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## G = D20⋊2Q8order 320 = 26·5

### 2nd semidirect product of D20 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20⋊2Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — D20⋊8C4 — D20⋊2Q8
 Lower central C5 — C10 — C2×C20 — D20⋊2Q8
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for D202Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a9, cbc-1=a13b, dbd-1=a8b, 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, C2.D8, C2.D8, C4×D4, C4⋊Q8, C52C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, D4⋊Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C10.D8, D206C4, C20.8Q8, D205C4, C5×C2.D8, C20⋊Q8, D208C4, D202Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, D8, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×D8, C8.C22, C22×D5, D4⋊Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D5×D8, Q16⋊D5, D202Q8

Smallest permutation representation of D202Q8
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 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 140)(138 139)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)
(1 59 129 64)(2 58 130 63)(3 57 131 62)(4 56 132 61)(5 55 133 80)(6 54 134 79)(7 53 135 78)(8 52 136 77)(9 51 137 76)(10 50 138 75)(11 49 139 74)(12 48 140 73)(13 47 121 72)(14 46 122 71)(15 45 123 70)(16 44 124 69)(17 43 125 68)(18 42 126 67)(19 41 127 66)(20 60 128 65)(21 98 144 120)(22 97 145 119)(23 96 146 118)(24 95 147 117)(25 94 148 116)(26 93 149 115)(27 92 150 114)(28 91 151 113)(29 90 152 112)(30 89 153 111)(31 88 154 110)(32 87 155 109)(33 86 156 108)(34 85 157 107)(35 84 158 106)(36 83 159 105)(37 82 160 104)(38 81 141 103)(39 100 142 102)(40 99 143 101)
(1 103 129 81)(2 112 130 90)(3 101 131 99)(4 110 132 88)(5 119 133 97)(6 108 134 86)(7 117 135 95)(8 106 136 84)(9 115 137 93)(10 104 138 82)(11 113 139 91)(12 102 140 100)(13 111 121 89)(14 120 122 98)(15 109 123 87)(16 118 124 96)(17 107 125 85)(18 116 126 94)(19 105 127 83)(20 114 128 92)(21 46 144 71)(22 55 145 80)(23 44 146 69)(24 53 147 78)(25 42 148 67)(26 51 149 76)(27 60 150 65)(28 49 151 74)(29 58 152 63)(30 47 153 72)(31 56 154 61)(32 45 155 70)(33 54 156 79)(34 43 157 68)(35 52 158 77)(36 41 159 66)(37 50 160 75)(38 59 141 64)(39 48 142 73)(40 57 143 62)```

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

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

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

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 D8 C4○D4 D10 D10 C4○D20 C8.C22 Q8×D5 D4×D5 D5×D8 Q16⋊D5 kernel D20⋊2Q8 C10.D8 D20⋊6C4 C20.8Q8 D20⋊5C4 C5×C2.D8 C20⋊Q8 D20⋊8C4 D20 C2×Dic5 C2.D8 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 D202Q8 in GL4(𝔽41) generated by

 0 1 0 0 40 0 0 0 0 0 35 40 0 0 1 0
,
 0 1 0 0 1 0 0 0 0 0 40 35 0 0 0 1
,
 29 12 0 0 12 12 0 0 0 0 21 21 0 0 18 20
,
 1 0 0 0 0 1 0 0 0 0 28 13 0 0 9 13
`G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,35,1,0,0,40,0],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,35,1],[29,12,0,0,12,12,0,0,0,0,21,18,0,0,21,20],[1,0,0,0,0,1,0,0,0,0,28,9,0,0,13,13] >;`

D202Q8 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_2Q_8`
`% in TeX`

`G:=Group("D20:2Q8");`
`// GroupNames label`

`G:=SmallGroup(320,517);`
`// by ID`

`G=gap.SmallGroup(320,517);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,268,1684,851,102,12550]);`
`// 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^13*b,d*b*d^-1=a^8*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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