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## G = C20.57D8order 320 = 26·5

### 11st non-split extension by C20 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.57D8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×C20 — C20⋊3C8 — C20.57D8
 Lower central C5 — C10 — C20 — C20.57D8
 Upper central C1 — C2×C4 — C42 — C4×D4

Generators and relations for C20.57D8
G = < a,b,c | a20=b8=1, c2=a5, bab-1=cac-1=a9, cbc-1=a5b-1 >

Subgroups: 214 in 82 conjugacy classes, 39 normal (35 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C4×C8, C4⋊C8, C4×D4, C52C8, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4⋊C8, C2×C52C8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C4×C52C8, C203C8, D4×C20, C20.57D8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), D8, SD16, Dic5, D10, C22⋊C8, D4⋊C4, C4≀C2, C52C8, C2×Dic5, C5⋊D4, D4⋊C8, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5, C23.D5, C20.55D4, D4⋊Dic5, D42Dic5, C20.57D8

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

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - - + - image C1 C2 C2 C2 C4 C4 C8 D4 D5 M4(2) D8 SD16 D10 Dic5 Dic5 C4≀C2 C5⋊D4 C5⋊2C8 C4.Dic5 D4⋊D5 D4.D5 D4⋊2Dic5 kernel C20.57D8 C4×C5⋊2C8 C20⋊3C8 D4×C20 C5×C4⋊C4 D4×C10 C5×D4 C2×C20 C4×D4 C20 C20 C20 C42 C4⋊C4 C2×D4 C10 C2×C4 D4 C4 C4 C4 C2 # reps 1 1 1 1 2 2 8 2 2 2 2 2 2 2 2 4 8 8 8 2 2 4

Matrix representation of C20.57D8 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 31 0 0 0 0 0 1 4 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 37 1 0 0 0 0 34 4 0 0 0 0 0 0 30 10 0 0 0 0 37 11 0 0 0 0 0 0 31 7 0 0 0 0 28 0
,
 37 1 0 0 0 0 16 4 0 0 0 0 0 0 30 10 0 0 0 0 37 11 0 0 0 0 0 0 31 7 0 0 0 0 28 10

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,31,1,0,0,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[37,34,0,0,0,0,1,4,0,0,0,0,0,0,30,37,0,0,0,0,10,11,0,0,0,0,0,0,31,28,0,0,0,0,7,0],[37,16,0,0,0,0,1,4,0,0,0,0,0,0,30,37,0,0,0,0,10,11,0,0,0,0,0,0,31,28,0,0,0,0,7,10] >;`

C20.57D8 in GAP, Magma, Sage, TeX

`C_{20}._{57}D_8`
`% in TeX`

`G:=Group("C20.57D8");`
`// GroupNames label`

`G:=SmallGroup(320,92);`
`// by ID`

`G=gap.SmallGroup(320,92);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,100,1123,570,136,12550]);`
`// Polycyclic`

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

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