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## G = C2×C25⋊D4order 400 = 24·52

### Direct product of C2 and C25⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C50 — C2×C25⋊D4
 Chief series C1 — C5 — C25 — C50 — D50 — C22×D25 — C2×C25⋊D4
 Lower central C25 — C50 — C2×C25⋊D4
 Upper central C1 — C22 — C23

Generators and relations for C2×C25⋊D4
G = < a,b,c,d | a2=b25=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 637 in 81 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C23, D5, C10, C10, C10, C2×D4, Dic5, D10, C2×C10, C2×C10, C2×C10, C25, C2×Dic5, C5⋊D4, C22×D5, C22×C10, D25, C50, C50, C50, C2×C5⋊D4, Dic25, D50, D50, C2×C50, C2×C50, C2×C50, C2×Dic25, C25⋊D4, C22×D25, C22×C50, C2×C25⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C5⋊D4, C22×D5, D25, C2×C5⋊D4, D50, C25⋊D4, C22×D25, C2×C25⋊D4

Smallest permutation representation of C2×C25⋊D4
On 200 points
Generators in S200
(1 121)(2 122)(3 123)(4 124)(5 125)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 147)(27 148)(28 149)(29 150)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 193)(77 194)(78 195)(79 196)(80 197)(81 198)(82 199)(83 200)(84 176)(85 177)(86 178)(87 179)(88 180)(89 181)(90 182)(91 183)(92 184)(93 185)(94 186)(95 187)(96 188)(97 189)(98 190)(99 191)(100 192)
(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 56 35 96)(2 55 36 95)(3 54 37 94)(4 53 38 93)(5 52 39 92)(6 51 40 91)(7 75 41 90)(8 74 42 89)(9 73 43 88)(10 72 44 87)(11 71 45 86)(12 70 46 85)(13 69 47 84)(14 68 48 83)(15 67 49 82)(16 66 50 81)(17 65 26 80)(18 64 27 79)(19 63 28 78)(20 62 29 77)(21 61 30 76)(22 60 31 100)(23 59 32 99)(24 58 33 98)(25 57 34 97)(101 163 136 183)(102 162 137 182)(103 161 138 181)(104 160 139 180)(105 159 140 179)(106 158 141 178)(107 157 142 177)(108 156 143 176)(109 155 144 200)(110 154 145 199)(111 153 146 198)(112 152 147 197)(113 151 148 196)(114 175 149 195)(115 174 150 194)(116 173 126 193)(117 172 127 192)(118 171 128 191)(119 170 129 190)(120 169 130 189)(121 168 131 188)(122 167 132 187)(123 166 133 186)(124 165 134 185)(125 164 135 184)
(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(13 14)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(45 50)(46 49)(47 48)(51 76)(52 100)(53 99)(54 98)(55 97)(56 96)(57 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(117 125)(118 124)(119 123)(120 122)(126 136)(127 135)(128 134)(129 133)(130 132)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(151 180)(152 179)(153 178)(154 177)(155 176)(156 200)(157 199)(158 198)(159 197)(160 196)(161 195)(162 194)(163 193)(164 192)(165 191)(166 190)(167 189)(168 188)(169 187)(170 186)(171 185)(172 184)(173 183)(174 182)(175 181)

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

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

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

106 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 10A ··· 10N 25A ··· 25J 50A ··· 50BR order 1 2 2 2 2 2 2 2 4 4 5 5 10 ··· 10 25 ··· 25 50 ··· 50 size 1 1 1 1 2 2 50 50 50 50 2 2 2 ··· 2 2 ··· 2 2 ··· 2

106 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D5 D10 C5⋊D4 D25 D50 C25⋊D4 kernel C2×C25⋊D4 C2×Dic25 C25⋊D4 C22×D25 C22×C50 C50 C22×C10 C2×C10 C10 C23 C22 C2 # reps 1 1 4 1 1 2 2 6 8 10 30 40

Matrix representation of C2×C25⋊D4 in GL3(𝔽101) generated by

 100 0 0 0 100 0 0 0 100
,
 1 0 0 0 39 64 0 37 97
,
 1 0 0 0 2 2 0 48 99
,
 100 0 0 0 78 23 0 100 23
G:=sub<GL(3,GF(101))| [100,0,0,0,100,0,0,0,100],[1,0,0,0,39,37,0,64,97],[1,0,0,0,2,48,0,2,99],[100,0,0,0,78,100,0,23,23] >;

C2×C25⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_{25}\rtimes D_4
% in TeX

G:=Group("C2xC25:D4");
// GroupNames label

G:=SmallGroup(400,44);
// by ID

G=gap.SmallGroup(400,44);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,218,4324,628,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^25=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
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