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## G = C2×C4×C5⋊D5order 400 = 24·52

### Direct product of C2×C4 and C5⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C4×C5⋊D5
 Chief series C1 — C5 — C52 — C5×C10 — C2×C5⋊D5 — C22×C5⋊D5 — C2×C4×C5⋊D5
 Lower central C52 — C2×C4×C5⋊D5
 Upper central C1 — C2×C4

Generators and relations for C2×C4×C5⋊D5
G = < a,b,c,d,e | a2=b4=c5=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1128 in 216 conjugacy classes, 83 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C22×C4, Dic5, C20, D10, C2×C10, C52, C4×D5, C2×Dic5, C2×C20, C22×D5, C5⋊D5, C5×C10, C5×C10, C2×C4×D5, C526C4, C5×C20, C2×C5⋊D5, C102, C4×C5⋊D5, C2×C526C4, C10×C20, C22×C5⋊D5, C2×C4×C5⋊D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C4×D5, C22×D5, C5⋊D5, C2×C4×D5, C2×C5⋊D5, C4×C5⋊D5, C22×C5⋊D5, C2×C4×C5⋊D5

Smallest permutation representation of C2×C4×C5⋊D5
On 200 points
Generators in S200
(1 129)(2 130)(3 126)(4 127)(5 128)(6 131)(7 132)(8 133)(9 134)(10 135)(11 136)(12 137)(13 138)(14 139)(15 140)(16 141)(17 142)(18 143)(19 144)(20 145)(21 146)(22 147)(23 148)(24 149)(25 150)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 176)(52 177)(53 178)(54 179)(55 180)(56 181)(57 182)(58 183)(59 184)(60 185)(61 186)(62 187)(63 188)(64 189)(65 190)(66 191)(67 192)(68 193)(69 194)(70 195)(71 196)(72 197)(73 198)(74 199)(75 200)(76 151)(77 152)(78 153)(79 154)(80 155)(81 156)(82 157)(83 158)(84 159)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 169)(95 170)(96 171)(97 172)(98 173)(99 174)(100 175)
(1 54 29 79)(2 55 30 80)(3 51 26 76)(4 52 27 77)(5 53 28 78)(6 56 31 81)(7 57 32 82)(8 58 33 83)(9 59 34 84)(10 60 35 85)(11 61 36 86)(12 62 37 87)(13 63 38 88)(14 64 39 89)(15 65 40 90)(16 66 41 91)(17 67 42 92)(18 68 43 93)(19 69 44 94)(20 70 45 95)(21 71 46 96)(22 72 47 97)(23 73 48 98)(24 74 49 99)(25 75 50 100)(101 151 126 176)(102 152 127 177)(103 153 128 178)(104 154 129 179)(105 155 130 180)(106 156 131 181)(107 157 132 182)(108 158 133 183)(109 159 134 184)(110 160 135 185)(111 161 136 186)(112 162 137 187)(113 163 138 188)(114 164 139 189)(115 165 140 190)(116 166 141 191)(117 167 142 192)(118 168 143 193)(119 169 144 194)(120 170 145 195)(121 171 146 196)(122 172 147 197)(123 173 148 198)(124 174 149 199)(125 175 150 200)
(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)
(1 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)(26 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)(51 71 66 61 56)(52 72 67 62 57)(53 73 68 63 58)(54 74 69 64 59)(55 75 70 65 60)(76 96 91 86 81)(77 97 92 87 82)(78 98 93 88 83)(79 99 94 89 84)(80 100 95 90 85)(101 121 116 111 106)(102 122 117 112 107)(103 123 118 113 108)(104 124 119 114 109)(105 125 120 115 110)(126 146 141 136 131)(127 147 142 137 132)(128 148 143 138 133)(129 149 144 139 134)(130 150 145 140 135)(151 171 166 161 156)(152 172 167 162 157)(153 173 168 163 158)(154 174 169 164 159)(155 175 170 165 160)(176 196 191 186 181)(177 197 192 187 182)(178 198 193 188 183)(179 199 194 189 184)(180 200 195 190 185)
(1 134)(2 133)(3 132)(4 131)(5 135)(6 127)(7 126)(8 130)(9 129)(10 128)(11 147)(12 146)(13 150)(14 149)(15 148)(16 142)(17 141)(18 145)(19 144)(20 143)(21 137)(22 136)(23 140)(24 139)(25 138)(26 107)(27 106)(28 110)(29 109)(30 108)(31 102)(32 101)(33 105)(34 104)(35 103)(36 122)(37 121)(38 125)(39 124)(40 123)(41 117)(42 116)(43 120)(44 119)(45 118)(46 112)(47 111)(48 115)(49 114)(50 113)(51 182)(52 181)(53 185)(54 184)(55 183)(56 177)(57 176)(58 180)(59 179)(60 178)(61 197)(62 196)(63 200)(64 199)(65 198)(66 192)(67 191)(68 195)(69 194)(70 193)(71 187)(72 186)(73 190)(74 189)(75 188)(76 157)(77 156)(78 160)(79 159)(80 158)(81 152)(82 151)(83 155)(84 154)(85 153)(86 172)(87 171)(88 175)(89 174)(90 173)(91 167)(92 166)(93 170)(94 169)(95 168)(96 162)(97 161)(98 165)(99 164)(100 163)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A ··· 5L 10A ··· 10AJ 20A ··· 20AV order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 1 1 25 25 25 25 1 1 1 1 25 25 25 25 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 D5 D10 D10 C4×D5 kernel C2×C4×C5⋊D5 C4×C5⋊D5 C2×C52⋊6C4 C10×C20 C22×C5⋊D5 C2×C5⋊D5 C2×C20 C20 C2×C10 C10 # reps 1 4 1 1 1 8 12 24 12 48

Matrix representation of C2×C4×C5⋊D5 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 32 0 0 0 0 32 0 0 0 0 32 0 0 0 0 32
,
 0 1 0 0 40 6 0 0 0 0 0 1 0 0 40 6
,
 35 35 0 0 6 40 0 0 0 0 40 6 0 0 35 35
,
 35 35 0 0 40 6 0 0 0 0 40 6 0 0 0 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[0,40,0,0,1,6,0,0,0,0,0,40,0,0,1,6],[35,6,0,0,35,40,0,0,0,0,40,35,0,0,6,35],[35,40,0,0,35,6,0,0,0,0,40,0,0,0,6,1] >;

C2×C4×C5⋊D5 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_5\rtimes D_5
% in TeX

G:=Group("C2xC4xC5:D5");
// GroupNames label

G:=SmallGroup(400,192);
// by ID

G=gap.SmallGroup(400,192);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,50,1924,11525]);
// Polycyclic

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

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