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## G = C20.9D8order 320 = 26·5

### 9th non-split extension by C20 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C20.9D8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×C20 — C20⋊3C8 — C20.9D8
 Lower central C5 — C2×C10 — C2×C20 — C20.9D8
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for C20.9D8
G = < a,b,c | a20=b8=1, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a5b-1 >

Subgroups: 270 in 84 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C10, C42, C2×C8, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, C4⋊C8, C41D4, C52C8, C2×C20, C2×C20, C5×D4, C22×C10, C4.D8, C2×C52C8, C4×C20, D4×C10, D4×C10, C203C8, C5×C41D4, C20.9D8
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, Dic5, D10, C4.D4, D4⋊C4, C2×Dic5, C5⋊D4, C4.D8, D4⋊D5, D4.D5, C23.D5, D4⋊Dic5, C20.D4, C20.9D8

Smallest permutation representation of C20.9D8
On 160 points
Generators in S160
```(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)
(1 86 54 69 128 148 111 39)(2 85 55 68 129 147 112 38)(3 84 56 67 130 146 113 37)(4 83 57 66 131 145 114 36)(5 82 58 65 132 144 115 35)(6 81 59 64 133 143 116 34)(7 100 60 63 134 142 117 33)(8 99 41 62 135 141 118 32)(9 98 42 61 136 160 119 31)(10 97 43 80 137 159 120 30)(11 96 44 79 138 158 101 29)(12 95 45 78 139 157 102 28)(13 94 46 77 140 156 103 27)(14 93 47 76 121 155 104 26)(15 92 48 75 122 154 105 25)(16 91 49 74 123 153 106 24)(17 90 50 73 124 152 107 23)(18 89 51 72 125 151 108 22)(19 88 52 71 126 150 109 21)(20 87 53 70 127 149 110 40)
(1 24 6 29 11 34 16 39)(2 33 7 38 12 23 17 28)(3 22 8 27 13 32 18 37)(4 31 9 36 14 21 19 26)(5 40 10 25 15 30 20 35)(41 156 46 141 51 146 56 151)(42 145 47 150 52 155 57 160)(43 154 48 159 53 144 58 149)(44 143 49 148 54 153 59 158)(45 152 50 157 55 142 60 147)(61 136 66 121 71 126 76 131)(62 125 67 130 72 135 77 140)(63 134 68 139 73 124 78 129)(64 123 69 128 74 133 79 138)(65 132 70 137 75 122 80 127)(81 106 86 111 91 116 96 101)(82 115 87 120 92 105 97 110)(83 104 88 109 93 114 98 119)(84 113 89 118 94 103 99 108)(85 102 90 107 95 112 100 117)```

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 5A 5B 8A ··· 8H 10A ··· 10F 10G ··· 10N 20A ··· 20L order 1 2 2 2 2 2 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 8 8 2 2 2 2 4 2 2 20 ··· 20 2 ··· 2 8 ··· 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + - image C1 C2 C2 C4 D4 D5 D8 SD16 D10 Dic5 C5⋊D4 C4.D4 D4⋊D5 D4.D5 C20.D4 kernel C20.9D8 C20⋊3C8 C5×C4⋊1D4 D4×C10 C2×C20 C4⋊1D4 C20 C20 C42 C2×D4 C2×C4 C10 C4 C4 C2 # reps 1 2 1 4 2 2 4 4 2 4 8 1 4 4 4

Matrix representation of C20.9D8 in GL6(𝔽41)

 0 1 0 0 0 0 40 0 0 0 0 0 0 0 7 1 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 26 15 0 0 0 0 15 15 0 0 0 0 0 0 29 14 0 0 0 0 16 12 0 0 0 0 0 0 15 15 0 0 0 0 26 15
,
 26 15 0 0 0 0 26 26 0 0 0 0 0 0 29 14 0 0 0 0 16 12 0 0 0 0 0 0 15 15 0 0 0 0 15 26

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,7,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[26,15,0,0,0,0,15,15,0,0,0,0,0,0,29,16,0,0,0,0,14,12,0,0,0,0,0,0,15,26,0,0,0,0,15,15],[26,26,0,0,0,0,15,26,0,0,0,0,0,0,29,16,0,0,0,0,14,12,0,0,0,0,0,0,15,15,0,0,0,0,15,26] >;`

C20.9D8 in GAP, Magma, Sage, TeX

`C_{20}._9D_8`
`% in TeX`

`G:=Group("C20.9D8");`
`// GroupNames label`

`G:=SmallGroup(320,102);`
`// by ID`

`G=gap.SmallGroup(320,102);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,100,1571,570,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^8=1,c^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^5*b^-1>;`
`// generators/relations`

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