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## G = C20.D10order 400 = 24·52

### 24th non-split extension by C20 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C20.D10
 Chief series C1 — C5 — C52 — C5×C10 — C2×C5⋊D5 — C4×C5⋊D5 — C20.D10
 Lower central C52 — C5×C10 — C20.D10
 Upper central C1 — C2 — D4

Generators and relations for C20.D10
G = < a,b,c | a20=b10=1, c2=a10, bab-1=a11, cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 808 in 160 conjugacy classes, 59 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, D4, D4, Q8, D5, C10, C10, C4○D4, Dic5, C20, D10, C2×C10, C52, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5⋊D5, C5×C10, C5×C10, D42D5, C526C4, C526C4, C5×C20, C2×C5⋊D5, C102, C524Q8, C4×C5⋊D5, C2×C526C4, C527D4, D4×C52, C20.D10
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, C5⋊D5, D42D5, C2×C5⋊D5, C22×C5⋊D5, C20.D10

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

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A ··· 5L 10A ··· 10L 10M ··· 10AJ 20A ··· 20L order 1 2 2 2 2 4 4 4 4 4 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 2 2 50 2 25 25 50 50 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D4⋊2D5 kernel C20.D10 C52⋊4Q8 C4×C5⋊D5 C2×C52⋊6C4 C52⋊7D4 D4×C52 C5×D4 C52 C20 C2×C10 C5 # reps 1 1 1 2 2 1 12 2 12 24 12

Matrix representation of C20.D10 in GL6(𝔽41)

 0 6 0 0 0 0 34 34 0 0 0 0 0 0 26 37 0 0 0 0 36 15 0 0 0 0 0 0 40 35 0 0 0 0 6 35
,
 7 1 0 0 0 0 33 40 0 0 0 0 0 0 1 35 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 26 23 0 0 0 0 17 15 0 0 0 0 0 0 29 26 0 0 0 0 37 12 0 0 0 0 0 0 5 38 0 0 0 0 8 36

`G:=sub<GL(6,GF(41))| [0,34,0,0,0,0,6,34,0,0,0,0,0,0,26,36,0,0,0,0,37,15,0,0,0,0,0,0,40,6,0,0,0,0,35,35],[7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,35,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,17,0,0,0,0,23,15,0,0,0,0,0,0,29,37,0,0,0,0,26,12,0,0,0,0,0,0,5,8,0,0,0,0,38,36] >;`

C20.D10 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(400,196);`
`// by ID`

`G=gap.SmallGroup(400,196);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,116,1924,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^10=1,c^2=a^10,b*a*b^-1=a^11,c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;`
`// generators/relations`

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