Copied to
clipboard

## G = C4.D40order 320 = 26·5

### 1st non-split extension by C4 of D40 acting via D40/D20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C4.D40
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×C20 — C20⋊4D4 — C4.D40
 Lower central C5 — C2×C10 — C2×C20 — C4.D40
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for C4.D40
G = < a,b,c | a4=b40=1, c2=a, bab-1=a-1, ac=ca, cbc-1=ab-1 >

Subgroups: 542 in 84 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, D4, C23, D5, C10, C42, C2×C8, C2×D4, C20, C20, D10, C2×C10, C4⋊C8, C4⋊C8, C41D4, C52C8, C40, D20, C2×C20, C22×D5, C4.D8, C2×C52C8, C4×C20, C2×C40, C2×D20, C2×D20, C203C8, C5×C4⋊C8, C204D4, C4.D40
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, C4.D4, D4⋊C4, C4×D5, D20, C5⋊D4, C4.D8, C40⋊C2, D40, D10⋊C4, D4⋊D5, Q8⋊D5, D206C4, D205C4, C20.46D4, C4.D40

Smallest permutation representation of C4.D40
On 160 points
Generators in S160
```(1 138 68 115)(2 116 69 139)(3 140 70 117)(4 118 71 141)(5 142 72 119)(6 120 73 143)(7 144 74 81)(8 82 75 145)(9 146 76 83)(10 84 77 147)(11 148 78 85)(12 86 79 149)(13 150 80 87)(14 88 41 151)(15 152 42 89)(16 90 43 153)(17 154 44 91)(18 92 45 155)(19 156 46 93)(20 94 47 157)(21 158 48 95)(22 96 49 159)(23 160 50 97)(24 98 51 121)(25 122 52 99)(26 100 53 123)(27 124 54 101)(28 102 55 125)(29 126 56 103)(30 104 57 127)(31 128 58 105)(32 106 59 129)(33 130 60 107)(34 108 61 131)(35 132 62 109)(36 110 63 133)(37 134 64 111)(38 112 65 135)(39 136 66 113)(40 114 67 137)
(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)
(1 114 138 67 68 137 115 40)(2 39 116 136 69 66 139 113)(3 112 140 65 70 135 117 38)(4 37 118 134 71 64 141 111)(5 110 142 63 72 133 119 36)(6 35 120 132 73 62 143 109)(7 108 144 61 74 131 81 34)(8 33 82 130 75 60 145 107)(9 106 146 59 76 129 83 32)(10 31 84 128 77 58 147 105)(11 104 148 57 78 127 85 30)(12 29 86 126 79 56 149 103)(13 102 150 55 80 125 87 28)(14 27 88 124 41 54 151 101)(15 100 152 53 42 123 89 26)(16 25 90 122 43 52 153 99)(17 98 154 51 44 121 91 24)(18 23 92 160 45 50 155 97)(19 96 156 49 46 159 93 22)(20 21 94 158 47 48 157 95)```

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

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

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

59 conjugacy classes

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

59 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D5 D8 SD16 D10 C4×D5 D20 C5⋊D4 C40⋊C2 D40 C4.D4 D4⋊D5 Q8⋊D5 C20.46D4 kernel C4.D40 C20⋊3C8 C5×C4⋊C8 C20⋊4D4 C2×D20 C2×C20 C4⋊C8 C20 C20 C42 C2×C4 C2×C4 C2×C4 C4 C4 C10 C4 C4 C2 # reps 1 1 1 1 4 2 2 4 4 2 4 4 4 8 8 1 2 2 4

Matrix representation of C4.D40 in GL4(𝔽41) generated by

 0 40 0 0 1 0 0 0 0 0 40 0 0 0 0 40
,
 15 15 0 0 15 26 0 0 0 0 28 18 0 0 23 25
,
 15 15 0 0 26 15 0 0 0 0 28 18 0 0 27 13
`G:=sub<GL(4,GF(41))| [0,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[15,15,0,0,15,26,0,0,0,0,28,23,0,0,18,25],[15,26,0,0,15,15,0,0,0,0,28,27,0,0,18,13] >;`

C4.D40 in GAP, Magma, Sage, TeX

`C_4.D_{40}`
`% in TeX`

`G:=Group("C4.D40");`
`// GroupNames label`

`G:=SmallGroup(320,43);`
`// by ID`

`G=gap.SmallGroup(320,43);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,100,1123,794,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^40=1,c^2=a,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a*b^-1>;`
`// generators/relations`

׿
×
𝔽