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## G = D20.5C8order 320 = 26·5

### 3rd non-split extension by D20 of C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D20.5C8
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D20.3C4 — D20.5C8
 Lower central C5 — C10 — D20.5C8
 Upper central C1 — C8 — M5(2)

Generators and relations for D20.5C8
G = < a,b,c | a20=b2=1, c8=a10, bab=a-1, cac-1=a11, cbc-1=a10b >

Subgroups: 214 in 84 conjugacy classes, 51 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C16, C16, C2×C8, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C2×C16, M5(2), M5(2), C8○D4, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, D4○C16, C52C16, C80, C8×D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, D5×C16, C80⋊C2, C2×C52C16, C5×M5(2), D20.3C4, D20.5C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, D10, C22×C8, C4×D5, C22×D5, D4○C16, C8×D5, C2×C4×D5, D5×C2×C8, D20.5C8

Smallest permutation representation of D20.5C8
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 27)(22 26)(23 25)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(101 111)(102 110)(103 109)(104 108)(105 107)(112 120)(113 119)(114 118)(115 117)(121 137)(122 136)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(138 140)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
(1 62 27 137 95 154 109 52 11 72 37 127 85 144 119 42)(2 73 28 128 96 145 110 43 12 63 38 138 86 155 120 53)(3 64 29 139 97 156 111 54 13 74 39 129 87 146 101 44)(4 75 30 130 98 147 112 45 14 65 40 140 88 157 102 55)(5 66 31 121 99 158 113 56 15 76 21 131 89 148 103 46)(6 77 32 132 100 149 114 47 16 67 22 122 90 159 104 57)(7 68 33 123 81 160 115 58 17 78 23 133 91 150 105 48)(8 79 34 134 82 151 116 49 18 69 24 124 92 141 106 59)(9 70 35 125 83 142 117 60 19 80 25 135 93 152 107 50)(10 61 36 136 84 153 118 51 20 71 26 126 94 143 108 41)```

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 10C 10D 16A ··· 16H 16I ··· 16P 16Q 16R 16S 16T 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I 40J 40K 40L 80A ··· 80P order 1 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 8 8 10 10 10 10 16 ··· 16 16 ··· 16 16 16 16 16 20 20 20 20 20 20 40 ··· 40 40 40 40 40 80 ··· 80 size 1 1 2 10 10 1 1 2 10 10 2 2 1 1 1 1 2 2 10 10 10 10 2 2 4 4 2 ··· 2 5 ··· 5 10 10 10 10 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D5 D10 D10 C4×D5 C4×D5 D4○C16 C8×D5 C8×D5 D20.5C8 kernel D20.5C8 D5×C16 C80⋊C2 C2×C5⋊2C16 C5×M5(2) D20.3C4 C8⋊D5 C4.Dic5 C4○D20 Dic10 D20 C5⋊D4 M5(2) C16 C2×C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 4 4 8 2 4 2 4 4 8 8 8 8

Matrix representation of D20.5C8 in GL4(𝔽241) generated by

 51 240 0 0 191 1 0 0 0 0 165 19 0 0 216 76
,
 240 240 0 0 0 1 0 0 0 0 240 0 0 0 233 1
,
 211 0 0 0 0 211 0 0 0 0 8 239 0 0 36 233
`G:=sub<GL(4,GF(241))| [51,191,0,0,240,1,0,0,0,0,165,216,0,0,19,76],[240,0,0,0,240,1,0,0,0,0,240,233,0,0,0,1],[211,0,0,0,0,211,0,0,0,0,8,36,0,0,239,233] >;`

D20.5C8 in GAP, Magma, Sage, TeX

`D_{20}._5C_8`
`% in TeX`

`G:=Group("D20.5C8");`
`// GroupNames label`

`G:=SmallGroup(320,534);`
`// by ID`

`G=gap.SmallGroup(320,534);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,58,80,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^2=1,c^8=a^10,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;`
`// generators/relations`

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