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G = C5×D44order 440 = 23·5·11

Direct product of C5 and D44

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

Aliases: C5×D44, C555D4, C2203C2, C445C10, C203D11, D224C10, C10.15D22, C110.15C22, C4⋊(C5×D11), C114(C5×D4), (C10×D11)⋊4C2, C2.4(C10×D11), C22.11(C2×C10), SmallGroup(440,26)

Series: Derived Chief Lower central Upper central

C1C22 — C5×D44
C1C11C22C110C10×D11 — C5×D44
C11C22 — C5×D44
C1C10C20

Generators and relations for C5×D44
 G = < a,b,c | a5=b44=c2=1, ab=ba, ac=ca, cbc=b-1 >

22C2
22C2
11C22
11C22
22C10
22C10
2D11
2D11
11D4
11C2×C10
11C2×C10
2C5×D11
2C5×D11
11C5×D4

Smallest permutation representation of C5×D44
On 220 points
Generators in S220
(1 133 115 85 214)(2 134 116 86 215)(3 135 117 87 216)(4 136 118 88 217)(5 137 119 45 218)(6 138 120 46 219)(7 139 121 47 220)(8 140 122 48 177)(9 141 123 49 178)(10 142 124 50 179)(11 143 125 51 180)(12 144 126 52 181)(13 145 127 53 182)(14 146 128 54 183)(15 147 129 55 184)(16 148 130 56 185)(17 149 131 57 186)(18 150 132 58 187)(19 151 89 59 188)(20 152 90 60 189)(21 153 91 61 190)(22 154 92 62 191)(23 155 93 63 192)(24 156 94 64 193)(25 157 95 65 194)(26 158 96 66 195)(27 159 97 67 196)(28 160 98 68 197)(29 161 99 69 198)(30 162 100 70 199)(31 163 101 71 200)(32 164 102 72 201)(33 165 103 73 202)(34 166 104 74 203)(35 167 105 75 204)(36 168 106 76 205)(37 169 107 77 206)(38 170 108 78 207)(39 171 109 79 208)(40 172 110 80 209)(41 173 111 81 210)(42 174 112 82 211)(43 175 113 83 212)(44 176 114 84 213)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(45 47)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(89 107)(90 106)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(108 132)(109 131)(110 130)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(133 143)(134 142)(135 141)(136 140)(137 139)(144 176)(145 175)(146 174)(147 173)(148 172)(149 171)(150 170)(151 169)(152 168)(153 167)(154 166)(155 165)(156 164)(157 163)(158 162)(159 161)(177 217)(178 216)(179 215)(180 214)(181 213)(182 212)(183 211)(184 210)(185 209)(186 208)(187 207)(188 206)(189 205)(190 204)(191 203)(192 202)(193 201)(194 200)(195 199)(196 198)(218 220)

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

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

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

125 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D10A10B10C10D10E···10L11A···11E20A20B20C20D22A···22E44A···44J55A···55T110A···110T220A···220AN
order1222455551010101010···1011···112020202022···2244···4455···55110···110220···220
size11222221111111122···222···222222···22···22···22···22···2

125 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C5C10C10D4D11C5×D4D22D44C5×D11C10×D11C5×D44
kernelC5×D44C220C10×D11D44C44D22C55C20C11C10C5C4C2C1
# reps112448154510202040

Matrix representation of C5×D44 in GL2(𝔽661) generated by

1970
0197
,
409377
284564
,
0660
6600
G:=sub<GL(2,GF(661))| [197,0,0,197],[409,284,377,564],[0,660,660,0] >;

C5×D44 in GAP, Magma, Sage, TeX

C_5\times D_{44}
% in TeX

G:=Group("C5xD44");
// GroupNames label

G:=SmallGroup(440,26);
// by ID

G=gap.SmallGroup(440,26);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-11,221,106,10004]);
// Polycyclic

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

Export

Subgroup lattice of C5×D44 in TeX

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