Copied to
clipboard

## G = C20⋊4D8order 320 = 26·5

### 1st semidirect product of C20 and D8 acting via D8/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C20⋊4D8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊4D4 — C20⋊4D8
 Lower central C5 — C10 — C2×C20 — C20⋊4D8
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C204D8
G = < a,b,c | a20=b8=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 1070 in 162 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C2×C8, D8, C2×D4, C20, D10, C2×C10, C4×C8, C41D4, C2×D8, C40, D20, C2×C20, C2×C20, C22×D5, C84D4, D40, C4×C20, C2×C40, C2×D20, C2×D20, C4×C40, C204D4, C2×D40, C204D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C41D4, C2×D8, D20, C22×D5, C84D4, D40, C2×D20, C204D4, C2×D40, C204D8

Smallest permutation representation of C204D8
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)
(1 158 38 114 85 135 57 65)(2 159 39 115 86 136 58 66)(3 160 40 116 87 137 59 67)(4 141 21 117 88 138 60 68)(5 142 22 118 89 139 41 69)(6 143 23 119 90 140 42 70)(7 144 24 120 91 121 43 71)(8 145 25 101 92 122 44 72)(9 146 26 102 93 123 45 73)(10 147 27 103 94 124 46 74)(11 148 28 104 95 125 47 75)(12 149 29 105 96 126 48 76)(13 150 30 106 97 127 49 77)(14 151 31 107 98 128 50 78)(15 152 32 108 99 129 51 79)(16 153 33 109 100 130 52 80)(17 154 34 110 81 131 53 61)(18 155 35 111 82 132 54 62)(19 156 36 112 83 133 55 63)(20 157 37 113 84 134 56 64)
(1 153)(2 152)(3 151)(4 150)(5 149)(6 148)(7 147)(8 146)(9 145)(10 144)(11 143)(12 142)(13 141)(14 160)(15 159)(16 158)(17 157)(18 156)(19 155)(20 154)(21 77)(22 76)(23 75)(24 74)(25 73)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 61)(38 80)(39 79)(40 78)(41 105)(42 104)(43 103)(44 102)(45 101)(46 120)(47 119)(48 118)(49 117)(50 116)(51 115)(52 114)(53 113)(54 112)(55 111)(56 110)(57 109)(58 108)(59 107)(60 106)(81 134)(82 133)(83 132)(84 131)(85 130)(86 129)(87 128)(88 127)(89 126)(90 125)(91 124)(92 123)(93 122)(94 121)(95 140)(96 139)(97 138)(98 137)(99 136)(100 135)```

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

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

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

86 conjugacy classes

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

86 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 D4 D4 D5 D8 D10 D10 D20 D20 D40 kernel C20⋊4D8 C4×C40 C20⋊4D4 C2×D40 C40 C2×C20 C4×C8 C20 C42 C2×C8 C8 C2×C4 C4 # reps 1 1 2 4 4 2 2 8 2 4 16 8 32

Matrix representation of C204D8 in GL4(𝔽41) generated by

 25 30 0 0 27 39 0 0 0 0 14 30 0 0 11 9
,
 5 15 0 0 34 12 0 0 0 0 1 0 0 0 0 1
,
 23 20 0 0 31 18 0 0 0 0 11 9 0 0 14 30
`G:=sub<GL(4,GF(41))| [25,27,0,0,30,39,0,0,0,0,14,11,0,0,30,9],[5,34,0,0,15,12,0,0,0,0,1,0,0,0,0,1],[23,31,0,0,20,18,0,0,0,0,11,14,0,0,9,30] >;`

C204D8 in GAP, Magma, Sage, TeX

`C_{20}\rtimes_4D_8`
`% in TeX`

`G:=Group("C20:4D8");`
`// GroupNames label`

`G:=SmallGroup(320,322);`
`// by ID`

`G=gap.SmallGroup(320,322);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,226,1123,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^8=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

׿
×
𝔽