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## G = C5×C11⋊D4order 440 = 23·5·11

### Direct product of C5 and C11⋊D4

Aliases: C5×C11⋊D4, C558D4, D225C10, C10.17D22, Dic114C10, C110.17C22, C115(C5×D4), C22⋊(C5×D11), (C2×C110)⋊4C2, (C2×C22)⋊6C10, (C2×C10)⋊1D11, (C10×D11)⋊5C2, C2.5(C10×D11), C22.13(C2×C10), (C5×Dic11)⋊4C2, SmallGroup(440,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C11⋊D4
 Chief series C1 — C11 — C22 — C110 — C10×D11 — C5×C11⋊D4
 Lower central C11 — C22 — C5×C11⋊D4
 Upper central C1 — C10 — C2×C10

Generators and relations for C5×C11⋊D4
G = < a,b,c,d | a5=b11=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C5×C11⋊D4
On 220 points
Generators in S220
(1 89 67 45 23)(2 90 68 46 24)(3 91 69 47 25)(4 92 70 48 26)(5 93 71 49 27)(6 94 72 50 28)(7 95 73 51 29)(8 96 74 52 30)(9 97 75 53 31)(10 98 76 54 32)(11 99 77 55 33)(12 100 78 56 34)(13 101 79 57 35)(14 102 80 58 36)(15 103 81 59 37)(16 104 82 60 38)(17 105 83 61 39)(18 106 84 62 40)(19 107 85 63 41)(20 108 86 64 42)(21 109 87 65 43)(22 110 88 66 44)(111 199 177 155 133)(112 200 178 156 134)(113 201 179 157 135)(114 202 180 158 136)(115 203 181 159 137)(116 204 182 160 138)(117 205 183 161 139)(118 206 184 162 140)(119 207 185 163 141)(120 208 186 164 142)(121 209 187 165 143)(122 210 188 166 144)(123 211 189 167 145)(124 212 190 168 146)(125 213 191 169 147)(126 214 192 170 148)(127 215 193 171 149)(128 216 194 172 150)(129 217 195 173 151)(130 218 196 174 152)(131 219 197 175 153)(132 220 198 176 154)
(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 12 122)(2 121 13 132)(3 120 14 131)(4 119 15 130)(5 118 16 129)(6 117 17 128)(7 116 18 127)(8 115 19 126)(9 114 20 125)(10 113 21 124)(11 112 22 123)(23 133 34 144)(24 143 35 154)(25 142 36 153)(26 141 37 152)(27 140 38 151)(28 139 39 150)(29 138 40 149)(30 137 41 148)(31 136 42 147)(32 135 43 146)(33 134 44 145)(45 155 56 166)(46 165 57 176)(47 164 58 175)(48 163 59 174)(49 162 60 173)(50 161 61 172)(51 160 62 171)(52 159 63 170)(53 158 64 169)(54 157 65 168)(55 156 66 167)(67 177 78 188)(68 187 79 198)(69 186 80 197)(70 185 81 196)(71 184 82 195)(72 183 83 194)(73 182 84 193)(74 181 85 192)(75 180 86 191)(76 179 87 190)(77 178 88 189)(89 199 100 210)(90 209 101 220)(91 208 102 219)(92 207 103 218)(93 206 104 217)(94 205 105 216)(95 204 106 215)(96 203 107 214)(97 202 108 213)(98 201 109 212)(99 200 110 211)
(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 18)(24 33)(25 32)(26 31)(27 30)(28 29)(35 44)(36 43)(37 42)(38 41)(39 40)(46 55)(47 54)(48 53)(49 52)(50 51)(57 66)(58 65)(59 64)(60 63)(61 62)(68 77)(69 76)(70 75)(71 74)(72 73)(79 88)(80 87)(81 86)(82 85)(83 84)(90 99)(91 98)(92 97)(93 96)(94 95)(101 110)(102 109)(103 108)(104 107)(105 106)(111 122)(112 132)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(121 123)(133 144)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(155 166)(156 176)(157 175)(158 174)(159 173)(160 172)(161 171)(162 170)(163 169)(164 168)(165 167)(177 188)(178 198)(179 197)(180 196)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)(199 210)(200 220)(201 219)(202 218)(203 217)(204 216)(205 215)(206 214)(207 213)(208 212)(209 211)

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

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

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

125 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 11A ··· 11E 20A 20B 20C 20D 22A ··· 22O 55A ··· 55T 110A ··· 110BH order 1 2 2 2 4 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 11 ··· 11 20 20 20 20 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 2 22 22 1 1 1 1 1 1 1 1 2 2 2 2 22 22 22 22 2 ··· 2 22 22 22 22 2 ··· 2 2 ··· 2 2 ··· 2

125 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D11 C5×D4 D22 C11⋊D4 C5×D11 C10×D11 C5×C11⋊D4 kernel C5×C11⋊D4 C5×Dic11 C10×D11 C2×C110 C11⋊D4 Dic11 D22 C2×C22 C55 C2×C10 C11 C10 C5 C22 C2 C1 # reps 1 1 1 1 4 4 4 4 1 5 4 5 10 20 20 40

Matrix representation of C5×C11⋊D4 in GL2(𝔽661) generated by

 247 0 0 247
,
 586 660 587 660
,
 402 297 493 259
,
 75 1 325 586
G:=sub<GL(2,GF(661))| [247,0,0,247],[586,587,660,660],[402,493,297,259],[75,325,1,586] >;

C5×C11⋊D4 in GAP, Magma, Sage, TeX

C_5\times C_{11}\rtimes D_4
% in TeX

G:=Group("C5xC11:D4");
// GroupNames label

G:=SmallGroup(440,28);
// by ID

G=gap.SmallGroup(440,28);
# by ID

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

G:=Group<a,b,c,d|a^5=b^11=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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