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## G = C4○D4×C25order 400 = 24·52

### Direct product of C25 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C25
 Chief series C1 — C5 — C10 — C50 — C2×C50 — D4×C25 — C4○D4×C25
 Lower central C1 — C2 — C4○D4×C25
 Upper central C1 — C100 — C4○D4×C25

Generators and relations for C4○D4×C25
G = < a,b,c,d | a25=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 69 in 60 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, D4, Q8, C10, C10, C4○D4, C20, C20, C2×C10, C25, C2×C20, C5×D4, C5×Q8, C50, C50, C5×C4○D4, C100, C100, C2×C50, C2×C100, D4×C25, Q8×C25, C4○D4×C25
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C2×C10, C25, C22×C10, C50, C5×C4○D4, C2×C50, C22×C50, C4○D4×C25

Smallest permutation representation of C4○D4×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 47 126 117)(2 48 127 118)(3 49 128 119)(4 50 129 120)(5 26 130 121)(6 27 131 122)(7 28 132 123)(8 29 133 124)(9 30 134 125)(10 31 135 101)(11 32 136 102)(12 33 137 103)(13 34 138 104)(14 35 139 105)(15 36 140 106)(16 37 141 107)(17 38 142 108)(18 39 143 109)(19 40 144 110)(20 41 145 111)(21 42 146 112)(22 43 147 113)(23 44 148 114)(24 45 149 115)(25 46 150 116)(51 197 152 83)(52 198 153 84)(53 199 154 85)(54 200 155 86)(55 176 156 87)(56 177 157 88)(57 178 158 89)(58 179 159 90)(59 180 160 91)(60 181 161 92)(61 182 162 93)(62 183 163 94)(63 184 164 95)(64 185 165 96)(65 186 166 97)(66 187 167 98)(67 188 168 99)(68 189 169 100)(69 190 170 76)(70 191 171 77)(71 192 172 78)(72 193 173 79)(73 194 174 80)(74 195 175 81)(75 196 151 82)
(1 100 126 189)(2 76 127 190)(3 77 128 191)(4 78 129 192)(5 79 130 193)(6 80 131 194)(7 81 132 195)(8 82 133 196)(9 83 134 197)(10 84 135 198)(11 85 136 199)(12 86 137 200)(13 87 138 176)(14 88 139 177)(15 89 140 178)(16 90 141 179)(17 91 142 180)(18 92 143 181)(19 93 144 182)(20 94 145 183)(21 95 146 184)(22 96 147 185)(23 97 148 186)(24 98 149 187)(25 99 150 188)(26 72 121 173)(27 73 122 174)(28 74 123 175)(29 75 124 151)(30 51 125 152)(31 52 101 153)(32 53 102 154)(33 54 103 155)(34 55 104 156)(35 56 105 157)(36 57 106 158)(37 58 107 159)(38 59 108 160)(39 60 109 161)(40 61 110 162)(41 62 111 163)(42 63 112 164)(43 64 113 165)(44 65 114 166)(45 66 115 167)(46 67 116 168)(47 68 117 169)(48 69 118 170)(49 70 119 171)(50 71 120 172)
(51 152)(52 153)(53 154)(54 155)(55 156)(56 157)(57 158)(58 159)(59 160)(60 161)(61 162)(62 163)(63 164)(64 165)(65 166)(66 167)(67 168)(68 169)(69 170)(70 171)(71 172)(72 173)(73 174)(74 175)(75 151)(76 190)(77 191)(78 192)(79 193)(80 194)(81 195)(82 196)(83 197)(84 198)(85 199)(86 200)(87 176)(88 177)(89 178)(90 179)(91 180)(92 181)(93 182)(94 183)(95 184)(96 185)(97 186)(98 187)(99 188)(100 189)

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

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

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

250 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 10A 10B 10C 10D 10E ··· 10P 20A ··· 20H 20I ··· 20T 25A ··· 25T 50A ··· 50T 50U ··· 50CB 100A ··· 100AN 100AO ··· 100CV order 1 2 2 2 2 4 4 4 4 4 5 5 5 5 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 25 ··· 25 50 ··· 50 50 ··· 50 100 ··· 100 100 ··· 100 size 1 1 2 2 2 1 1 2 2 2 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 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 C2 C5 C10 C10 C10 C25 C50 C50 C50 C4○D4 C5×C4○D4 C4○D4×C25 kernel C4○D4×C25 C2×C100 D4×C25 Q8×C25 C5×C4○D4 C2×C20 C5×D4 C5×Q8 C4○D4 C2×C4 D4 Q8 C25 C5 C1 # reps 1 3 3 1 4 12 12 4 20 60 60 20 2 8 40

Matrix representation of C4○D4×C25 in GL2(𝔽101) generated by

 78 0 0 78
,
 10 0 0 10
,
 15 65 68 86
,
 1 56 0 100
G:=sub<GL(2,GF(101))| [78,0,0,78],[10,0,0,10],[15,68,65,86],[1,0,56,100] >;

C4○D4×C25 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{25}
% in TeX

G:=Group("C4oD4xC25");
// GroupNames label

G:=SmallGroup(400,48);
// by ID

G=gap.SmallGroup(400,48);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,505,194,261]);
// Polycyclic

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

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