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## G = C20.59D10order 400 = 24·52

### 20th non-split extension by C20 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C20.59D10
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C52⋊7C8 — C20.59D10
 Lower central C52 — C5×C10 — C20.59D10
 Upper central C1 — C4 — C2×C4

Generators and relations for C20.59D10
G = < a,b,c | a20=b10=1, c2=a15, ab=ba, cac-1=a9, cbc-1=a10b-1 >

Subgroups: 232 in 80 conjugacy classes, 51 normal (13 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C10, C10, M4(2), C20, C2×C10, C52, C52C8, C2×C20, C5×C10, C5×C10, C4.Dic5, C5×C20, C102, C527C8, C10×C20, C20.59D10
Quotients: C1, C2, C4, C22, C2×C4, D5, M4(2), Dic5, D10, C2×Dic5, C5⋊D5, C4.Dic5, C526C4, C2×C5⋊D5, C2×C526C4, C20.59D10

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

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

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

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

106 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A ··· 5L 8A 8B 8C 8D 10A ··· 10AJ 20A ··· 20AV order 1 2 2 4 4 4 5 ··· 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 2 1 1 2 2 ··· 2 50 50 50 50 2 ··· 2 2 ··· 2

106 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 D5 M4(2) Dic5 D10 Dic5 C4.Dic5 kernel C20.59D10 C52⋊7C8 C10×C20 C5×C20 C102 C2×C20 C52 C20 C20 C2×C10 C5 # reps 1 2 1 2 2 12 2 12 12 12 48

Matrix representation of C20.59D10 in GL4(𝔽41) generated by

 39 0 0 0 0 21 0 0 0 0 32 0 0 0 0 32
,
 1 0 0 0 0 40 0 0 0 0 10 0 0 0 0 4
,
 0 1 0 0 32 0 0 0 0 0 0 37 0 0 8 0
`G:=sub<GL(4,GF(41))| [39,0,0,0,0,21,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,40,0,0,0,0,10,0,0,0,0,4],[0,32,0,0,1,0,0,0,0,0,0,8,0,0,37,0] >;`

C20.59D10 in GAP, Magma, Sage, TeX

`C_{20}._{59}D_{10}`
`% in TeX`

`G:=Group("C20.59D10");`
`// GroupNames label`

`G:=SmallGroup(400,98);`
`// by ID`

`G=gap.SmallGroup(400,98);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,1924,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^10=1,c^2=a^15,a*b=b*a,c*a*c^-1=a^9,c*b*c^-1=a^10*b^-1>;`
`// generators/relations`

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