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## G = C102⋊11C4order 400 = 24·52

### 3rd semidirect product of C102 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C102⋊11C4
 Chief series C1 — C5 — C52 — C5×C10 — C102 — C2×C52⋊6C4 — C102⋊11C4
 Lower central C52 — C5×C10 — C102⋊11C4
 Upper central C1 — C22 — C23

Generators and relations for C10211C4
G = < a,b,c | a10=b10=c4=1, ab=ba, cac-1=a-1b5, cbc-1=b-1 >

Subgroups: 520 in 136 conjugacy classes, 67 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C23, C10, C10, C22⋊C4, Dic5, C2×C10, C2×C10, C52, C2×Dic5, C22×C10, C5×C10, C5×C10, C5×C10, C23.D5, C526C4, C102, C102, C102, C2×C526C4, C2×C102, C10211C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C2×Dic5, C5⋊D4, C5⋊D5, C23.D5, C526C4, C2×C5⋊D5, C2×C526C4, C527D4, C10211C4

Smallest permutation representation of C10211C4
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)
(1 169 156 99 85 40 142 103 47 132)(2 170 157 100 86 31 143 104 48 133)(3 161 158 91 87 32 144 105 49 134)(4 162 159 92 88 33 145 106 50 135)(5 163 160 93 89 34 146 107 41 136)(6 164 151 94 90 35 147 108 42 137)(7 165 152 95 81 36 148 109 43 138)(8 166 153 96 82 37 149 110 44 139)(9 167 154 97 83 38 150 101 45 140)(10 168 155 98 84 39 141 102 46 131)(11 199 175 114 62 78 56 183 122 24)(12 200 176 115 63 79 57 184 123 25)(13 191 177 116 64 80 58 185 124 26)(14 192 178 117 65 71 59 186 125 27)(15 193 179 118 66 72 60 187 126 28)(16 194 180 119 67 73 51 188 127 29)(17 195 171 120 68 74 52 189 128 30)(18 196 172 111 69 75 53 190 129 21)(19 197 173 112 70 76 54 181 130 22)(20 198 174 113 61 77 55 182 121 23)
(1 200 6 52)(2 56 7 194)(3 198 8 60)(4 54 9 192)(5 196 10 58)(11 148 73 143)(12 164 74 169)(13 146 75 141)(14 162 76 167)(15 144 77 149)(16 170 78 165)(17 142 79 147)(18 168 80 163)(19 150 71 145)(20 166 72 161)(21 155 64 160)(22 101 65 106)(23 153 66 158)(24 109 67 104)(25 151 68 156)(26 107 69 102)(27 159 70 154)(28 105 61 110)(29 157 62 152)(30 103 63 108)(31 199 36 51)(32 55 37 193)(33 197 38 59)(34 53 39 191)(35 195 40 57)(41 111 46 124)(42 128 47 115)(43 119 48 122)(44 126 49 113)(45 117 50 130)(81 188 86 175)(82 179 87 182)(83 186 88 173)(84 177 89 190)(85 184 90 171)(91 121 96 118)(92 112 97 125)(93 129 98 116)(94 120 99 123)(95 127 100 114)(131 185 136 172)(132 176 137 189)(133 183 138 180)(134 174 139 187)(135 181 140 178)```

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

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

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

106 conjugacy classes

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

106 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C4 D4 D5 Dic5 D10 C5⋊D4 kernel C102⋊11C4 C2×C52⋊6C4 C2×C102 C102 C5×C10 C22×C10 C2×C10 C2×C10 C10 # reps 1 2 1 4 2 12 24 12 48

Matrix representation of C10211C4 in GL4(𝔽41) generated by

 37 0 0 0 0 31 0 0 0 0 25 0 0 0 0 18
,
 4 0 0 0 0 31 0 0 0 0 40 0 0 0 0 40
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 40 0
`G:=sub<GL(4,GF(41))| [37,0,0,0,0,31,0,0,0,0,25,0,0,0,0,18],[4,0,0,0,0,31,0,0,0,0,40,0,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,0,40,0,0,1,0] >;`

C10211C4 in GAP, Magma, Sage, TeX

`C_{10}^2\rtimes_{11}C_4`
`% in TeX`

`G:=Group("C10^2:11C4");`
`// GroupNames label`

`G:=SmallGroup(400,107);`
`// by ID`

`G=gap.SmallGroup(400,107);`
`# by ID`

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

`G:=Group<a,b,c|a^10=b^10=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^5,c*b*c^-1=b^-1>;`
`// generators/relations`

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