Copied to
clipboard

G = C8×D25order 400 = 24·52

Direct product of C8 and D25

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C8×D25, C2003C2, C40.6D5, D50.4C4, C4.12D50, C20.51D10, Dic25.4C4, C100.12C22, C5.(C8×D5), C253(C2×C8), C252C86C2, C50.8(C2×C4), C2.1(C4×D25), (C4×D25).7C2, C10.12(C4×D5), SmallGroup(400,5)

Series: Derived Chief Lower central Upper central

C1C25 — C8×D25
C1C5C25C50C100C4×D25 — C8×D25
C25 — C8×D25
C1C8

Generators and relations for C8×D25
 G = < a,b,c | a8=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >

25C2
25C2
25C22
25C4
5D5
5D5
25C2×C4
25C8
5Dic5
5D10
25C2×C8
5C52C8
5C4×D5
5C8×D5

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

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

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

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

112 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B8A8B8C8D8E8F8G8H10A10B20A20B20C20D25A···25J40A···40H50A···50J100A···100T200A···200AN
order12224444558888888810102020202025···2540···4050···50100···100200···200
size112525112525221111252525252222222···22···22···22···22···2

112 irreducible representations

dim111111122222222
type++++++++
imageC1C2C2C2C4C4C8D5D10C4×D5D25C8×D5D50C4×D25C8×D25
kernelC8×D25C252C8C200C4×D25Dic25D50D25C40C20C10C8C5C4C2C1
# reps1111228224108102040

Matrix representation of C8×D25 in GL3(𝔽401) generated by

35600
03810
00381
,
100
062373
028162
,
100
062373
037339
G:=sub<GL(3,GF(401))| [356,0,0,0,381,0,0,0,381],[1,0,0,0,62,28,0,373,162],[1,0,0,0,62,37,0,373,339] >;

C8×D25 in GAP, Magma, Sage, TeX

C_8\times D_{25}
% in TeX

G:=Group("C8xD25");
// GroupNames label

G:=SmallGroup(400,5);
// by ID

G=gap.SmallGroup(400,5);
# by ID

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

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

Export

Subgroup lattice of C8×D25 in TeX

׿
×
𝔽