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## G = C22⋊C4×C25order 400 = 24·52

### Direct product of C25 and C22⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22⋊C4×C25
 Chief series C1 — C5 — C10 — C2×C10 — C2×C50 — C2×C100 — C22⋊C4×C25
 Lower central C1 — C2 — C22⋊C4×C25
 Upper central C1 — C2×C50 — C22⋊C4×C25

Generators and relations for C22⋊C4×C25
G = < a,b,c,d | a25=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Smallest permutation representation of C22⋊C4×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 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 51)(23 52)(24 53)(25 54)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 93)(50 94)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 183)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(115 190)(116 191)(117 192)(118 193)(119 194)(120 195)(121 196)(122 197)(123 198)(124 199)(125 200)(126 168)(127 169)(128 170)(129 171)(130 172)(131 173)(132 174)(133 175)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)(141 158)(142 159)(143 160)(144 161)(145 162)(146 163)(147 164)(148 165)(149 166)(150 167)
(1 48)(2 49)(3 50)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)(61 98)(62 99)(63 100)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)(73 85)(74 86)(75 87)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 126)(124 127)(125 128)(151 181)(152 182)(153 183)(154 184)(155 185)(156 186)(157 187)(158 188)(159 189)(160 190)(161 191)(162 192)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 199)(170 200)(171 176)(172 177)(173 178)(174 179)(175 180)
(1 148 55 195)(2 149 56 196)(3 150 57 197)(4 126 58 198)(5 127 59 199)(6 128 60 200)(7 129 61 176)(8 130 62 177)(9 131 63 178)(10 132 64 179)(11 133 65 180)(12 134 66 181)(13 135 67 182)(14 136 68 183)(15 137 69 184)(16 138 70 185)(17 139 71 186)(18 140 72 187)(19 141 73 188)(20 142 74 189)(21 143 75 190)(22 144 51 191)(23 145 52 192)(24 146 53 193)(25 147 54 194)(26 123 95 168)(27 124 96 169)(28 125 97 170)(29 101 98 171)(30 102 99 172)(31 103 100 173)(32 104 76 174)(33 105 77 175)(34 106 78 151)(35 107 79 152)(36 108 80 153)(37 109 81 154)(38 110 82 155)(39 111 83 156)(40 112 84 157)(41 113 85 158)(42 114 86 159)(43 115 87 160)(44 116 88 161)(45 117 89 162)(46 118 90 163)(47 119 91 164)(48 120 92 165)(49 121 93 166)(50 122 94 167)

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

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

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

250 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 10A ··· 10L 10M ··· 10T 20A ··· 20P 25A ··· 25T 50A ··· 50BH 50BI ··· 50CV 100A ··· 100CB order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 25 ··· 25 50 ··· 50 50 ··· 50 100 ··· 100 size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

250 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + image C1 C2 C2 C4 C5 C10 C10 C20 C25 C50 C50 C100 D4 C5×D4 D4×C25 kernel C22⋊C4×C25 C2×C100 C22×C50 C2×C50 C5×C22⋊C4 C2×C20 C22×C10 C2×C10 C22⋊C4 C2×C4 C23 C22 C50 C10 C2 # reps 1 2 1 4 4 8 4 16 20 40 20 80 2 8 40

Matrix representation of C22⋊C4×C25 in GL3(𝔽101) generated by

 1 0 0 0 68 0 0 0 68
,
 1 0 0 0 1 24 0 0 100
,
 1 0 0 0 100 0 0 0 100
,
 10 0 0 0 1 0 0 42 100
G:=sub<GL(3,GF(101))| [1,0,0,0,68,0,0,0,68],[1,0,0,0,1,0,0,24,100],[1,0,0,0,100,0,0,0,100],[10,0,0,0,1,42,0,0,100] >;

C22⋊C4×C25 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times C_{25}
% in TeX

G:=Group("C2^2:C4xC25");
// GroupNames label

G:=SmallGroup(400,21);
// by ID

G=gap.SmallGroup(400,21);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,240,265,374]);
// Polycyclic

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

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