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## G = C100.C4order 400 = 24·52

### 1st non-split extension by C100 of C4 acting faithfully

Aliases: C20.1F5, C100.1C4, D50.3C4, C251M4(2), Dic25.6C22, C25⋊C81C2, 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

 Derived series C1 — C50 — C100.C4
 Chief series C1 — C5 — C25 — C50 — Dic25 — C25⋊C8 — C100.C4
 Lower central C25 — C50 — C100.C4
 Upper central C1 — C2 — C4

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

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

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

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

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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