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## G = C10.11D20order 400 = 24·52

### 11st non-split extension by C10 of D20 acting via D20/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C10.11D20
 Chief series C1 — C5 — C52 — C5×C10 — C102 — C22×C5⋊D5 — C10.11D20
 Lower central C52 — C5×C10 — C10.11D20
 Upper central C1 — C22 — C2×C4

Generators and relations for C10.11D20
G = < a,b,c | a10=b20=1, c2=a5, ab=ba, cac-1=a-1, cbc-1=a5b-1 >

Subgroups: 904 in 136 conjugacy classes, 53 normal (15 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C22⋊C4, Dic5, C20, D10, C2×C10, C52, C2×Dic5, C2×C20, C22×D5, C5⋊D5, C5×C10, D10⋊C4, C526C4, C5×C20, C2×C5⋊D5, C2×C5⋊D5, C102, C2×C526C4, C10×C20, C22×C5⋊D5, C10.11D20
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D10, C4×D5, D20, C5⋊D4, C5⋊D5, D10⋊C4, C2×C5⋊D5, C4×C5⋊D5, C20⋊D5, C527D4, C10.11D20

Smallest permutation representation of C10.11D20
On 200 points
Generators in S200
```(1 95 148 22 66 106 136 200 164 43)(2 96 149 23 67 107 137 181 165 44)(3 97 150 24 68 108 138 182 166 45)(4 98 151 25 69 109 139 183 167 46)(5 99 152 26 70 110 140 184 168 47)(6 100 153 27 71 111 121 185 169 48)(7 81 154 28 72 112 122 186 170 49)(8 82 155 29 73 113 123 187 171 50)(9 83 156 30 74 114 124 188 172 51)(10 84 157 31 75 115 125 189 173 52)(11 85 158 32 76 116 126 190 174 53)(12 86 159 33 77 117 127 191 175 54)(13 87 160 34 78 118 128 192 176 55)(14 88 141 35 79 119 129 193 177 56)(15 89 142 36 80 120 130 194 178 57)(16 90 143 37 61 101 131 195 179 58)(17 91 144 38 62 102 132 196 180 59)(18 92 145 39 63 103 133 197 161 60)(19 93 146 40 64 104 134 198 162 41)(20 94 147 21 65 105 135 199 163 42)
(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)
(1 115 106 10)(2 9 107 114)(3 113 108 8)(4 7 109 112)(5 111 110 6)(11 105 116 20)(12 19 117 104)(13 103 118 18)(14 17 119 102)(15 101 120 16)(21 190 163 158)(22 157 164 189)(23 188 165 156)(24 155 166 187)(25 186 167 154)(26 153 168 185)(27 184 169 152)(28 151 170 183)(29 182 171 150)(30 149 172 181)(31 200 173 148)(32 147 174 199)(33 198 175 146)(34 145 176 197)(35 196 177 144)(36 143 178 195)(37 194 179 142)(38 141 180 193)(39 192 161 160)(40 159 162 191)(41 127 64 86)(42 85 65 126)(43 125 66 84)(44 83 67 124)(45 123 68 82)(46 81 69 122)(47 121 70 100)(48 99 71 140)(49 139 72 98)(50 97 73 138)(51 137 74 96)(52 95 75 136)(53 135 76 94)(54 93 77 134)(55 133 78 92)(56 91 79 132)(57 131 80 90)(58 89 61 130)(59 129 62 88)(60 87 63 128)```

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

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

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

106 conjugacy classes

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

106 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 D4 D5 D10 C4×D5 D20 C5⋊D4 kernel C10.11D20 C2×C52⋊6C4 C10×C20 C22×C5⋊D5 C2×C5⋊D5 C5×C10 C2×C20 C2×C10 C10 C10 C10 # reps 1 1 1 1 4 2 12 12 24 24 24

Matrix representation of C10.11D20 in GL6(𝔽41)

 0 1 0 0 0 0 40 34 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 34 34 0 0 0 0 7 1 0 0 0 0 0 0 40 9 0 0 0 0 0 1 0 0 0 0 0 0 25 2 0 0 0 0 39 13
,
 34 34 0 0 0 0 1 7 0 0 0 0 0 0 40 9 0 0 0 0 18 1 0 0 0 0 0 0 2 25 0 0 0 0 13 39

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,0,0,0,0,0,9,1,0,0,0,0,0,0,25,39,0,0,0,0,2,13],[34,1,0,0,0,0,34,7,0,0,0,0,0,0,40,18,0,0,0,0,9,1,0,0,0,0,0,0,2,13,0,0,0,0,25,39] >;`

C10.11D20 in GAP, Magma, Sage, TeX

`C_{10}._{11}D_{20}`
`% in TeX`

`G:=Group("C10.11D20");`
`// GroupNames label`

`G:=SmallGroup(400,102);`
`// by ID`

`G=gap.SmallGroup(400,102);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,31,1924,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=b^20=1,c^2=a^5,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^5*b^-1>;`
`// generators/relations`

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