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G = C20.26D10order 400 = 24·52

26th non-split extension by C20 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C20.26D10
 Chief series C1 — C5 — C52 — C5×C10 — C2×C5⋊D5 — C4×C5⋊D5 — C20.26D10
 Lower central C52 — C5×C10 — C20.26D10
 Upper central C1 — C2 — Q8

Generators and relations for C20.26D10
G = < a,b,c | a20=1, b10=c2=a10, bab-1=a11, cac-1=a9, cbc-1=b9 >

Subgroups: 1000 in 160 conjugacy classes, 59 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, D4, Q8, D5, C10, C4○D4, Dic5, C20, D10, C52, C4×D5, D20, C5×Q8, C5⋊D5, C5×C10, Q82D5, C526C4, C5×C20, C2×C5⋊D5, C4×C5⋊D5, C20⋊D5, Q8×C52, C20.26D10
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, C5⋊D5, Q82D5, C2×C5⋊D5, C22×C5⋊D5, C20.26D10

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

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 2 2 2 4 type + + + + + + + image C1 C2 C2 C2 D5 C4○D4 D10 Q8⋊2D5 kernel C20.26D10 C4×C5⋊D5 C20⋊D5 Q8×C52 C5×Q8 C52 C20 C5 # reps 1 3 3 1 12 2 36 12

Matrix representation of C20.26D10 in GL6(𝔽41)

 40 40 0 0 0 0 36 35 0 0 0 0 0 0 1 34 0 0 0 0 7 34 0 0 0 0 0 0 32 0 0 0 0 0 15 9
,
 0 7 0 0 0 0 35 35 0 0 0 0 0 0 34 7 0 0 0 0 34 1 0 0 0 0 0 0 35 1 0 0 0 0 4 6
,
 1 0 0 0 0 0 5 40 0 0 0 0 0 0 7 40 0 0 0 0 7 34 0 0 0 0 0 0 32 0 0 0 0 0 0 32

`G:=sub<GL(6,GF(41))| [40,36,0,0,0,0,40,35,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,0,0,0,0,32,15,0,0,0,0,0,9],[0,35,0,0,0,0,7,35,0,0,0,0,0,0,34,34,0,0,0,0,7,1,0,0,0,0,0,0,35,4,0,0,0,0,1,6],[1,5,0,0,0,0,0,40,0,0,0,0,0,0,7,7,0,0,0,0,40,34,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;`

C20.26D10 in GAP, Magma, Sage, TeX

`C_{20}._{26}D_{10}`
`% in TeX`

`G:=Group("C20.26D10");`
`// GroupNames label`

`G:=SmallGroup(400,198);`
`// by ID`

`G=gap.SmallGroup(400,198);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,116,50,1924,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^10=c^2=a^10,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=b^9>;`
`// generators/relations`

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