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G = C19×D12order 456 = 23·3·19

Direct product of C19 and D12

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

Aliases: C19×D12, C576D4, C763S3, C121C38, C2285C2, D61C38, C38.15D6, C114.20C22, C4⋊(S3×C19), C31(D4×C19), (S3×C38)⋊4C2, C2.4(S3×C38), C6.3(C2×C38), SmallGroup(456,31)

Series: Derived Chief Lower central Upper central

C1C6 — C19×D12
C1C3C6C114S3×C38 — C19×D12
C3C6 — C19×D12
C1C38C76

Generators and relations for C19×D12
 G = < a,b,c | a19=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

6C2
6C2
3C22
3C22
2S3
2S3
6C38
6C38
3D4
3C2×C38
3C2×C38
2S3×C19
2S3×C19
3D4×C19

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

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

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

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

171 conjugacy classes

class 1 2A2B2C 3  4  6 12A12B19A···19R38A···38R38S···38BB57A···57R76A···76R114A···114R228A···228AJ
order1222346121219···1938···3838···3857···5776···76114···114228···228
size1166222221···11···16···62···22···22···22···2

171 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C19C38C38S3D4D6D12S3×C19D4×C19S3×C38C19×D12
kernelC19×D12C228S3×C38D12C12D6C76C57C38C19C4C3C2C1
# reps112181836111218181836

Matrix representation of C19×D12 in GL4(𝔽229) generated by

225000
022500
00440
00044
,
17422300
1995500
0011
002280
,
228000
171100
0010
00228228
G:=sub<GL(4,GF(229))| [225,0,0,0,0,225,0,0,0,0,44,0,0,0,0,44],[174,199,0,0,223,55,0,0,0,0,1,228,0,0,1,0],[228,171,0,0,0,1,0,0,0,0,1,228,0,0,0,228] >;

C19×D12 in GAP, Magma, Sage, TeX

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

G:=Group("C19xD12");
// GroupNames label

G:=SmallGroup(456,31);
// by ID

G=gap.SmallGroup(456,31);
# by ID

G:=PCGroup([5,-2,-2,-19,-2,-3,781,386,7604]);
// Polycyclic

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

Export

Subgroup lattice of C19×D12 in TeX

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