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## G = C20.58D8order 320 = 26·5

### 12nd non-split extension by C20 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C20.58D8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C40.6C4 — C20.58D8
 Lower central C5 — C10 — C20 — C40 — C20.58D8
 Upper central C1 — C4 — C2×C4 — C2×C8 — C4○D8

Generators and relations for C20.58D8
G = < a,b,c | a20=1, b8=a10, c2=a5, bab-1=cac-1=a9, cbc-1=a15b7 >

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 10E 10F 10G 10H 16A ··· 16H 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 40A ··· 40H order 1 2 2 2 4 4 4 4 5 5 8 8 8 8 8 8 10 10 10 10 10 10 10 10 16 ··· 16 20 20 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 2 8 1 1 2 8 2 2 2 2 2 2 40 40 2 2 4 4 8 8 8 8 10 ··· 10 2 2 2 2 4 4 8 8 8 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - - + - image C1 C2 C2 C2 C4 C4 D4 D4 D5 D8 SD16 D10 Dic5 Dic5 C5⋊D4 C5⋊D4 D8.C4 D4⋊D5 D4.D5 C20.58D8 kernel C20.58D8 C2×C5⋊2C16 C40.6C4 C5×C4○D8 C5×D8 C5×Q16 C40 C2×C20 C4○D8 C20 C2×C10 C2×C8 D8 Q16 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 2 2 2 2 2 2 4 4 8 2 2 8

Matrix representation of C20.58D8 in GL4(𝔽241) generated by

 64 0 0 0 0 64 0 0 0 0 98 0 0 0 0 91
,
 18 25 0 0 191 68 0 0 0 0 0 1 0 0 240 0
,
 223 216 0 0 155 18 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(241))| [64,0,0,0,0,64,0,0,0,0,98,0,0,0,0,91],[18,191,0,0,25,68,0,0,0,0,0,240,0,0,1,0],[223,155,0,0,216,18,0,0,0,0,0,1,0,0,1,0] >;`

C20.58D8 in GAP, Magma, Sage, TeX

`C_{20}._{58}D_8`
`% in TeX`

`G:=Group("C20.58D8");`
`// GroupNames label`

`G:=SmallGroup(320,125);`
`// by ID`

`G=gap.SmallGroup(320,125);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,184,675,346,192,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^8=a^10,c^2=a^5,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^15*b^7>;`
`// generators/relations`

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