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G = C11×Dic5order 220 = 22·5·11

Direct product of C11 and Dic5

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

Aliases: C11×Dic5, C555C4, C52C44, C10.C22, C22.2D5, C110.3C2, C2.(D5×C11), SmallGroup(220,3)

Series: Derived Chief Lower central Upper central

C1C5 — C11×Dic5
C1C5C10C110 — C11×Dic5
C5 — C11×Dic5
C1C22

Generators and relations for C11×Dic5
 G = < a,b,c | a11=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

5C4
5C44

Smallest permutation representation of C11×Dic5
Regular action on 220 points
Generators in S220
(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)
(1 159 133 208 111 186 124 188 173 146)(2 160 134 209 112 187 125 189 174 147)(3 161 135 199 113 177 126 190 175 148)(4 162 136 200 114 178 127 191 176 149)(5 163 137 201 115 179 128 192 166 150)(6 164 138 202 116 180 129 193 167 151)(7 165 139 203 117 181 130 194 168 152)(8 155 140 204 118 182 131 195 169 153)(9 156 141 205 119 183 132 196 170 154)(10 157 142 206 120 184 122 197 171 144)(11 158 143 207 121 185 123 198 172 145)(12 109 219 96 32 47 74 34 61 87)(13 110 220 97 33 48 75 35 62 88)(14 100 210 98 23 49 76 36 63 78)(15 101 211 99 24 50 77 37 64 79)(16 102 212 89 25 51 67 38 65 80)(17 103 213 90 26 52 68 39 66 81)(18 104 214 91 27 53 69 40 56 82)(19 105 215 92 28 54 70 41 57 83)(20 106 216 93 29 55 71 42 58 84)(21 107 217 94 30 45 72 43 59 85)(22 108 218 95 31 46 73 44 60 86)
(1 36 186 210)(2 37 187 211)(3 38 177 212)(4 39 178 213)(5 40 179 214)(6 41 180 215)(7 42 181 216)(8 43 182 217)(9 44 183 218)(10 34 184 219)(11 35 185 220)(12 142 47 197)(13 143 48 198)(14 133 49 188)(15 134 50 189)(16 135 51 190)(17 136 52 191)(18 137 53 192)(19 138 54 193)(20 139 55 194)(21 140 45 195)(22 141 46 196)(23 173 78 208)(24 174 79 209)(25 175 80 199)(26 176 81 200)(27 166 82 201)(28 167 83 202)(29 168 84 203)(30 169 85 204)(31 170 86 205)(32 171 87 206)(33 172 88 207)(56 115 91 150)(57 116 92 151)(58 117 93 152)(59 118 94 153)(60 119 95 154)(61 120 96 144)(62 121 97 145)(63 111 98 146)(64 112 99 147)(65 113 89 148)(66 114 90 149)(67 126 102 161)(68 127 103 162)(69 128 104 163)(70 129 105 164)(71 130 106 165)(72 131 107 155)(73 132 108 156)(74 122 109 157)(75 123 110 158)(76 124 100 159)(77 125 101 160)

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

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

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

C11×Dic5 is a maximal subgroup of   C55⋊C8  D552C4  C5⋊D44  C55⋊Q8  D5×C44

88 conjugacy classes

class 1  2 4A4B5A5B10A10B11A···11J22A···22J44A···44T55A···55T110A···110T
order124455101011···1122···2244···4455···55110···110
size115522221···11···15···52···22···2

88 irreducible representations

dim1111112222
type+++-
imageC1C2C4C11C22C44D5Dic5D5×C11C11×Dic5
kernelC11×Dic5C110C55Dic5C10C5C22C11C2C1
# reps112101020222020

Matrix representation of C11×Dic5 in GL2(𝔽661) generated by

810
081
,
1660
59603
,
0568
4620
G:=sub<GL(2,GF(661))| [81,0,0,81],[1,59,660,603],[0,462,568,0] >;

C11×Dic5 in GAP, Magma, Sage, TeX

C_{11}\times {\rm Dic}_5
% in TeX

G:=Group("C11xDic5");
// GroupNames label

G:=SmallGroup(220,3);
// by ID

G=gap.SmallGroup(220,3);
# by ID

G:=PCGroup([4,-2,-11,-2,-5,88,2819]);
// Polycyclic

G:=Group<a,b,c|a^11=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C11×Dic5 in TeX

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