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G = C8⋊D25order 400 = 24·52

3rd semidirect product of C8 and D25 acting via D25/C25=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D25, C2004C2, C40.7D5, D50.1C4, C4.13D50, C253M4(2), C20.52D10, Dic25.1C4, C100.13C22, C252C84C2, C50.9(C2×C4), C5.(C8⋊D5), C2.3(C4×D25), (C4×D25).2C2, C10.13(C4×D5), SmallGroup(400,6)

Series: Derived Chief Lower central Upper central

C1C50 — C8⋊D25
C1C5C25C50C100C4×D25 — C8⋊D25
C25C50 — C8⋊D25
C1C4C8

Generators and relations for C8⋊D25
 G = < a,b,c | a8=b25=c2=1, ab=ba, cac=a5, cbc=b-1 >

50C2
25C4
25C22
10D5
25C8
25C2×C4
5D10
5Dic5
2D25
25M4(2)
5C4×D5
5C52C8
5C8⋊D5

Smallest permutation representation of C8⋊D25
On 200 points
Generators in S200
(1 188 92 134 45 172 65 110)(2 189 93 135 46 173 66 111)(3 190 94 136 47 174 67 112)(4 191 95 137 48 175 68 113)(5 192 96 138 49 151 69 114)(6 193 97 139 50 152 70 115)(7 194 98 140 26 153 71 116)(8 195 99 141 27 154 72 117)(9 196 100 142 28 155 73 118)(10 197 76 143 29 156 74 119)(11 198 77 144 30 157 75 120)(12 199 78 145 31 158 51 121)(13 200 79 146 32 159 52 122)(14 176 80 147 33 160 53 123)(15 177 81 148 34 161 54 124)(16 178 82 149 35 162 55 125)(17 179 83 150 36 163 56 101)(18 180 84 126 37 164 57 102)(19 181 85 127 38 165 58 103)(20 182 86 128 39 166 59 104)(21 183 87 129 40 167 60 105)(22 184 88 130 41 168 61 106)(23 185 89 131 42 169 62 107)(24 186 90 132 43 170 63 108)(25 187 91 133 44 171 64 109)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(51 53)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(76 82)(77 81)(78 80)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 142)(102 141)(103 140)(104 139)(105 138)(106 137)(107 136)(108 135)(109 134)(110 133)(111 132)(112 131)(113 130)(114 129)(115 128)(116 127)(117 126)(118 150)(119 149)(120 148)(121 147)(122 146)(123 145)(124 144)(125 143)(151 183)(152 182)(153 181)(154 180)(155 179)(156 178)(157 177)(158 176)(159 200)(160 199)(161 198)(162 197)(163 196)(164 195)(165 194)(166 193)(167 192)(168 191)(169 190)(170 189)(171 188)(172 187)(173 186)(174 185)(175 184)

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

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

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

106 conjugacy classes

class 1 2A2B4A4B4C5A5B8A8B8C8D10A10B20A20B20C20D25A···25J40A···40H50A···50J100A···100T200A···200AN
order12244455888810102020202025···2540···4050···50100···100200···200
size11501150222250502222222···22···22···22···22···2

106 irreducible representations

dim111111222222222
type++++++++
imageC1C2C2C2C4C4D5M4(2)D10C4×D5D25C8⋊D5D50C4×D25C8⋊D25
kernelC8⋊D25C252C8C200C4×D25Dic25D50C40C25C20C10C8C5C4C2C1
# reps1111222224108102040

Matrix representation of C8⋊D25 in GL2(𝔽401) generated by

227185
348174
,
311339
18999
,
329366
17172
G:=sub<GL(2,GF(401))| [227,348,185,174],[311,189,339,99],[329,171,366,72] >;

C8⋊D25 in GAP, Magma, Sage, TeX

C_8\rtimes D_{25}
% in TeX

G:=Group("C8:D25");
// GroupNames label

G:=SmallGroup(400,6);
// by ID

G=gap.SmallGroup(400,6);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,31,50,4324,628,11525]);
// Polycyclic

G:=Group<a,b,c|a^8=b^25=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D25 in TeX

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