Copied to
clipboard

## G = D4.5D20order 320 = 26·5

### 5th non-split extension by D4 of D20 acting via D20/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.5D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C2×Dic20 — D4.5D20
 Lower central C5 — C10 — C2×C20 — D4.5D20
 Upper central C1 — C2 — C2×C4 — C8○D4

Generators and relations for D4.5D20
G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c19 >

Subgroups: 350 in 100 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C52C8, C40, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, D4.5D4, Dic20, C4.Dic5, D4.D5, C5⋊Q16, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C5×C4○D4, C40.6C4, C4.12D20, C2×Dic20, D4.9D10, C5×C8○D4, D4.5D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C5⋊D4, C22×D5, D4.5D4, C2×D20, C4○D20, C2×C5⋊D4, C207D4, D4.5D20

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 8E 8F 8G 10A 10B 10C ··· 10H 20A 20B 20C 20D 20E ··· 20J 40A ··· 40H 40I ··· 40T order 1 2 2 2 4 4 4 4 4 5 5 8 8 8 8 8 8 8 10 10 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 2 4 2 2 4 40 40 2 2 2 2 4 4 4 40 40 2 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 D20 D20 C4○D20 D4.5D4 D4.5D20 kernel D4.5D20 C40.6C4 C4.12D20 C2×Dic20 D4.9D10 C5×C8○D4 C40 C5×D4 C5×Q8 C8○D4 C2×C10 C2×C8 M4(2) C4○D4 C8 D4 Q8 C22 C5 C1 # reps 1 1 2 1 2 1 2 1 1 2 2 2 2 2 8 4 4 8 2 8

Matrix representation of D4.5D20 in GL4(𝔽41) generated by

 1 0 15 17 5 40 39 0 0 1 18 22 24 16 17 23
,
 1 0 15 17 5 40 39 0 0 0 18 22 0 0 17 23
,
 13 0 0 0 10 19 0 0 16 0 26 38 10 0 3 6
,
 25 31 33 5 13 3 5 5 7 35 16 28 35 30 10 38
`G:=sub<GL(4,GF(41))| [1,5,0,24,0,40,1,16,15,39,18,17,17,0,22,23],[1,5,0,0,0,40,0,0,15,39,18,17,17,0,22,23],[13,10,16,10,0,19,0,0,0,0,26,3,0,0,38,6],[25,13,7,35,31,3,35,30,33,5,16,10,5,5,28,38] >;`

D4.5D20 in GAP, Magma, Sage, TeX

`D_4._5D_{20}`
`% in TeX`

`G:=Group("D4.5D20");`
`// GroupNames label`

`G:=SmallGroup(320,770);`
`// by ID`

`G=gap.SmallGroup(320,770);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,344,254,1123,297,136,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^20=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=a^2*c^19>;`
`// generators/relations`

׿
×
𝔽