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## G = D20⋊5Q8order 320 = 26·5

### 3rd semidirect product of D20 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20⋊5Q8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4×D20 — D20⋊5Q8
 Lower central C5 — C10 — C2×C20 — D20⋊5Q8
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for D205Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 438 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C52C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D42Q8, C2×C52C8, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C203C8, C20.Q8, D206C4, C4×D20, C5×C4⋊Q8, D205Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8⋊C22, C5⋊D4, C22×D5, D42Q8, Q8⋊D5, Q8×D5, Q82D5, C2×C5⋊D4, D4.D10, C2×Q8⋊D5, D103Q8, D205Q8

Smallest permutation representation of D205Q8
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 97)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 81)(18 100)(19 99)(20 98)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(121 154)(122 153)(123 152)(124 151)(125 150)(126 149)(127 148)(128 147)(129 146)(130 145)(131 144)(132 143)(133 142)(134 141)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)
(1 49 88 108)(2 60 89 119)(3 51 90 110)(4 42 91 101)(5 53 92 112)(6 44 93 103)(7 55 94 114)(8 46 95 105)(9 57 96 116)(10 48 97 107)(11 59 98 118)(12 50 99 109)(13 41 100 120)(14 52 81 111)(15 43 82 102)(16 54 83 113)(17 45 84 104)(18 56 85 115)(19 47 86 106)(20 58 87 117)(21 127 75 146)(22 138 76 157)(23 129 77 148)(24 140 78 159)(25 131 79 150)(26 122 80 141)(27 133 61 152)(28 124 62 143)(29 135 63 154)(30 126 64 145)(31 137 65 156)(32 128 66 147)(33 139 67 158)(34 130 68 149)(35 121 69 160)(36 132 70 151)(37 123 71 142)(38 134 72 153)(39 125 73 144)(40 136 74 155)
(1 74 88 40)(2 75 89 21)(3 76 90 22)(4 77 91 23)(5 78 92 24)(6 79 93 25)(7 80 94 26)(8 61 95 27)(9 62 96 28)(10 63 97 29)(11 64 98 30)(12 65 99 31)(13 66 100 32)(14 67 81 33)(15 68 82 34)(16 69 83 35)(17 70 84 36)(18 71 85 37)(19 72 86 38)(20 73 87 39)(41 128 120 147)(42 129 101 148)(43 130 102 149)(44 131 103 150)(45 132 104 151)(46 133 105 152)(47 134 106 153)(48 135 107 154)(49 136 108 155)(50 137 109 156)(51 138 110 157)(52 139 111 158)(53 140 112 159)(54 121 113 160)(55 122 114 141)(56 123 115 142)(57 124 116 143)(58 125 117 144)(59 126 118 145)(60 127 119 146)```

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + - + + + + + + - + image C1 C2 C2 C2 C2 C2 Q8 D4 D5 SD16 C4○D4 D10 D10 C5⋊D4 C8⋊C22 Q8⋊D5 Q8×D5 Q8⋊2D5 D4.D10 kernel D20⋊5Q8 C20⋊3C8 C20.Q8 D20⋊6C4 C4×D20 C5×C4⋊Q8 D20 C2×C20 C4⋊Q8 C20 C20 C42 C4⋊C4 C2×C4 C10 C4 C4 C4 C2 # reps 1 1 2 2 1 1 2 2 2 4 2 2 4 8 1 4 2 2 4

Matrix representation of D205Q8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 1 0 0 0 0 5 40 0 0 0 0 0 0 40 4 0 0 0 0 20 1
,
 1 0 0 0 0 0 32 40 0 0 0 0 0 0 40 40 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 21 40
,
 2 5 0 0 0 0 40 39 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 19 0 0 0 0 13 30
,
 32 0 0 0 0 0 40 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 37 0 0 0 0 21 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,20,0,0,0,0,4,1],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,21,0,0,0,0,0,40],[2,40,0,0,0,0,5,39,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,13,0,0,0,0,19,30],[32,40,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,37,40] >;`

D205Q8 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_5Q_8`
`% in TeX`

`G:=Group("D20:5Q8");`
`// GroupNames label`

`G:=SmallGroup(320,711);`
`// by ID`

`G=gap.SmallGroup(320,711);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,268,1123,297,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;`
`// generators/relations`

׿
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