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## G = C11×C5⋊D4order 440 = 23·5·11

### Direct product of C11 and C5⋊D4

Aliases: C11×C5⋊D4, C559D4, Dic5⋊C22, D102C22, C22.17D10, C110.22C22, C52(D4×C11), (C2×C22)⋊1D5, C22⋊(D5×C11), (C2×C110)⋊6C2, (C2×C10)⋊2C22, (D5×C22)⋊5C2, C2.5(D5×C22), C10.5(C2×C22), (C11×Dic5)⋊4C2, SmallGroup(440,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C11×C5⋊D4
 Chief series C1 — C5 — C10 — C110 — D5×C22 — C11×C5⋊D4
 Lower central C5 — C10 — C11×C5⋊D4
 Upper central C1 — C22 — C2×C22

Generators and relations for C11×C5⋊D4
G = < a,b,c,d | a11=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

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

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

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

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

143 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 10A ··· 10F 11A ··· 11J 22A ··· 22J 22K ··· 22T 22U ··· 22AD 44A ··· 44J 55A ··· 55T 110A ··· 110BH order 1 2 2 2 4 5 5 10 ··· 10 11 ··· 11 22 ··· 22 22 ··· 22 22 ··· 22 44 ··· 44 55 ··· 55 110 ··· 110 size 1 1 2 10 10 2 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 10 ··· 10 10 ··· 10 2 ··· 2 2 ··· 2

143 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 C11 C22 C22 C22 D4 D5 D10 C5⋊D4 D4×C11 D5×C11 D5×C22 C11×C5⋊D4 kernel C11×C5⋊D4 C11×Dic5 D5×C22 C2×C110 C5⋊D4 Dic5 D10 C2×C10 C55 C2×C22 C22 C11 C5 C22 C2 C1 # reps 1 1 1 1 10 10 10 10 1 2 2 4 10 20 20 40

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

 418 0 0 418
,
 603 660 1 0
,
 44 389 481 617
,
 1 0 603 660
G:=sub<GL(2,GF(661))| [418,0,0,418],[603,1,660,0],[44,481,389,617],[1,603,0,660] >;

C11×C5⋊D4 in GAP, Magma, Sage, TeX

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

G:=Group("C11xC5:D4");
// GroupNames label

G:=SmallGroup(440,33);
// by ID

G=gap.SmallGroup(440,33);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-5,461,8804]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^5=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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