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G = C4×D55order 440 = 23·5·11

Direct product of C4 and D55

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

Aliases: C4×D55, C442D5, C2202C2, C202D11, C10.9D22, C2.1D110, C22.9D10, Dic555C2, D110.2C2, C110.9C22, C557(C2×C4), C53(C4×D11), C112(C4×D5), SmallGroup(440,35)

Series: Derived Chief Lower central Upper central

C1C55 — C4×D55
C1C11C55C110D110 — C4×D55
C55 — C4×D55
C1C4

Generators and relations for C4×D55
 G = < a,b,c | a4=b55=c2=1, ab=ba, ac=ca, cbc=b-1 >

55C2
55C2
55C4
55C22
11D5
11D5
5D11
5D11
55C2×C4
11Dic5
11D10
5D22
5Dic11
11C4×D5
5C4×D11

Smallest permutation representation of C4×D55
On 220 points
Generators in S220
(1 209 70 121)(2 210 71 122)(3 211 72 123)(4 212 73 124)(5 213 74 125)(6 214 75 126)(7 215 76 127)(8 216 77 128)(9 217 78 129)(10 218 79 130)(11 219 80 131)(12 220 81 132)(13 166 82 133)(14 167 83 134)(15 168 84 135)(16 169 85 136)(17 170 86 137)(18 171 87 138)(19 172 88 139)(20 173 89 140)(21 174 90 141)(22 175 91 142)(23 176 92 143)(24 177 93 144)(25 178 94 145)(26 179 95 146)(27 180 96 147)(28 181 97 148)(29 182 98 149)(30 183 99 150)(31 184 100 151)(32 185 101 152)(33 186 102 153)(34 187 103 154)(35 188 104 155)(36 189 105 156)(37 190 106 157)(38 191 107 158)(39 192 108 159)(40 193 109 160)(41 194 110 161)(42 195 56 162)(43 196 57 163)(44 197 58 164)(45 198 59 165)(46 199 60 111)(47 200 61 112)(48 201 62 113)(49 202 63 114)(50 203 64 115)(51 204 65 116)(52 205 66 117)(53 206 67 118)(54 207 68 119)(55 208 69 120)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 99)(27 98)(28 97)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 88)(38 87)(39 86)(40 85)(41 84)(42 83)(43 82)(44 81)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(111 218)(112 217)(113 216)(114 215)(115 214)(116 213)(117 212)(118 211)(119 210)(120 209)(121 208)(122 207)(123 206)(124 205)(125 204)(126 203)(127 202)(128 201)(129 200)(130 199)(131 198)(132 197)(133 196)(134 195)(135 194)(136 193)(137 192)(138 191)(139 190)(140 189)(141 188)(142 187)(143 186)(144 185)(145 184)(146 183)(147 182)(148 181)(149 180)(150 179)(151 178)(152 177)(153 176)(154 175)(155 174)(156 173)(157 172)(158 171)(159 170)(160 169)(161 168)(162 167)(163 166)(164 220)(165 219)

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

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

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

116 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B10A10B11A···11E20A20B20C20D22A···22E44A···44J55A···55T110A···110T220A···220AN
order1222444455101011···112020202022···2244···4455···55110···110220···220
size11555511555522222···222222···22···22···22···22···2

116 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4D5D10D11C4×D5D22C4×D11D55D110C4×D55
kernelC4×D55Dic55C220D110D55C44C22C20C11C10C5C4C2C1
# reps111142254510202040

Matrix representation of C4×D55 in GL2(𝔽661) generated by

1060
0106
,
50482
179549
,
611179
31150
G:=sub<GL(2,GF(661))| [106,0,0,106],[50,179,482,549],[611,311,179,50] >;

C4×D55 in GAP, Magma, Sage, TeX

C_4\times D_{55}
% in TeX

G:=Group("C4xD55");
// GroupNames label

G:=SmallGroup(440,35);
// by ID

G=gap.SmallGroup(440,35);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,26,643,10004]);
// Polycyclic

G:=Group<a,b,c|a^4=b^55=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D55 in TeX

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