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G = C11×D20order 440 = 23·5·11

Direct product of C11 and D20

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

Aliases: C11×D20, C556D4, C443D5, C201C22, C2205C2, D101C22, C22.15D10, C110.20C22, C4⋊(D5×C11), C51(D4×C11), (D5×C22)⋊4C2, C2.4(D5×C22), C10.3(C2×C22), SmallGroup(440,31)

Series: Derived Chief Lower central Upper central

C1C10 — C11×D20
C1C5C10C110D5×C22 — C11×D20
C5C10 — C11×D20
C1C22C44

Generators and relations for C11×D20
 G = < a,b,c | a11=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

10C2
10C2
5C22
5C22
2D5
2D5
10C22
10C22
5D4
5C2×C22
5C2×C22
2D5×C11
2D5×C11
5D4×C11

Smallest permutation representation of C11×D20
On 220 points
Generators in S220
(1 128 154 207 82 191 115 165 40 49 67)(2 129 155 208 83 192 116 166 21 50 68)(3 130 156 209 84 193 117 167 22 51 69)(4 131 157 210 85 194 118 168 23 52 70)(5 132 158 211 86 195 119 169 24 53 71)(6 133 159 212 87 196 120 170 25 54 72)(7 134 160 213 88 197 101 171 26 55 73)(8 135 141 214 89 198 102 172 27 56 74)(9 136 142 215 90 199 103 173 28 57 75)(10 137 143 216 91 200 104 174 29 58 76)(11 138 144 217 92 181 105 175 30 59 77)(12 139 145 218 93 182 106 176 31 60 78)(13 140 146 219 94 183 107 177 32 41 79)(14 121 147 220 95 184 108 178 33 42 80)(15 122 148 201 96 185 109 179 34 43 61)(16 123 149 202 97 186 110 180 35 44 62)(17 124 150 203 98 187 111 161 36 45 63)(18 125 151 204 99 188 112 162 37 46 64)(19 126 152 205 100 189 113 163 38 47 65)(20 127 153 206 81 190 114 164 39 48 66)
(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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(121 129)(122 128)(123 127)(124 126)(130 140)(131 139)(132 138)(133 137)(134 136)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)(161 163)(164 180)(165 179)(166 178)(167 177)(168 176)(169 175)(170 174)(171 173)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)(196 200)(197 199)(201 207)(202 206)(203 205)(208 220)(209 219)(210 218)(211 217)(212 216)(213 215)

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

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

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

143 conjugacy classes

class 1 2A2B2C 4 5A5B10A10B11A···11J20A20B20C20D22A···22J22K···22AD44A···44J55A···55T110A···110T220A···220AN
order1222455101011···112020202022···2222···2244···4455···55110···110220···220
size111010222221···122221···110···102···22···22···22···2

143 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C11C22C22D4D5D10D20D4×C11D5×C11D5×C22C11×D20
kernelC11×D20C220D5×C22D20C20D10C55C44C22C11C5C4C2C1
# reps112101020122410202040

Matrix representation of C11×D20 in GL2(𝔽661) generated by

680
068
,
572624
37409
,
0660
6600
G:=sub<GL(2,GF(661))| [68,0,0,68],[572,37,624,409],[0,660,660,0] >;

C11×D20 in GAP, Magma, Sage, TeX

C_{11}\times D_{20}
% in TeX

G:=Group("C11xD20");
// GroupNames label

G:=SmallGroup(440,31);
// by ID

G=gap.SmallGroup(440,31);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-5,461,226,8804]);
// Polycyclic

G:=Group<a,b,c|a^11=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C11×D20 in TeX

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