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## G = D20.4Q8order 320 = 26·5

### 2nd non-split extension by D20 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.4Q8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4×D20 — D20.4Q8
 Lower central C5 — C10 — C2×C20 — D20.4Q8
 Upper central C1 — C22 — C42 — C42.C2

Generators and relations for D20.4Q8
G = < a,b,c,d | a20=b2=c4=1, d2=a10c2, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 422 in 102 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, D4.Q8, C2×C52C8, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C203C8, C10.D8, C20.Q8, D206C4, C4×D20, C5×C42.C2, D20.4Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4.Q8, Q8×D5, Q82D5, C2×C5⋊D4, D103Q8, D4⋊D10, D4.8D10, D20.4Q8

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

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

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

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

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 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 Q8 D4 D5 C4○D4 D10 D10 C4○D8 C5⋊D4 C8⋊C22 Q8×D5 Q8⋊2D5 D4⋊D10 D4.8D10 kernel D20.4Q8 C20⋊3C8 C10.D8 C20.Q8 D20⋊6C4 C4×D20 C5×C42.C2 D20 C2×C20 C42.C2 C20 C42 C4⋊C4 C10 C2×C4 C10 C4 C4 C2 C2 # reps 1 1 1 1 2 1 1 2 2 2 2 2 4 4 8 1 2 2 4 4

Matrix representation of D20.4Q8 in GL6(𝔽41)

 0 1 0 0 0 0 40 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 39 0 0 0 0 1 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 2 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 24 17 0 0 0 0 12 17
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 6 39 0 0 0 0 39 35 0 0 0 0 0 0 32 0 0 0 0 0 0 32

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,24,12,0,0,0,0,17,17],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,39,0,0,0,0,39,35,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;`

D20.4Q8 in GAP, Magma, Sage, TeX

`D_{20}._4Q_8`
`% in TeX`

`G:=Group("D20.4Q8");`
`// GroupNames label`

`G:=SmallGroup(320,693);`
`// by ID`

`G=gap.SmallGroup(320,693);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,100,1123,297,136,12550]);`
`// Polycyclic`

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

׿
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