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G = D8×C25order 400 = 24·52

Direct product of C25 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: D8×C25, D4⋊C50, C81C50, C2005C2, C40.3C10, C50.14D4, C100.17C22, C5.(C5×D8), (C5×D8).C5, (D4×C25)⋊4C2, C4.1(C2×C50), C2.3(D4×C25), (C5×D4).2C10, C10.14(C5×D4), C20.17(C2×C10), SmallGroup(400,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C25
 Chief series C1 — C2 — C10 — C20 — C100 — D4×C25 — D8×C25
 Lower central C1 — C2 — C4 — D8×C25
 Upper central C1 — C50 — C100 — D8×C25

Generators and relations for D8×C25
G = < a,b,c | a25=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D8×C25
On 200 points
Generators in S200
(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)
(1 134 71 90 122 152 37 196)(2 135 72 91 123 153 38 197)(3 136 73 92 124 154 39 198)(4 137 74 93 125 155 40 199)(5 138 75 94 101 156 41 200)(6 139 51 95 102 157 42 176)(7 140 52 96 103 158 43 177)(8 141 53 97 104 159 44 178)(9 142 54 98 105 160 45 179)(10 143 55 99 106 161 46 180)(11 144 56 100 107 162 47 181)(12 145 57 76 108 163 48 182)(13 146 58 77 109 164 49 183)(14 147 59 78 110 165 50 184)(15 148 60 79 111 166 26 185)(16 149 61 80 112 167 27 186)(17 150 62 81 113 168 28 187)(18 126 63 82 114 169 29 188)(19 127 64 83 115 170 30 189)(20 128 65 84 116 171 31 190)(21 129 66 85 117 172 32 191)(22 130 67 86 118 173 33 192)(23 131 68 87 119 174 34 193)(24 132 69 88 120 175 35 194)(25 133 70 89 121 151 36 195)
(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 75)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)(49 58)(50 59)(76 163)(77 164)(78 165)(79 166)(80 167)(81 168)(82 169)(83 170)(84 171)(85 172)(86 173)(87 174)(88 175)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(97 159)(98 160)(99 161)(100 162)(126 188)(127 189)(128 190)(129 191)(130 192)(131 193)(132 194)(133 195)(134 196)(135 197)(136 198)(137 199)(138 200)(139 176)(140 177)(141 178)(142 179)(143 180)(144 181)(145 182)(146 183)(147 184)(148 185)(149 186)(150 187)

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

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

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

175 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E ··· 10L 20A 20B 20C 20D 25A ··· 25T 40A ··· 40H 50A ··· 50T 50U ··· 50BH 100A ··· 100T 200A ··· 200AN order 1 2 2 2 4 5 5 5 5 8 8 10 10 10 10 10 ··· 10 20 20 20 20 25 ··· 25 40 ··· 40 50 ··· 50 50 ··· 50 100 ··· 100 200 ··· 200 size 1 1 4 4 2 1 1 1 1 2 2 1 1 1 1 4 ··· 4 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

175 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 C25 C50 C50 D4 D8 C5×D4 C5×D8 D4×C25 D8×C25 kernel D8×C25 C200 D4×C25 C5×D8 C40 C5×D4 D8 C8 D4 C50 C25 C10 C5 C2 C1 # reps 1 1 2 4 4 8 20 20 40 1 2 4 8 20 40

Matrix representation of D8×C25 in GL2(𝔽401) generated by

 385 0 0 385
,
 0 348 227 348
,
 1 0 1 400
G:=sub<GL(2,GF(401))| [385,0,0,385],[0,227,348,348],[1,1,0,400] >;

D8×C25 in GAP, Magma, Sage, TeX

D_8\times C_{25}
% in TeX

G:=Group("D8xC25");
// GroupNames label

G:=SmallGroup(400,25);
// by ID

G=gap.SmallGroup(400,25);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-5,-2,265,194,5283,2649,261]);
// Polycyclic

G:=Group<a,b,c|a^25=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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