metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C20.1F5, C100.1C4, D50.3C4, C25⋊1M4(2), Dic25.6C22, C25⋊C8⋊1C2, C4.(C25⋊C4), C5.(C4.F5), C50.2(C2×C4), C10.7(C2×F5), (C4×D25).3C2, C2.4(C2×C25⋊C4), SmallGroup(400,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C100.C4
G = < a,b | a100=1, b4=a50, bab-1=a43 >
Smallest permutation representation of C100.C4
►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)
(1 135 76 160 51 185 26 110)(2 142 25 103 52 192 75 153)(3 149 74 146 53 199 24 196)(4 156 23 189 54 106 73 139)(5 163 72 132 55 113 22 182)(6 170 21 175 56 120 71 125)(7 177 70 118 57 127 20 168)(8 184 19 161 58 134 69 111)(9 191 68 104 59 141 18 154)(10 198 17 147 60 148 67 197)(11 105 66 190 61 155 16 140)(12 112 15 133 62 162 65 183)(13 119 64 176 63 169 14 126)(27 117 50 178 77 167 100 128)(28 124 99 121 78 174 49 171)(29 131 48 164 79 181 98 114)(30 138 97 107 80 188 47 157)(31 145 46 150 81 195 96 200)(32 152 95 193 82 102 45 143)(33 159 44 136 83 109 94 186)(34 166 93 179 84 116 43 129)(35 173 42 122 85 123 92 172)(36 180 91 165 86 130 41 115)(37 187 40 108 87 137 90 158)(38 194 89 151 88 144 39 101)
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), (1,135,76,160,51,185,26,110)(2,142,25,103,52,192,75,153)(3,149,74,146,53,199,24,196)(4,156,23,189,54,106,73,139)(5,163,72,132,55,113,22,182)(6,170,21,175,56,120,71,125)(7,177,70,118,57,127,20,168)(8,184,19,161,58,134,69,111)(9,191,68,104,59,141,18,154)(10,198,17,147,60,148,67,197)(11,105,66,190,61,155,16,140)(12,112,15,133,62,162,65,183)(13,119,64,176,63,169,14,126)(27,117,50,178,77,167,100,128)(28,124,99,121,78,174,49,171)(29,131,48,164,79,181,98,114)(30,138,97,107,80,188,47,157)(31,145,46,150,81,195,96,200)(32,152,95,193,82,102,45,143)(33,159,44,136,83,109,94,186)(34,166,93,179,84,116,43,129)(35,173,42,122,85,123,92,172)(36,180,91,165,86,130,41,115)(37,187,40,108,87,137,90,158)(38,194,89,151,88,144,39,101)>;
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), (1,135,76,160,51,185,26,110)(2,142,25,103,52,192,75,153)(3,149,74,146,53,199,24,196)(4,156,23,189,54,106,73,139)(5,163,72,132,55,113,22,182)(6,170,21,175,56,120,71,125)(7,177,70,118,57,127,20,168)(8,184,19,161,58,134,69,111)(9,191,68,104,59,141,18,154)(10,198,17,147,60,148,67,197)(11,105,66,190,61,155,16,140)(12,112,15,133,62,162,65,183)(13,119,64,176,63,169,14,126)(27,117,50,178,77,167,100,128)(28,124,99,121,78,174,49,171)(29,131,48,164,79,181,98,114)(30,138,97,107,80,188,47,157)(31,145,46,150,81,195,96,200)(32,152,95,193,82,102,45,143)(33,159,44,136,83,109,94,186)(34,166,93,179,84,116,43,129)(35,173,42,122,85,123,92,172)(36,180,91,165,86,130,41,115)(37,187,40,108,87,137,90,158)(38,194,89,151,88,144,39,101) );
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)], [(1,135,76,160,51,185,26,110),(2,142,25,103,52,192,75,153),(3,149,74,146,53,199,24,196),(4,156,23,189,54,106,73,139),(5,163,72,132,55,113,22,182),(6,170,21,175,56,120,71,125),(7,177,70,118,57,127,20,168),(8,184,19,161,58,134,69,111),(9,191,68,104,59,141,18,154),(10,198,17,147,60,148,67,197),(11,105,66,190,61,155,16,140),(12,112,15,133,62,162,65,183),(13,119,64,176,63,169,14,126),(27,117,50,178,77,167,100,128),(28,124,99,121,78,174,49,171),(29,131,48,164,79,181,98,114),(30,138,97,107,80,188,47,157),(31,145,46,150,81,195,96,200),(32,152,95,193,82,102,45,143),(33,159,44,136,83,109,94,186),(34,166,93,179,84,116,43,129),(35,173,42,122,85,123,92,172),(36,180,91,165,86,130,41,115),(37,187,40,108,87,137,90,158),(38,194,89,151,88,144,39,101)])
34 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 10 | 20A | 20B | 25A | ··· | 25E | 50A | ··· | 50E | 100A | ··· | 100J |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 20 | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 50 | 2 | 25 | 25 | 4 | 50 | 50 | 50 | 50 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | M4(2) | F5 | C2×F5 | C4.F5 | C25⋊C4 | C2×C25⋊C4 | C100.C4 |
| kernel | C100.C4 | C25⋊C8 | C4×D25 | C100 | D50 | C25 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 5 | 5 | 10 |
Matrix representation of C100.C4 ►in GL4(𝔽7) generated by
| 4 | 6 | 5 | 6 |
| 2 | 1 | 4 | 6 |
| 1 | 4 | 0 | 5 |
| 2 | 1 | 3 | 4 |
| 5 | 0 | 1 | 2 |
| 1 | 5 | 6 | 0 |
| 3 | 4 | 1 | 5 |
| 3 | 3 | 0 | 3 |
G:=sub<GL(4,GF(7))| [4,2,1,2,6,1,4,1,5,4,0,3,6,6,5,4],[5,1,3,3,0,5,4,3,1,6,1,0,2,0,5,3] >;
C100.C4 in GAP, Magma, Sage, TeX
C_{100}.C_4 % in TeX
G:=Group("C100.C4"); // GroupNames label
G:=SmallGroup(400,29);
// by ID
G=gap.SmallGroup(400,29);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,50,3364,2896,178,5765,2897]);
// Polycyclic
G:=Group<a,b|a^100=1,b^4=a^50,b*a*b^-1=a^43>;
// generators/relations
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