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G = C5×Dic11order 220 = 22·5·11

Direct product of C5 and Dic11

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

Aliases: C5×Dic11, C554C4, C113C20, C22.3C10, C110.2C2, C10.2D11, C2.(C5×D11), SmallGroup(220,4)

Series: Derived Chief Lower central Upper central

C1C11 — C5×Dic11
C1C11C22C110 — C5×Dic11
C11 — C5×Dic11
C1C10

Generators and relations for C5×Dic11
 G = < a,b,c | a5=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >

11C4
11C20

Smallest permutation representation of C5×Dic11
Regular action on 220 points
Generators in S220
(1 92 83 65 31)(2 93 84 66 32)(3 94 85 45 33)(4 95 86 46 34)(5 96 87 47 35)(6 97 88 48 36)(7 98 67 49 37)(8 99 68 50 38)(9 100 69 51 39)(10 101 70 52 40)(11 102 71 53 41)(12 103 72 54 42)(13 104 73 55 43)(14 105 74 56 44)(15 106 75 57 23)(16 107 76 58 24)(17 108 77 59 25)(18 109 78 60 26)(19 110 79 61 27)(20 89 80 62 28)(21 90 81 63 29)(22 91 82 64 30)(111 199 188 155 144)(112 200 189 156 145)(113 201 190 157 146)(114 202 191 158 147)(115 203 192 159 148)(116 204 193 160 149)(117 205 194 161 150)(118 206 195 162 151)(119 207 196 163 152)(120 208 197 164 153)(121 209 198 165 154)(122 210 177 166 133)(123 211 178 167 134)(124 212 179 168 135)(125 213 180 169 136)(126 214 181 170 137)(127 215 182 171 138)(128 216 183 172 139)(129 217 184 173 140)(130 218 185 174 141)(131 219 186 175 142)(132 220 187 176 143)
(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)
(1 111 12 122)(2 132 13 121)(3 131 14 120)(4 130 15 119)(5 129 16 118)(6 128 17 117)(7 127 18 116)(8 126 19 115)(9 125 20 114)(10 124 21 113)(11 123 22 112)(23 152 34 141)(24 151 35 140)(25 150 36 139)(26 149 37 138)(27 148 38 137)(28 147 39 136)(29 146 40 135)(30 145 41 134)(31 144 42 133)(32 143 43 154)(33 142 44 153)(45 175 56 164)(46 174 57 163)(47 173 58 162)(48 172 59 161)(49 171 60 160)(50 170 61 159)(51 169 62 158)(52 168 63 157)(53 167 64 156)(54 166 65 155)(55 165 66 176)(67 182 78 193)(68 181 79 192)(69 180 80 191)(70 179 81 190)(71 178 82 189)(72 177 83 188)(73 198 84 187)(74 197 85 186)(75 196 86 185)(76 195 87 184)(77 194 88 183)(89 202 100 213)(90 201 101 212)(91 200 102 211)(92 199 103 210)(93 220 104 209)(94 219 105 208)(95 218 106 207)(96 217 107 206)(97 216 108 205)(98 215 109 204)(99 214 110 203)

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

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

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

C5×Dic11 is a maximal subgroup of   D552C4  C11⋊D20  C55⋊Q8  C20×D11

70 conjugacy classes

class 1  2 4A4B5A5B5C5D10A10B10C10D11A···11E20A···20H22A···22E55A···55T110A···110T
order124455551010101011···1120···2022···2255···55110···110
size111111111111112···211···112···22···22···2

70 irreducible representations

dim1111112222
type+++-
imageC1C2C4C5C10C20D11Dic11C5×D11C5×Dic11
kernelC5×Dic11C110C55Dic11C22C11C10C5C2C1
# reps112448552020

Matrix representation of C5×Dic11 in GL3(𝔽661) generated by

100
04710
00471
,
66000
001
0660585
,
55500
0511116
0279150
G:=sub<GL(3,GF(661))| [1,0,0,0,471,0,0,0,471],[660,0,0,0,0,660,0,1,585],[555,0,0,0,511,279,0,116,150] >;

C5×Dic11 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{11}
% in TeX

G:=Group("C5xDic11");
// GroupNames label

G:=SmallGroup(220,4);
// by ID

G=gap.SmallGroup(220,4);
# by ID

G:=PCGroup([4,-2,-5,-2,-11,40,3203]);
// Polycyclic

G:=Group<a,b,c|a^5=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×Dic11 in TeX

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