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G = D4×C55order 440 = 23·5·11

Direct product of C55 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C55, C4⋊C110, C2207C2, C203C22, C447C10, C22⋊C110, C110.23C22, (C2×C22)⋊5C10, (C2×C110)⋊1C2, (C2×C10)⋊1C22, C10.6(C2×C22), C2.1(C2×C110), C22.14(C2×C10), SmallGroup(440,40)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C55
C1C2C22C110C2×C110 — D4×C55
C1C2 — D4×C55
C1C110 — D4×C55

Generators and relations for D4×C55
 G = < a,b,c | a55=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C10
2C10
2C22
2C22
2C110
2C110

Smallest permutation representation of D4×C55
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)
(1 111 204 69)(2 112 205 70)(3 113 206 71)(4 114 207 72)(5 115 208 73)(6 116 209 74)(7 117 210 75)(8 118 211 76)(9 119 212 77)(10 120 213 78)(11 121 214 79)(12 122 215 80)(13 123 216 81)(14 124 217 82)(15 125 218 83)(16 126 219 84)(17 127 220 85)(18 128 166 86)(19 129 167 87)(20 130 168 88)(21 131 169 89)(22 132 170 90)(23 133 171 91)(24 134 172 92)(25 135 173 93)(26 136 174 94)(27 137 175 95)(28 138 176 96)(29 139 177 97)(30 140 178 98)(31 141 179 99)(32 142 180 100)(33 143 181 101)(34 144 182 102)(35 145 183 103)(36 146 184 104)(37 147 185 105)(38 148 186 106)(39 149 187 107)(40 150 188 108)(41 151 189 109)(42 152 190 110)(43 153 191 56)(44 154 192 57)(45 155 193 58)(46 156 194 59)(47 157 195 60)(48 158 196 61)(49 159 197 62)(50 160 198 63)(51 161 199 64)(52 162 200 65)(53 163 201 66)(54 164 202 67)(55 165 203 68)
(56 153)(57 154)(58 155)(59 156)(60 157)(61 158)(62 159)(63 160)(64 161)(65 162)(66 163)(67 164)(68 165)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 121)(80 122)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 145)(104 146)(105 147)(106 148)(107 149)(108 150)(109 151)(110 152)

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

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

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

275 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D10A10B10C10D10E···10L11A···11J20A20B20C20D22A···22J22K···22AD44A···44J55A···55AN110A···110AN110AO···110DP220A···220AN
order1222455551010101010···1011···112020202022···2222···2244···4455···55110···110110···110220···220
size11222111111112···21···122221···12···22···21···11···12···22···2

275 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C5C10C10C11C22C22C55C110C110D4C5×D4D4×C11D4×C55
kernelD4×C55C220C2×C110D4×C11C44C2×C22C5×D4C20C2×C10D4C4C22C55C11C5C1
# reps112448101020404080141040

Matrix representation of D4×C55 in GL3(𝔽661) generated by

40600
06340
00634
,
66000
04602
0290201
,
66000
010
0201660
G:=sub<GL(3,GF(661))| [406,0,0,0,634,0,0,0,634],[660,0,0,0,460,290,0,2,201],[660,0,0,0,1,201,0,0,660] >;

D4×C55 in GAP, Magma, Sage, TeX

D_4\times C_{55}
% in TeX

G:=Group("D4xC55");
// GroupNames label

G:=SmallGroup(440,40);
// by ID

G=gap.SmallGroup(440,40);
# by ID

G:=PCGroup([5,-2,-2,-5,-11,-2,2221]);
// Polycyclic

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

Export

Subgroup lattice of D4×C55 in TeX

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