direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C25, C8⋊3C50, C4.C100, C200⋊7C2, C100.7C4, C20.8C20, C40.7C10, C22.C100, C100.22C22, C4.5(C2×C50), (C2×C4).2C50, (C2×C50).3C4, C5.(C5×M4(2)), (C2×C100).8C2, (C2×C10).4C20, C2.3(C2×C100), C50.19(C2×C4), (C5×M4(2)).C5, C20.28(C2×C10), (C2×C20).10C10, C10.19(C2×C20), SmallGroup(400,24)
Series: Derived ►Chief ►Lower central ►Upper central
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 81 198 137 72 156 118 35)(2 82 199 138 73 157 119 36)(3 83 200 139 74 158 120 37)(4 84 176 140 75 159 121 38)(5 85 177 141 51 160 122 39)(6 86 178 142 52 161 123 40)(7 87 179 143 53 162 124 41)(8 88 180 144 54 163 125 42)(9 89 181 145 55 164 101 43)(10 90 182 146 56 165 102 44)(11 91 183 147 57 166 103 45)(12 92 184 148 58 167 104 46)(13 93 185 149 59 168 105 47)(14 94 186 150 60 169 106 48)(15 95 187 126 61 170 107 49)(16 96 188 127 62 171 108 50)(17 97 189 128 63 172 109 26)(18 98 190 129 64 173 110 27)(19 99 191 130 65 174 111 28)(20 100 192 131 66 175 112 29)(21 76 193 132 67 151 113 30)(22 77 194 133 68 152 114 31)(23 78 195 134 69 153 115 32)(24 79 196 135 70 154 116 33)(25 80 197 136 71 155 117 34)
(26 128)(27 129)(28 130)(29 131)(30 132)(31 133)(32 134)(33 135)(34 136)(35 137)(36 138)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 126)(50 127)(76 151)(77 152)(78 153)(79 154)(80 155)(81 156)(82 157)(83 158)(84 159)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 169)(95 170)(96 171)(97 172)(98 173)(99 174)(100 175)
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), (1,81,198,137,72,156,118,35)(2,82,199,138,73,157,119,36)(3,83,200,139,74,158,120,37)(4,84,176,140,75,159,121,38)(5,85,177,141,51,160,122,39)(6,86,178,142,52,161,123,40)(7,87,179,143,53,162,124,41)(8,88,180,144,54,163,125,42)(9,89,181,145,55,164,101,43)(10,90,182,146,56,165,102,44)(11,91,183,147,57,166,103,45)(12,92,184,148,58,167,104,46)(13,93,185,149,59,168,105,47)(14,94,186,150,60,169,106,48)(15,95,187,126,61,170,107,49)(16,96,188,127,62,171,108,50)(17,97,189,128,63,172,109,26)(18,98,190,129,64,173,110,27)(19,99,191,130,65,174,111,28)(20,100,192,131,66,175,112,29)(21,76,193,132,67,151,113,30)(22,77,194,133,68,152,114,31)(23,78,195,134,69,153,115,32)(24,79,196,135,70,154,116,33)(25,80,197,136,71,155,117,34), (26,128)(27,129)(28,130)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,137)(36,138)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,126)(50,127)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,171)(97,172)(98,173)(99,174)(100,175)>;
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), (1,81,198,137,72,156,118,35)(2,82,199,138,73,157,119,36)(3,83,200,139,74,158,120,37)(4,84,176,140,75,159,121,38)(5,85,177,141,51,160,122,39)(6,86,178,142,52,161,123,40)(7,87,179,143,53,162,124,41)(8,88,180,144,54,163,125,42)(9,89,181,145,55,164,101,43)(10,90,182,146,56,165,102,44)(11,91,183,147,57,166,103,45)(12,92,184,148,58,167,104,46)(13,93,185,149,59,168,105,47)(14,94,186,150,60,169,106,48)(15,95,187,126,61,170,107,49)(16,96,188,127,62,171,108,50)(17,97,189,128,63,172,109,26)(18,98,190,129,64,173,110,27)(19,99,191,130,65,174,111,28)(20,100,192,131,66,175,112,29)(21,76,193,132,67,151,113,30)(22,77,194,133,68,152,114,31)(23,78,195,134,69,153,115,32)(24,79,196,135,70,154,116,33)(25,80,197,136,71,155,117,34), (26,128)(27,129)(28,130)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,137)(36,138)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,126)(50,127)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,171)(97,172)(98,173)(99,174)(100,175) );
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)], [(1,81,198,137,72,156,118,35),(2,82,199,138,73,157,119,36),(3,83,200,139,74,158,120,37),(4,84,176,140,75,159,121,38),(5,85,177,141,51,160,122,39),(6,86,178,142,52,161,123,40),(7,87,179,143,53,162,124,41),(8,88,180,144,54,163,125,42),(9,89,181,145,55,164,101,43),(10,90,182,146,56,165,102,44),(11,91,183,147,57,166,103,45),(12,92,184,148,58,167,104,46),(13,93,185,149,59,168,105,47),(14,94,186,150,60,169,106,48),(15,95,187,126,61,170,107,49),(16,96,188,127,62,171,108,50),(17,97,189,128,63,172,109,26),(18,98,190,129,64,173,110,27),(19,99,191,130,65,174,111,28),(20,100,192,131,66,175,112,29),(21,76,193,132,67,151,113,30),(22,77,194,133,68,152,114,31),(23,78,195,134,69,153,115,32),(24,79,196,135,70,154,116,33),(25,80,197,136,71,155,117,34)], [(26,128),(27,129),(28,130),(29,131),(30,132),(31,133),(32,134),(33,135),(34,136),(35,137),(36,138),(37,139),(38,140),(39,141),(40,142),(41,143),(42,144),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,126),(50,127),(76,151),(77,152),(78,153),(79,154),(80,155),(81,156),(82,157),(83,158),(84,159),(85,160),(86,161),(87,162),(88,163),(89,164),(90,165),(91,166),(92,167),(93,168),(94,169),(95,170),(96,171),(97,172),(98,173),(99,174),(100,175)]])
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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