 218 54 118 86 202)(13 165 113 185 32 146 74 217 55 117 87 201)(14 164 114 184 33 145 75 216 56 116 88 200)(15 163 96 183 34 144 76 215 57 115 89 199)(16 162 97 182 35 143 58 214 39 133 90 198)(17 161 98 181 36 142 59 213 40 132 91 197)(18 160 99 180 37 141 60 212 41 131 92 196)(19 159 100 179 38 140 61 211 42 130 93 195)
(1 43)(2 42)(3 41)(4 40)(5 39)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(58 105)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 114)(69 113)(70 112)(71 111)(72 110)(73 109)(74 108)(75 107)(76 106)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)(93 95)(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)(133 209)(134 183)(135 182)(136 181)(137 180)(138 179)(139 178)(140 177)(141 176)(142 175)(143 174)(144 173)(145 172)(146 190)(147 189)(148 188)(149 187)(150 186)(151 185)(152 184)(153 215)(154 214)(155 213)(156 212)(157 211)(158 210)(159 228)(160 227)(161 226)(162 225)(163 224)(164 223)(165 222)(166 221)(167 220)(168 219)(169 218)(170 217)(171 216)
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), (1,158,101,178,20,139,62,210,43,129,94,194)(2,157,102,177,21,138,63,228,44,128,95,193)(3,156,103,176,22,137,64,227,45,127,77,192)(4,155,104,175,23,136,65,226,46,126,78,191)(5,154,105,174,24,135,66,225,47,125,79,209)(6,153,106,173,25,134,67,224,48,124,80,208)(7,171,107,172,26,152,68,223,49,123,81,207)(8,170,108,190,27,151,69,222,50,122,82,206)(9,169,109,189,28,150,70,221,51,121,83,205)(10,168,110,188,29,149,71,220,52,120,84,204)(11,167,111,187,30,148,72,219,53,119,85,203)(12,166,112,186,31,147,73,218,54,118,86,202)(13,165,113,185,32,146,74,217,55,117,87,201)(14,164,114,184,33,145,75,216,56,116,88,200)(15,163,96,183,34,144,76,215,57,115,89,199)(16,162,97,182,35,143,58,214,39,133,90,198)(17,161,98,181,36,142,59,213,40,132,91,197)(18,160,99,180,37,141,60,212,41,131,92,196)(19,159,100,179,38,140,61,211,42,130,93,195), (1,43)(2,42)(3,41)(4,40)(5,39)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,114)(69,113)(70,112)(71,111)(72,110)(73,109)(74,108)(75,107)(76,106)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(93,95)(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)(133,209)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,190)(147,189)(148,188)(149,187)(150,186)(151,185)(152,184)(153,215)(154,214)(155,213)(156,212)(157,211)(158,210)(159,228)(160,227)(161,226)(162,225)(163,224)(164,223)(165,222)(166,221)(167,220)(168,219)(169,218)(170,217)(171,216)>;
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), (1,158,101,178,20,139,62,210,43,129,94,194)(2,157,102,177,21,138,63,228,44,128,95,193)(3,156,103,176,22,137,64,227,45,127,77,192)(4,155,104,175,23,136,65,226,46,126,78,191)(5,154,105,174,24,135,66,225,47,125,79,209)(6,153,106,173,25,134,67,224,48,124,80,208)(7,171,107,172,26,152,68,223,49,123,81,207)(8,170,108,190,27,151,69,222,50,122,82,206)(9,169,109,189,28,150,70,221,51,121,83,205)(10,168,110,188,29,149,71,220,52,120,84,204)(11,167,111,187,30,148,72,219,53,119,85,203)(12,166,112,186,31,147,73,218,54,118,86,202)(13,165,113,185,32,146,74,217,55,117,87,201)(14,164,114,184,33,145,75,216,56,116,88,200)(15,163,96,183,34,144,76,215,57,115,89,199)(16,162,97,182,35,143,58,214,39,133,90,198)(17,161,98,181,36,142,59,213,40,132,91,197)(18,160,99,180,37,141,60,212,41,131,92,196)(19,159,100,179,38,140,61,211,42,130,93,195), (1,43)(2,42)(3,41)(4,40)(5,39)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,114)(69,113)(70,112)(71,111)(72,110)(73,109)(74,108)(75,107)(76,106)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(93,95)(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)(133,209)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,190)(147,189)(148,188)(149,187)(150,186)(151,185)(152,184)(153,215)(154,214)(155,213)(156,212)(157,211)(158,210)(159,228)(160,227)(161,226)(162,225)(163,224)(164,223)(165,222)(166,221)(167,220)(168,219)(169,218)(170,217)(171,216) );
