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## G = D4×C5×C10order 400 = 24·52

### Direct product of C5×C10 and D4

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C5×C10
 Chief series C1 — C2 — C10 — C5×C10 — C102 — D4×C52 — D4×C5×C10
 Lower central C1 — C2 — D4×C5×C10
 Upper central C1 — C102 — D4×C5×C10

Generators and relations for D4×C5×C10
G = < a,b,c,d | a5=b10=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 280 in 216 conjugacy classes, 152 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C10, C10, C2×D4, C20, C2×C10, C2×C10, C52, C2×C20, C5×D4, C22×C10, C5×C10, C5×C10, C5×C10, D4×C10, C5×C20, C102, C102, C102, C10×C20, D4×C52, C2×C102, D4×C5×C10
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C52, C5×D4, C22×C10, C5×C10, D4×C10, C102, D4×C52, C2×C102, D4×C5×C10

Smallest permutation representation of D4×C5×C10
On 200 points
Generators in S200
(1 108 97 42 81)(2 109 98 43 82)(3 110 99 44 83)(4 101 100 45 84)(5 102 91 46 85)(6 103 92 47 86)(7 104 93 48 87)(8 105 94 49 88)(9 106 95 50 89)(10 107 96 41 90)(11 75 30 69 33)(12 76 21 70 34)(13 77 22 61 35)(14 78 23 62 36)(15 79 24 63 37)(16 80 25 64 38)(17 71 26 65 39)(18 72 27 66 40)(19 73 28 67 31)(20 74 29 68 32)(51 197 142 181 136)(52 198 143 182 137)(53 199 144 183 138)(54 200 145 184 139)(55 191 146 185 140)(56 192 147 186 131)(57 193 148 187 132)(58 194 149 188 133)(59 195 150 189 134)(60 196 141 190 135)(111 160 172 127 166)(112 151 173 128 167)(113 152 174 129 168)(114 153 175 130 169)(115 154 176 121 170)(116 155 177 122 161)(117 156 178 123 162)(118 157 179 124 163)(119 158 180 125 164)(120 159 171 126 165)
(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)
(1 131 16 153)(2 132 17 154)(3 133 18 155)(4 134 19 156)(5 135 20 157)(6 136 11 158)(7 137 12 159)(8 138 13 160)(9 139 14 151)(10 140 15 152)(21 126 93 198)(22 127 94 199)(23 128 95 200)(24 129 96 191)(25 130 97 192)(26 121 98 193)(27 122 99 194)(28 123 100 195)(29 124 91 196)(30 125 92 197)(31 117 84 189)(32 118 85 190)(33 119 86 181)(34 120 87 182)(35 111 88 183)(36 112 89 184)(37 113 90 185)(38 114 81 186)(39 115 82 187)(40 116 83 188)(41 146 63 168)(42 147 64 169)(43 148 65 170)(44 149 66 161)(45 150 67 162)(46 141 68 163)(47 142 69 164)(48 143 70 165)(49 144 61 166)(50 145 62 167)(51 75 180 103)(52 76 171 104)(53 77 172 105)(54 78 173 106)(55 79 174 107)(56 80 175 108)(57 71 176 109)(58 72 177 110)(59 73 178 101)(60 74 179 102)
(1 153)(2 154)(3 155)(4 156)(5 157)(6 158)(7 159)(8 160)(9 151)(10 152)(11 136)(12 137)(13 138)(14 139)(15 140)(16 131)(17 132)(18 133)(19 134)(20 135)(21 198)(22 199)(23 200)(24 191)(25 192)(26 193)(27 194)(28 195)(29 196)(30 197)(31 189)(32 190)(33 181)(34 182)(35 183)(36 184)(37 185)(38 186)(39 187)(40 188)(41 168)(42 169)(43 170)(44 161)(45 162)(46 163)(47 164)(48 165)(49 166)(50 167)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 71)(58 72)(59 73)(60 74)(61 144)(62 145)(63 146)(64 147)(65 148)(66 149)(67 150)(68 141)(69 142)(70 143)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 111)(89 112)(90 113)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(97 130)(98 121)(99 122)(100 123)(101 178)(102 179)(103 180)(104 171)(105 172)(106 173)(107 174)(108 175)(109 176)(110 177)

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

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

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

250 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A ··· 5X 10A ··· 10BT 10BU ··· 10FL 20A ··· 20AV order 1 2 2 2 2 2 2 2 4 4 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

250 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 C5×D4 kernel D4×C5×C10 C10×C20 D4×C52 C2×C102 D4×C10 C2×C20 C5×D4 C22×C10 C5×C10 C10 # reps 1 1 4 2 24 24 96 48 2 48

Matrix representation of D4×C5×C10 in GL3(𝔽41) generated by

 18 0 0 0 1 0 0 0 1
,
 23 0 0 0 23 0 0 0 23
,
 40 0 0 0 4 12 0 2 37
,
 1 0 0 0 37 28 0 39 4
G:=sub<GL(3,GF(41))| [18,0,0,0,1,0,0,0,1],[23,0,0,0,23,0,0,0,23],[40,0,0,0,4,2,0,12,37],[1,0,0,0,37,39,0,28,4] >;

D4×C5×C10 in GAP, Magma, Sage, TeX

D_4\times C_5\times C_{10}
% in TeX

G:=Group("D4xC5xC10");
// GroupNames label

G:=SmallGroup(400,202);
// by ID

G=gap.SmallGroup(400,202);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-5,-2,2425]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^10=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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