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## G = C4.5D40order 320 = 26·5

### 5th non-split extension by C4 of D40 acting via D40/C40=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C4.5D40
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊4D4 — C4.5D40
 Lower central C5 — C10 — C2×C20 — C4.5D40
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C4.5D40
G = < a,b,c | a4=b40=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 686 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C4⋊C4, C2×C8, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C4×C8, D4⋊C4, C41D4, C4⋊Q8, C40, Dic10, D20, C2×Dic5, C2×C20, C22×D5, C4.4D8, C4⋊Dic5, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C2×D20, C2×D20, D205C4, C4×C40, C202Q8, C204D4, C4.5D40
Quotients: C1, C2, C22, D4, C23, D5, D8, SD16, C2×D4, C4○D4, D10, C4.4D4, C2×D8, C2×SD16, D20, C22×D5, C4.4D8, C40⋊C2, D40, C2×D20, C4○D20, C4.D20, C2×C40⋊C2, C2×D40, C4.5D40

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

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

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

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

86 conjugacy classes

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

86 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D5 D8 SD16 C4○D4 D10 D10 D20 C40⋊C2 D40 C4○D20 kernel C4.5D40 D20⋊5C4 C4×C40 C20⋊2Q8 C20⋊4D4 C2×C20 C4×C8 C20 C20 C20 C42 C2×C8 C2×C4 C4 C4 C4 # reps 1 4 1 1 1 2 2 4 4 4 2 4 8 16 16 16

Matrix representation of C4.5D40 in GL4(𝔽41) generated by

 30 28 0 0 22 11 0 0 0 0 40 0 0 0 0 40
,
 22 13 0 0 19 0 0 0 0 0 0 11 0 0 26 17
,
 19 32 0 0 22 22 0 0 0 0 0 11 0 0 15 0
`G:=sub<GL(4,GF(41))| [30,22,0,0,28,11,0,0,0,0,40,0,0,0,0,40],[22,19,0,0,13,0,0,0,0,0,0,26,0,0,11,17],[19,22,0,0,32,22,0,0,0,0,0,15,0,0,11,0] >;`

C4.5D40 in GAP, Magma, Sage, TeX

`C_4._5D_{40}`
`% in TeX`

`G:=Group("C4.5D40");`
`// GroupNames label`

`G:=SmallGroup(320,321);`
`// by ID`

`G=gap.SmallGroup(320,321);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,142,1123,136,12550]);`
`// Polycyclic`

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

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