metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C25⋊2C8, C50.2C4, C4.2D25, C20.4D5, C2.Dic25, C100.2C2, C10.2Dic5, C5.(C5⋊2C8), SmallGroup(200,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C25⋊2C8 |
Generators and relations for C25⋊2C8
G = < a,b | a25=b8=1, bab-1=a-1 >
Smallest permutation representation of C25⋊2C8
►Regular action 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)
(1 182 89 150 31 167 64 103)(2 181 90 149 32 166 65 102)(3 180 91 148 33 165 66 101)(4 179 92 147 34 164 67 125)(5 178 93 146 35 163 68 124)(6 177 94 145 36 162 69 123)(7 176 95 144 37 161 70 122)(8 200 96 143 38 160 71 121)(9 199 97 142 39 159 72 120)(10 198 98 141 40 158 73 119)(11 197 99 140 41 157 74 118)(12 196 100 139 42 156 75 117)(13 195 76 138 43 155 51 116)(14 194 77 137 44 154 52 115)(15 193 78 136 45 153 53 114)(16 192 79 135 46 152 54 113)(17 191 80 134 47 151 55 112)(18 190 81 133 48 175 56 111)(19 189 82 132 49 174 57 110)(20 188 83 131 50 173 58 109)(21 187 84 130 26 172 59 108)(22 186 85 129 27 171 60 107)(23 185 86 128 28 170 61 106)(24 184 87 127 29 169 62 105)(25 183 88 126 30 168 63 104)
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), (1,182,89,150,31,167,64,103)(2,181,90,149,32,166,65,102)(3,180,91,148,33,165,66,101)(4,179,92,147,34,164,67,125)(5,178,93,146,35,163,68,124)(6,177,94,145,36,162,69,123)(7,176,95,144,37,161,70,122)(8,200,96,143,38,160,71,121)(9,199,97,142,39,159,72,120)(10,198,98,141,40,158,73,119)(11,197,99,140,41,157,74,118)(12,196,100,139,42,156,75,117)(13,195,76,138,43,155,51,116)(14,194,77,137,44,154,52,115)(15,193,78,136,45,153,53,114)(16,192,79,135,46,152,54,113)(17,191,80,134,47,151,55,112)(18,190,81,133,48,175,56,111)(19,189,82,132,49,174,57,110)(20,188,83,131,50,173,58,109)(21,187,84,130,26,172,59,108)(22,186,85,129,27,171,60,107)(23,185,86,128,28,170,61,106)(24,184,87,127,29,169,62,105)(25,183,88,126,30,168,63,104)>;
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), (1,182,89,150,31,167,64,103)(2,181,90,149,32,166,65,102)(3,180,91,148,33,165,66,101)(4,179,92,147,34,164,67,125)(5,178,93,146,35,163,68,124)(6,177,94,145,36,162,69,123)(7,176,95,144,37,161,70,122)(8,200,96,143,38,160,71,121)(9,199,97,142,39,159,72,120)(10,198,98,141,40,158,73,119)(11,197,99,140,41,157,74,118)(12,196,100,139,42,156,75,117)(13,195,76,138,43,155,51,116)(14,194,77,137,44,154,52,115)(15,193,78,136,45,153,53,114)(16,192,79,135,46,152,54,113)(17,191,80,134,47,151,55,112)(18,190,81,133,48,175,56,111)(19,189,82,132,49,174,57,110)(20,188,83,131,50,173,58,109)(21,187,84,130,26,172,59,108)(22,186,85,129,27,171,60,107)(23,185,86,128,28,170,61,106)(24,184,87,127,29,169,62,105)(25,183,88,126,30,168,63,104) );
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)], [(1,182,89,150,31,167,64,103),(2,181,90,149,32,166,65,102),(3,180,91,148,33,165,66,101),(4,179,92,147,34,164,67,125),(5,178,93,146,35,163,68,124),(6,177,94,145,36,162,69,123),(7,176,95,144,37,161,70,122),(8,200,96,143,38,160,71,121),(9,199,97,142,39,159,72,120),(10,198,98,141,40,158,73,119),(11,197,99,140,41,157,74,118),(12,196,100,139,42,156,75,117),(13,195,76,138,43,155,51,116),(14,194,77,137,44,154,52,115),(15,193,78,136,45,153,53,114),(16,192,79,135,46,152,54,113),(17,191,80,134,47,151,55,112),(18,190,81,133,48,175,56,111),(19,189,82,132,49,174,57,110),(20,188,83,131,50,173,58,109),(21,187,84,130,26,172,59,108),(22,186,85,129,27,171,60,107),(23,185,86,128,28,170,61,106),(24,184,87,127,29,169,62,105),(25,183,88,126,30,168,63,104)]])
C25⋊2C8 is a maximal subgroup of
C25⋊C16 C8×D25 C8⋊D25 C4.Dic25 D4.D25 D4⋊D25 C25⋊Q16 Q8⋊D25
C25⋊2C8 is a maximal quotient of
C25⋊2C16
56 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 20A | 20B | 20C | 20D | 25A | ··· | 25J | 50A | ··· | 50J | 100A | ··· | 100T |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 20 | 20 | 20 | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 25 | 25 | 25 | 25 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||
| image | C1 | C2 | C4 | C8 | D5 | Dic5 | C5⋊2C8 | D25 | Dic25 | C25⋊2C8 |
| kernel | C25⋊2C8 | C100 | C50 | C25 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 10 | 10 | 20 |
Matrix representation of C25⋊2C8 ►in GL2(𝔽401) generated by
| 267 | 162 |
| 140 | 127 |
| 384 | 192 |
| 126 | 17 |
G:=sub<GL(2,GF(401))| [267,140,162,127],[384,126,192,17] >;
C25⋊2C8 in GAP, Magma, Sage, TeX
C_{25}\rtimes_2C_8 % in TeX
G:=Group("C25:2C8"); // GroupNames label
G:=SmallGroup(200,1);
// by ID
G=gap.SmallGroup(200,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,10,26,1443,418,4004]);
// Polycyclic
G:=Group<a,b|a^25=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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