Copied to
clipboard

## G = C2×C4×D25order 400 = 24·52

### Direct product of C2×C4 and D25

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C2×C4×D25
 Chief series C1 — C5 — C25 — C50 — D50 — C22×D25 — C2×C4×D25
 Lower central C25 — C2×C4×D25
 Upper central C1 — C2×C4

Generators and relations for C2×C4×D25
G = < a,b,c,d | a2=b4=c25=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 613 in 81 conjugacy classes, 43 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C10, C22×C4, Dic5, C20, D10, C2×C10, C25, C4×D5, C2×Dic5, C2×C20, C22×D5, D25, C50, C50, C2×C4×D5, Dic25, C100, D50, C2×C50, C4×D25, C2×Dic25, C2×C100, C22×D25, C2×C4×D25
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C4×D5, C22×D5, D25, C2×C4×D5, D50, C4×D25, C22×D25, C2×C4×D25

Smallest permutation representation of C2×C4×D25
On 200 points
Generators in S200
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 121)(13 122)(14 123)(15 124)(16 125)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 138)(27 139)(28 140)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 126)(40 127)(41 128)(42 129)(43 130)(44 131)(45 132)(46 133)(47 134)(48 135)(49 136)(50 137)(51 172)(52 173)(53 174)(54 175)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 167)(72 168)(73 169)(74 170)(75 171)(76 176)(77 177)(78 178)(79 179)(80 180)(81 181)(82 182)(83 183)(84 184)(85 185)(86 186)(87 187)(88 188)(89 189)(90 190)(91 191)(92 192)(93 193)(94 194)(95 195)(96 196)(97 197)(98 198)(99 199)(100 200)
(1 77 35 52)(2 78 36 53)(3 79 37 54)(4 80 38 55)(5 81 39 56)(6 82 40 57)(7 83 41 58)(8 84 42 59)(9 85 43 60)(10 86 44 61)(11 87 45 62)(12 88 46 63)(13 89 47 64)(14 90 48 65)(15 91 49 66)(16 92 50 67)(17 93 26 68)(18 94 27 69)(19 95 28 70)(20 96 29 71)(21 97 30 72)(22 98 31 73)(23 99 32 74)(24 100 33 75)(25 76 34 51)(101 193 138 164)(102 194 139 165)(103 195 140 166)(104 196 141 167)(105 197 142 168)(106 198 143 169)(107 199 144 170)(108 200 145 171)(109 176 146 172)(110 177 147 173)(111 178 148 174)(112 179 149 175)(113 180 150 151)(114 181 126 152)(115 182 127 153)(116 183 128 154)(117 184 129 155)(118 185 130 156)(119 186 131 157)(120 187 132 158)(121 188 133 159)(122 189 134 160)(123 190 135 161)(124 191 136 162)(125 192 137 163)
(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 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(44 50)(45 49)(46 48)(51 52)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(76 77)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 110)(119 125)(120 124)(121 123)(126 142)(127 141)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(143 150)(144 149)(145 148)(146 147)(151 169)(152 168)(153 167)(154 166)(155 165)(156 164)(157 163)(158 162)(159 161)(170 175)(171 174)(172 173)(176 177)(178 200)(179 199)(180 198)(181 197)(182 196)(183 195)(184 194)(185 193)(186 192)(187 191)(188 190)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 10A ··· 10F 20A ··· 20H 25A ··· 25J 50A ··· 50AD 100A ··· 100AN order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 1 1 25 25 25 25 1 1 1 1 25 25 25 25 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 D5 D10 D10 C4×D5 D25 D50 D50 C4×D25 kernel C2×C4×D25 C4×D25 C2×Dic25 C2×C100 C22×D25 D50 C2×C20 C20 C2×C10 C10 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 2 4 2 8 10 20 10 40

Matrix representation of C2×C4×D25 in GL4(𝔽101) generated by

 100 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 91 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 46 52 0 0 49 62
,
 100 0 0 0 0 100 0 0 0 0 46 52 0 0 4 55
G:=sub<GL(4,GF(101))| [100,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,46,49,0,0,52,62],[100,0,0,0,0,100,0,0,0,0,46,4,0,0,52,55] >;

C2×C4×D25 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_{25}
% in TeX

G:=Group("C2xC4xD25");
// GroupNames label

G:=SmallGroup(400,36);
// by ID

G=gap.SmallGroup(400,36);
# by ID

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

G:=Group<a,b,c,d|a^2=b^4=c^25=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

׿
×
𝔽