Copied to
clipboard

## G = D4.2D20order 320 = 26·5

### 2nd non-split extension by D4 of D20 acting via D20/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.2D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C20⋊2Q8 — D4.2D20
 Lower central C5 — C10 — C2×C20 — D4.2D20
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for D4.2D20
G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 406 in 120 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C52C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4.D4, C2×C52C8, C4⋊Dic5, D4.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C203C8, C10.Q16, C202Q8, C2×D4.D5, D4×C20, D4.2D20
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8.C22, D20, C5⋊D4, C22×D5, D4.D4, D4.D5, C2×D20, C4○D20, C2×C5⋊D4, C207D4, C2×D4.D5, D4.9D10, D4.2D20

Smallest permutation representation of D4.2D20
On 160 points
Generators in S160
```(1 72 36 106)(2 73 37 107)(3 74 38 108)(4 75 39 109)(5 76 40 110)(6 77 21 111)(7 78 22 112)(8 79 23 113)(9 80 24 114)(10 61 25 115)(11 62 26 116)(12 63 27 117)(13 64 28 118)(14 65 29 119)(15 66 30 120)(16 67 31 101)(17 68 32 102)(18 69 33 103)(19 70 34 104)(20 71 35 105)(41 142 87 125)(42 143 88 126)(43 144 89 127)(44 145 90 128)(45 146 91 129)(46 147 92 130)(47 148 93 131)(48 149 94 132)(49 150 95 133)(50 151 96 134)(51 152 97 135)(52 153 98 136)(53 154 99 137)(54 155 100 138)(55 156 81 139)(56 157 82 140)(57 158 83 121)(58 159 84 122)(59 160 85 123)(60 141 86 124)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 97)(42 98)(43 99)(44 100)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(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 43 36 89)(2 42 37 88)(3 41 38 87)(4 60 39 86)(5 59 40 85)(6 58 21 84)(7 57 22 83)(8 56 23 82)(9 55 24 81)(10 54 25 100)(11 53 26 99)(12 52 27 98)(13 51 28 97)(14 50 29 96)(15 49 30 95)(16 48 31 94)(17 47 32 93)(18 46 33 92)(19 45 34 91)(20 44 35 90)(61 138 115 155)(62 137 116 154)(63 136 117 153)(64 135 118 152)(65 134 119 151)(66 133 120 150)(67 132 101 149)(68 131 102 148)(69 130 103 147)(70 129 104 146)(71 128 105 145)(72 127 106 144)(73 126 107 143)(74 125 108 142)(75 124 109 141)(76 123 110 160)(77 122 111 159)(78 121 112 158)(79 140 113 157)(80 139 114 156)```

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

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

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

59 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 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 4 2 2 2 2 4 4 4 40 40 2 2 20 20 20 20 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 D4 D4 D5 SD16 C4○D4 D10 D10 D10 C5⋊D4 D20 C4○D20 C8.C22 D4.D5 D4.9D10 kernel D4.2D20 C20⋊3C8 C10.Q16 C20⋊2Q8 C2×D4.D5 D4×C20 C2×C20 C5×D4 C4×D4 C20 C20 C42 C4⋊C4 C2×D4 C2×C4 D4 C4 C10 C4 C2 # reps 1 1 2 1 2 1 2 2 2 4 2 2 2 2 8 8 8 1 4 4

Matrix representation of D4.2D20 in GL6(𝔽41)

 40 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 0 2 22 0 0 0 0 24 39
,
 1 0 0 0 0 0 8 40 0 0 0 0 0 0 40 0 0 0 0 0 39 1 0 0 0 0 0 0 2 22 0 0 0 0 39 39
,
 32 0 0 0 0 0 10 9 0 0 0 0 0 0 16 0 0 0 0 0 39 18 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 23 25 0 0 0 0 33 18 0 0 0 0 0 0 28 13 0 0 0 0 6 13 0 0 0 0 0 0 15 2 0 0 0 0 10 26

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,24,0,0,0,0,22,39],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,40,39,0,0,0,0,0,1,0,0,0,0,0,0,2,39,0,0,0,0,22,39],[32,10,0,0,0,0,0,9,0,0,0,0,0,0,16,39,0,0,0,0,0,18,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[23,33,0,0,0,0,25,18,0,0,0,0,0,0,28,6,0,0,0,0,13,13,0,0,0,0,0,0,15,10,0,0,0,0,2,26] >;`

D4.2D20 in GAP, Magma, Sage, TeX

`D_4._2D_{20}`
`% in TeX`

`G:=Group("D4.2D20");`
`// GroupNames label`

`G:=SmallGroup(320,646);`
`// by ID`

`G=gap.SmallGroup(320,646);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,1123,297,136,12550]);`
`// Polycyclic`

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

׿
×
𝔽