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G = C11×D19order 418 = 2·11·19

Direct product of C11 and D19

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

Aliases: C11×D19, C19⋊C22, C2092C2, SmallGroup(418,2)

Series: Derived Chief Lower central Upper central

C1C19 — C11×D19
C1C19C209 — C11×D19
C19 — C11×D19
C1C11

Generators and relations for C11×D19
 G = < a,b,c | a11=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

19C2
19C22

Smallest permutation representation of C11×D19
On 209 points
Generators in S209
(1 193 190 169 150 115 101 91 68 41 27)(2 194 172 170 151 116 102 92 69 42 28)(3 195 173 171 152 117 103 93 70 43 29)(4 196 174 153 134 118 104 94 71 44 30)(5 197 175 154 135 119 105 95 72 45 31)(6 198 176 155 136 120 106 77 73 46 32)(7 199 177 156 137 121 107 78 74 47 33)(8 200 178 157 138 122 108 79 75 48 34)(9 201 179 158 139 123 109 80 76 49 35)(10 202 180 159 140 124 110 81 58 50 36)(11 203 181 160 141 125 111 82 59 51 37)(12 204 182 161 142 126 112 83 60 52 38)(13 205 183 162 143 127 113 84 61 53 20)(14 206 184 163 144 128 114 85 62 54 21)(15 207 185 164 145 129 96 86 63 55 22)(16 208 186 165 146 130 97 87 64 56 23)(17 209 187 166 147 131 98 88 65 57 24)(18 191 188 167 148 132 99 89 66 39 25)(19 192 189 168 149 133 100 90 67 40 26)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(34 38)(35 37)(39 42)(40 41)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(77 85)(78 84)(79 83)(80 82)(86 95)(87 94)(88 93)(89 92)(90 91)(96 105)(97 104)(98 103)(99 102)(100 101)(106 114)(107 113)(108 112)(109 111)(115 133)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)(147 152)(148 151)(149 150)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(166 171)(167 170)(168 169)(172 188)(173 187)(174 186)(175 185)(176 184)(177 183)(178 182)(179 181)(189 190)(191 194)(192 193)(195 209)(196 208)(197 207)(198 206)(199 205)(200 204)(201 203)

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

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

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

121 conjugacy classes

class 1  2 11A···11J19A···19I22A···22J209A···209CL
order1211···1119···1922···22209···209
size1191···12···219···192···2

121 irreducible representations

dim111122
type+++
imageC1C2C11C22D19C11×D19
kernelC11×D19C209D19C19C11C1
# reps111010990

Matrix representation of C11×D19 in GL2(𝔽419) generated by

1520
0152
,
2121
127135
,
39914
18120
G:=sub<GL(2,GF(419))| [152,0,0,152],[212,127,1,135],[399,181,14,20] >;

C11×D19 in GAP, Magma, Sage, TeX

C_{11}\times D_{19}
% in TeX

G:=Group("C11xD19");
// GroupNames label

G:=SmallGroup(418,2);
// by ID

G=gap.SmallGroup(418,2);
# by ID

G:=PCGroup([3,-2,-11,-19,3566]);
// Polycyclic

G:=Group<a,b,c|a^11=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C11×D19 in TeX

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