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## G = M4(2)×C25order 400 = 24·52

### Direct product of C25 and M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C25
 Chief series C1 — C2 — C10 — C20 — C100 — C200 — M4(2)×C25
 Lower central C1 — C2 — M4(2)×C25
 Upper central C1 — C100 — M4(2)×C25

Generators and relations for M4(2)×C25
G = < a,b,c | a25=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Smallest permutation representation of M4(2)×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 88 194 154 148 27 123 71)(2 89 195 155 149 28 124 72)(3 90 196 156 150 29 125 73)(4 91 197 157 126 30 101 74)(5 92 198 158 127 31 102 75)(6 93 199 159 128 32 103 51)(7 94 200 160 129 33 104 52)(8 95 176 161 130 34 105 53)(9 96 177 162 131 35 106 54)(10 97 178 163 132 36 107 55)(11 98 179 164 133 37 108 56)(12 99 180 165 134 38 109 57)(13 100 181 166 135 39 110 58)(14 76 182 167 136 40 111 59)(15 77 183 168 137 41 112 60)(16 78 184 169 138 42 113 61)(17 79 185 170 139 43 114 62)(18 80 186 171 140 44 115 63)(19 81 187 172 141 45 116 64)(20 82 188 173 142 46 117 65)(21 83 189 174 143 47 118 66)(22 84 190 175 144 48 119 67)(23 85 191 151 145 49 120 68)(24 86 192 152 146 50 121 69)(25 87 193 153 147 26 122 70)
(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 159)(52 160)(53 161)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 175)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)

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

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

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

250 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 20A ··· 20H 20I 20J 20K 20L 25A ··· 25T 40A ··· 40P 50A ··· 50T 50U ··· 50AN 100A ··· 100AN 100AO ··· 100BH 200A ··· 200CB order 1 2 2 4 4 4 5 5 5 5 8 8 8 8 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 25 ··· 25 40 ··· 40 50 ··· 50 50 ··· 50 100 ··· 100 100 ··· 100 200 ··· 200 size 1 1 2 1 1 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

250 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 C25 C50 C50 C100 C100 M4(2) C5×M4(2) M4(2)×C25 kernel M4(2)×C25 C200 C2×C100 C100 C2×C50 C5×M4(2) C40 C2×C20 C20 C2×C10 M4(2) C8 C2×C4 C4 C22 C25 C5 C1 # reps 1 2 1 2 2 4 8 4 8 8 20 40 20 40 40 2 8 40

Matrix representation of M4(2)×C25 in GL2(𝔽401) generated by

 63 0 0 63
,
 0 1 381 0
,
 1 0 0 400
G:=sub<GL(2,GF(401))| [63,0,0,63],[0,381,1,0],[1,0,0,400] >;

M4(2)×C25 in GAP, Magma, Sage, TeX

M_4(2)\times C_{25}
% in TeX

G:=Group("M4(2)xC25");
// GroupNames label

G:=SmallGroup(400,24);
// by ID

G=gap.SmallGroup(400,24);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-5,-2,120,2425,194,261]);
// Polycyclic

G:=Group<a,b,c|a^25=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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