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G = D5×C44order 440 = 23·5·11

Direct product of C44 and D5

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

Aliases: D5×C44, C202C22, C2206C2, Dic52C22, D10.2C22, C22.14D10, C110.19C22, C559(C2×C4), C52(C2×C44), C2.1(D5×C22), C10.2(C2×C22), (D5×C22).4C2, (C11×Dic5)⋊5C2, SmallGroup(440,30)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C44
C1C5C10C110D5×C22 — D5×C44
C5 — D5×C44
C1C44

Generators and relations for D5×C44
 G = < a,b,c | a44=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C4
5C22
5C22
5C22
5C2×C4
5C44
5C2×C22
5C2×C44

Smallest permutation representation of D5×C44
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)
(1 198 115 136 75)(2 199 116 137 76)(3 200 117 138 77)(4 201 118 139 78)(5 202 119 140 79)(6 203 120 141 80)(7 204 121 142 81)(8 205 122 143 82)(9 206 123 144 83)(10 207 124 145 84)(11 208 125 146 85)(12 209 126 147 86)(13 210 127 148 87)(14 211 128 149 88)(15 212 129 150 45)(16 213 130 151 46)(17 214 131 152 47)(18 215 132 153 48)(19 216 89 154 49)(20 217 90 155 50)(21 218 91 156 51)(22 219 92 157 52)(23 220 93 158 53)(24 177 94 159 54)(25 178 95 160 55)(26 179 96 161 56)(27 180 97 162 57)(28 181 98 163 58)(29 182 99 164 59)(30 183 100 165 60)(31 184 101 166 61)(32 185 102 167 62)(33 186 103 168 63)(34 187 104 169 64)(35 188 105 170 65)(36 189 106 171 66)(37 190 107 172 67)(38 191 108 173 68)(39 192 109 174 69)(40 193 110 175 70)(41 194 111 176 71)(42 195 112 133 72)(43 196 113 134 73)(44 197 114 135 74)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(133 195)(134 196)(135 197)(136 198)(137 199)(138 200)(139 201)(140 202)(141 203)(142 204)(143 205)(144 206)(145 207)(146 208)(147 209)(148 210)(149 211)(150 212)(151 213)(152 214)(153 215)(154 216)(155 217)(156 218)(157 219)(158 220)(159 177)(160 178)(161 179)(162 180)(163 181)(164 182)(165 183)(166 184)(167 185)(168 186)(169 187)(170 188)(171 189)(172 190)(173 191)(174 192)(175 193)(176 194)

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

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

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

176 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B10A10B11A···11J20A20B20C20D22A···22J22K···22AD44A···44T44U···44AN55A···55T110A···110T220A···220AN
order1222444455101011···112020202022···2222···2244···4444···4455···55110···110220···220
size1155115522221···122221···15···51···15···52···22···22···2

176 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C4C11C22C22C22C44D5D10C4×D5D5×C11D5×C22D5×C44
kernelD5×C44C11×Dic5C220D5×C22D5×C11C4×D5Dic5C20D10D5C44C22C11C4C2C1
# reps111141010101040224202040

Matrix representation of D5×C44 in GL2(𝔽661) generated by

3790
0379
,
6601
56604
,
6600
561
G:=sub<GL(2,GF(661))| [379,0,0,379],[660,56,1,604],[660,56,0,1] >;

D5×C44 in GAP, Magma, Sage, TeX

D_5\times C_{44}
% in TeX

G:=Group("D5xC44");
// GroupNames label

G:=SmallGroup(440,30);
// by ID

G=gap.SmallGroup(440,30);
# by ID

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

G:=Group<a,b,c|a^44=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D5×C44 in TeX

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