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## G = Q8.D20order 320 = 26·5

### 2nd non-split extension by Q8 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Q8.D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D10⋊2Q8 — Q8.D20
 Lower central C5 — C10 — C2×C20 — Q8.D20
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Q8.D20
G = < a,b,c,d | a4=c20=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 702 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C52C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×D5, D4.7D4, C40⋊C2, C2×C52C8, C4⋊Dic5, D10⋊C4, C5⋊Q16, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q82D5, Q8×C10, D206C4, D101C8, C5×Q8⋊C4, D102Q8, C2×C40⋊C2, C2×C5⋊Q16, C2×Q82D5, Q8.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8.C22, D20, C22×D5, D4.7D4, C2×D20, D4×D5, C22⋊D20, SD163D5, Q16⋊D5, Q8.D20

Smallest permutation representation of Q8.D20
On 160 points
Generators in S160
```(1 88 40 106)(2 107 21 89)(3 90 22 108)(4 109 23 91)(5 92 24 110)(6 111 25 93)(7 94 26 112)(8 113 27 95)(9 96 28 114)(10 115 29 97)(11 98 30 116)(12 117 31 99)(13 100 32 118)(14 119 33 81)(15 82 34 120)(16 101 35 83)(17 84 36 102)(18 103 37 85)(19 86 38 104)(20 105 39 87)(41 64 125 141)(42 142 126 65)(43 66 127 143)(44 144 128 67)(45 68 129 145)(46 146 130 69)(47 70 131 147)(48 148 132 71)(49 72 133 149)(50 150 134 73)(51 74 135 151)(52 152 136 75)(53 76 137 153)(54 154 138 77)(55 78 139 155)(56 156 140 79)(57 80 121 157)(58 158 122 61)(59 62 123 159)(60 160 124 63)
(1 127 40 43)(2 144 21 67)(3 129 22 45)(4 146 23 69)(5 131 24 47)(6 148 25 71)(7 133 26 49)(8 150 27 73)(9 135 28 51)(10 152 29 75)(11 137 30 53)(12 154 31 77)(13 139 32 55)(14 156 33 79)(15 121 34 57)(16 158 35 61)(17 123 36 59)(18 160 37 63)(19 125 38 41)(20 142 39 65)(42 87 126 105)(44 89 128 107)(46 91 130 109)(48 93 132 111)(50 95 134 113)(52 97 136 115)(54 99 138 117)(56 81 140 119)(58 83 122 101)(60 85 124 103)(62 102 159 84)(64 104 141 86)(66 106 143 88)(68 108 145 90)(70 110 147 92)(72 112 149 94)(74 114 151 96)(76 116 153 98)(78 118 155 100)(80 120 157 82)
(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 52 40 136)(2 135 21 51)(3 50 22 134)(4 133 23 49)(5 48 24 132)(6 131 25 47)(7 46 26 130)(8 129 27 45)(9 44 28 128)(10 127 29 43)(11 42 30 126)(12 125 31 41)(13 60 32 124)(14 123 33 59)(15 58 34 122)(16 121 35 57)(17 56 36 140)(18 139 37 55)(19 54 38 138)(20 137 39 53)(61 120 158 82)(62 81 159 119)(63 118 160 100)(64 99 141 117)(65 116 142 98)(66 97 143 115)(67 114 144 96)(68 95 145 113)(69 112 146 94)(70 93 147 111)(71 110 148 92)(72 91 149 109)(73 108 150 90)(74 89 151 107)(75 106 152 88)(76 87 153 105)(77 104 154 86)(78 85 155 103)(79 102 156 84)(80 83 157 101)```

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 2 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 20 2 2 4 4 8 10 10 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 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D5 D10 D10 D10 C4○D8 D20 C8.C22 D4×D5 D4×D5 SD16⋊3D5 Q16⋊D5 kernel Q8.D20 D20⋊6C4 D10⋊1C8 C5×Q8⋊C4 D10⋊2Q8 C2×C40⋊C2 C2×C5⋊Q16 C2×Q8⋊2D5 D20 C2×Dic5 C5×Q8 C22×D5 Q8⋊C4 C4⋊C4 C2×C8 C2×Q8 C10 Q8 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 2 1 2 2 2 2 4 8 1 2 2 4 4

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

 1 0 0 0 0 1 0 0 0 0 1 9 0 0 18 40
,
 1 0 0 0 0 1 0 0 0 0 9 40 0 0 0 32
,
 14 39 0 0 16 30 0 0 0 0 17 15 0 0 30 24
,
 30 2 0 0 22 11 0 0 0 0 0 12 0 0 17 0
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,18,0,0,9,40],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,40,32],[14,16,0,0,39,30,0,0,0,0,17,30,0,0,15,24],[30,22,0,0,2,11,0,0,0,0,0,17,0,0,12,0] >;`

Q8.D20 in GAP, Magma, Sage, TeX

`Q_8.D_{20}`
`% in TeX`

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

`G:=SmallGroup(320,437);`
`// by ID`

`G=gap.SmallGroup(320,437);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,758,135,184,570,297,136,12550]);`
`// Polycyclic`

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

׿
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