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G = D40.5C4order 320 = 26·5

3rd non-split extension by D40 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D40.5C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — D40⋊7C2 — D40.5C4
 Lower central C5 — C10 — C20 — C40 — D40.5C4
 Upper central C1 — C4 — C2×C4 — C2×C8 — C8.C4

Generators and relations for D40.5C4
G = < a,b,c | a40=b2=1, c4=a20, bab=a-1, cac-1=a31, cbc-1=a15b >

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 16A ··· 16H 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I ··· 40P order 1 2 2 2 4 4 4 4 5 5 8 8 8 8 8 8 10 10 10 10 16 ··· 16 20 20 20 20 20 20 40 ··· 40 40 ··· 40 size 1 1 2 40 1 1 2 40 2 2 2 2 2 2 8 8 2 2 4 4 10 ··· 10 2 2 2 2 4 4 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 D5 D8 SD16 D10 C4×D5 D20 C5⋊D4 D8.C4 D4⋊D5 Q8⋊D5 D40.5C4 kernel D40.5C4 C2×C5⋊2C16 C5×C8.C4 D40⋊7C2 D40 Dic20 C40 C2×C20 C8.C4 C20 C2×C10 C2×C8 C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 2 2 2 2 4 4 4 8 2 2 8

Matrix representation of D40.5C4 in GL4(𝔽241) generated by

 0 219 0 0 11 219 0 0 0 0 190 190 0 0 51 240
,
 0 219 0 0 230 0 0 0 0 0 190 190 0 0 240 51
,
 173 50 0 0 198 68 0 0 0 0 177 0 0 0 0 177
`G:=sub<GL(4,GF(241))| [0,11,0,0,219,219,0,0,0,0,190,51,0,0,190,240],[0,230,0,0,219,0,0,0,0,0,190,240,0,0,190,51],[173,198,0,0,50,68,0,0,0,0,177,0,0,0,0,177] >;`

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

`D_{40}._5C_4`
`% in TeX`

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

`G:=SmallGroup(320,55);`
`// by ID`

`G=gap.SmallGroup(320,55);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,184,675,346,192,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^2=1,c^4=a^20,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^15*b>;`
`// generators/relations`

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