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## G = D20⋊Q8order 320 = 26·5

### 1st semidirect product of D20 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20⋊Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — D20⋊8C4 — D20⋊Q8
 Lower central C5 — C10 — C2×C20 — D20⋊Q8
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for D20⋊Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a9, cbc-1=a3b, dbd-1=a18b, dcd-1=c-1 >

Subgroups: 502 in 108 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C4.Q8, C4×D4, C4⋊Q8, C52C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, D42Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C20.Q8, D206C4, C20.8Q8, D205C4, C5×C4.Q8, C20⋊Q8, D208C4, D20⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8⋊C22, C22×D5, D42Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D5×SD16, D40⋊C2, D20⋊Q8

Smallest permutation representation of D20⋊Q8
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(101 107)(102 106)(103 105)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(135 140)(136 139)(137 138)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)(159 160)
(1 112 138 26)(2 111 139 25)(3 110 140 24)(4 109 121 23)(5 108 122 22)(6 107 123 21)(7 106 124 40)(8 105 125 39)(9 104 126 38)(10 103 127 37)(11 102 128 36)(12 101 129 35)(13 120 130 34)(14 119 131 33)(15 118 132 32)(16 117 133 31)(17 116 134 30)(18 115 135 29)(19 114 136 28)(20 113 137 27)(41 146 82 72)(42 145 83 71)(43 144 84 70)(44 143 85 69)(45 142 86 68)(46 141 87 67)(47 160 88 66)(48 159 89 65)(49 158 90 64)(50 157 91 63)(51 156 92 62)(52 155 93 61)(53 154 94 80)(54 153 95 79)(55 152 96 78)(56 151 97 77)(57 150 98 76)(58 149 99 75)(59 148 100 74)(60 147 81 73)
(1 145 138 71)(2 154 139 80)(3 143 140 69)(4 152 121 78)(5 141 122 67)(6 150 123 76)(7 159 124 65)(8 148 125 74)(9 157 126 63)(10 146 127 72)(11 155 128 61)(12 144 129 70)(13 153 130 79)(14 142 131 68)(15 151 132 77)(16 160 133 66)(17 149 134 75)(18 158 135 64)(19 147 136 73)(20 156 137 62)(21 98 107 57)(22 87 108 46)(23 96 109 55)(24 85 110 44)(25 94 111 53)(26 83 112 42)(27 92 113 51)(28 81 114 60)(29 90 115 49)(30 99 116 58)(31 88 117 47)(32 97 118 56)(33 86 119 45)(34 95 120 54)(35 84 101 43)(36 93 102 52)(37 82 103 41)(38 91 104 50)(39 100 105 59)(40 89 106 48)```

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

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

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

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 20B 20C 20D 20E ··· 20L 40A ··· 40H 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 20 ··· 20 40 ··· 40 size 1 1 1 1 20 20 2 2 4 4 8 10 10 20 40 2 2 4 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

47 irreducible representations

 dim 1 1 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 C2 C2 Q8 D4 D5 SD16 C4○D4 D10 D10 C4○D20 C8⋊C22 Q8×D5 D4×D5 D5×SD16 D40⋊C2 kernel D20⋊Q8 C20.Q8 D20⋊6C4 C20.8Q8 D20⋊5C4 C5×C4.Q8 C20⋊Q8 D20⋊8C4 D20 C2×Dic5 C4.Q8 Dic5 C20 C4⋊C4 C2×C8 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 4 2 4 2 8 1 2 2 4 4

Matrix representation of D20⋊Q8 in GL4(𝔽41) generated by

 34 33 0 0 1 1 0 0 0 0 28 10 0 0 24 13
,
 1 8 0 0 0 40 0 0 0 0 28 10 0 0 16 13
,
 38 37 0 0 23 3 0 0 0 0 25 27 0 0 33 16
,
 19 19 0 0 9 22 0 0 0 0 13 31 0 0 17 28
`G:=sub<GL(4,GF(41))| [34,1,0,0,33,1,0,0,0,0,28,24,0,0,10,13],[1,0,0,0,8,40,0,0,0,0,28,16,0,0,10,13],[38,23,0,0,37,3,0,0,0,0,25,33,0,0,27,16],[19,9,0,0,19,22,0,0,0,0,13,17,0,0,31,28] >;`

D20⋊Q8 in GAP, Magma, Sage, TeX

`D_{20}\rtimes Q_8`
`% in TeX`

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

`G:=SmallGroup(320,497);`
`// by ID`

`G=gap.SmallGroup(320,497);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,100,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^9,c*b*c^-1=a^3*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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