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)], [(1,158,101,178,20,139,62,210,43,129,94,194),(2,157,102,177,21,138,63,228,44,128,95,193),(3,156,103,176,22,137,64,227,45,127,77,192),(4,155,104,175,23,136,65,226,46,126,78,191),(5,154,105,174,24,135,66,225,47,125,79,209),(6,153,106,173,25,134,67,224,48,124,80,208),(7,171,107,172,26,152,68,223,49,123,81,207),(8,170,108,190,27,151,69,222,50,122,82,206),(9,169,109,189,28,150,70,221,51,121,83,205),(10,168,110,188,29,149,71,220,52,120,84,204),(11,167,111,187,30,148,72,219,53,119,85,203),(12,166,112,186,31,147,73,218,54,118,86,202),(13,165,113,185,32,146,74,217,55,117,87,201),(14,164,114,184,33,145,75,216,56,116,88,200),(15,163,96,183,34,144,76,215,57,115,89,199),(16,162,97,182,35,143,58,214,39,133,90,198),(17,161,98,181,36,142,59,213,40,132,91,197),(18,160,99,180,37,141,60,212,41,131,92,196),(19,159,100,179,38,140,61,211,42,130,93,195)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,57),(7,56),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(58,105),(59,104),(60,103),(61,102),(62,101),(63,100),(64,99),(65,98),(66,97),(67,96),(68,114),(69,113),(70,112),(71,111),(72,110),(73,109),(74,108),(75,107),(76,106),(77,92),(78,91),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85),(93,95),(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),(133,209),(134,183),(135,182),(136,181),(137,180),(138,179),(139,178),(140,177),(141,176),(142,175),(143,174),(144,173),(145,172),(146,190),(147,189),(148,188),(149,187),(150,186),(151,185),(152,184),(153,215),(154,214),(155,213),(156,212),(157,211),(158,210),(159,228),(160,227),(161,226),(162,225),(163,224),(164,223),(165,222),(166,221),(167,220),(168,219),(169,218),(170,217),(171,216)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38I | 38J | ··· | 38AA | 57A | ··· | 57I | 114A | ··· | 114I |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 6 | 114 | 2 | 38 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 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 | D12 | D19 | D38 | C19⋊D4 | S3×D19 | C19⋊D12 |
| kernel | C19⋊D12 | C3×Dic19 | S3×C38 | D114 | Dic19 | C57 | C38 | C19 | D6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 9 | 9 |
Matrix representation of C19⋊D12 ►in GL4(𝔽229) generated by
| 212 | 1 | 0 | 0 |
| 91 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 178 | 98 | 0 | 0 |
| 109 | 51 | 0 | 0 |
| 0 | 0 | 71 | 25 |
| 0 | 0 | 174 | 0 |
| 178 | 98 | 0 | 0 |
| 109 | 51 | 0 | 0 |
| 0 | 0 | 2 | 172 |
| 0 | 0 | 217 | 227 |
G:=sub<GL(4,GF(229))| [212,91,0,0,1,35,0,0,0,0,1,0,0,0,0,1],[178,109,0,0,98,51,0,0,0,0,71,174,0,0,25,0],[178,109,0,0,98,51,0,0,0,0,2,217,0,0,172,227] >;
C19⋊D12 in GAP, Magma, Sage, TeX
C_{19}\rtimes D_{12} % in TeX
G:=Group("C19:D12"); // GroupNames label
G:=SmallGroup(456,17);
// by ID
G=gap.SmallGroup(456,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,20,61,168,10804]);
// Polycyclic
G:=Group<a,b,c|a^19=b^12=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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