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## G = C20.50D10order 400 = 24·52

### 11st non-split extension by C20 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C20.50D10
 Chief series C1 — C5 — C52 — C5×C10 — C2×C5⋊D5 — C4×C5⋊D5 — C20.50D10
 Lower central C52 — C5×C10 — C20.50D10
 Upper central C1 — C4 — C2×C4

Generators and relations for C20.50D10
G = < a,b,c | a20=1, b10=c2=a10, ab=ba, cac-1=a-1, cbc-1=b9 >

Subgroups: 904 in 160 conjugacy classes, 59 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, C20, D10, C2×C10, C52, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5⋊D5, C5×C10, C5×C10, C4○D20, C526C4, C5×C20, C2×C5⋊D5, C102, C524Q8, C4×C5⋊D5, C20⋊D5, C527D4, C10×C20, C20.50D10
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, C5⋊D5, C4○D20, C2×C5⋊D5, C22×C5⋊D5, C20.50D10

Smallest permutation representation of C20.50D10
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 49 65 83 117 6 54 70 88 102 11 59 75 93 107 16 44 80 98 112)(2 50 66 84 118 7 55 71 89 103 12 60 76 94 108 17 45 61 99 113)(3 51 67 85 119 8 56 72 90 104 13 41 77 95 109 18 46 62 100 114)(4 52 68 86 120 9 57 73 91 105 14 42 78 96 110 19 47 63 81 115)(5 53 69 87 101 10 58 74 92 106 15 43 79 97 111 20 48 64 82 116)(21 133 144 182 177 36 128 159 197 172 31 123 154 192 167 26 138 149 187 162)(22 134 145 183 178 37 129 160 198 173 32 124 155 193 168 27 139 150 188 163)(23 135 146 184 179 38 130 141 199 174 33 125 156 194 169 28 140 151 189 164)(24 136 147 185 180 39 131 142 200 175 34 126 157 195 170 29 121 152 190 165)(25 137 148 186 161 40 132 143 181 176 35 127 158 196 171 30 122 153 191 166)
(1 170 11 180)(2 169 12 179)(3 168 13 178)(4 167 14 177)(5 166 15 176)(6 165 16 175)(7 164 17 174)(8 163 18 173)(9 162 19 172)(10 161 20 171)(21 120 31 110)(22 119 32 109)(23 118 33 108)(24 117 34 107)(25 116 35 106)(26 115 36 105)(27 114 37 104)(28 113 38 103)(29 112 39 102)(30 111 40 101)(41 193 51 183)(42 192 52 182)(43 191 53 181)(44 190 54 200)(45 189 55 199)(46 188 56 198)(47 187 57 197)(48 186 58 196)(49 185 59 195)(50 184 60 194)(61 141 71 151)(62 160 72 150)(63 159 73 149)(64 158 74 148)(65 157 75 147)(66 156 76 146)(67 155 77 145)(68 154 78 144)(69 153 79 143)(70 152 80 142)(81 138 91 128)(82 137 92 127)(83 136 93 126)(84 135 94 125)(85 134 95 124)(86 133 96 123)(87 132 97 122)(88 131 98 121)(89 130 99 140)(90 129 100 139)```

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

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

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

106 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A ··· 5L 10A ··· 10AJ 20A ··· 20AV order 1 2 2 2 2 4 4 4 4 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 2 50 50 1 1 2 50 50 2 ··· 2 2 ··· 2 2 ··· 2

106 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 C4○D20 kernel C20.50D10 C52⋊4Q8 C4×C5⋊D5 C20⋊D5 C52⋊7D4 C10×C20 C2×C20 C52 C20 C2×C10 C5 # reps 1 1 2 1 2 1 12 2 24 12 48

Matrix representation of C20.50D10 in GL4(𝔽41) generated by

 21 0 0 0 39 2 0 0 0 0 5 0 0 0 0 33
,
 8 0 0 0 4 5 0 0 0 0 5 0 0 0 0 8
,
 27 31 0 0 32 14 0 0 0 0 0 8 0 0 5 0
`G:=sub<GL(4,GF(41))| [21,39,0,0,0,2,0,0,0,0,5,0,0,0,0,33],[8,4,0,0,0,5,0,0,0,0,5,0,0,0,0,8],[27,32,0,0,31,14,0,0,0,0,0,5,0,0,8,0] >;`

C20.50D10 in GAP, Magma, Sage, TeX

`C_{20}._{50}D_{10}`
`% in TeX`

`G:=Group("C20.50D10");`
`// GroupNames label`

`G:=SmallGroup(400,194);`
`// by ID`

`G=gap.SmallGroup(400,194);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,1924,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^10=c^2=a^10,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^9>;`
`// generators/relations`

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