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## G = C40.5D4order 320 = 26·5

### 5th non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C40.5D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C2×D40 — C40.5D4
 Lower central C5 — C10 — C20 — C40 — C40.5D4
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2.D8

Generators and relations for C40.5D4
G = < a,b,c | a40=b4=1, c2=a25, bab-1=a31, cac-1=a9, cbc-1=a25b-1 >

Subgroups: 430 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C16, C4⋊C4, C2×C8, D8, C2×D4, C20, C20, D10, C2×C10, C2.D8, C2×C16, C2×D8, C40, D20, C2×C20, C2×C20, C22×D5, C2.D16, C52C16, D40, D40, C5×C4⋊C4, C2×C40, C2×D20, C2×C52C16, C5×C2.D8, C2×D40, C40.5D4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, D16, SD32, C4×D5, D20, C5⋊D4, C2.D16, D10⋊C4, D4⋊D5, Q8⋊D5, D206C4, C5⋊D16, C5⋊SD32, C40.5D4

Smallest permutation representation of C40.5D4
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)
(1 106 137 43)(2 97 138 74)(3 88 139 65)(4 119 140 56)(5 110 141 47)(6 101 142 78)(7 92 143 69)(8 83 144 60)(9 114 145 51)(10 105 146 42)(11 96 147 73)(12 87 148 64)(13 118 149 55)(14 109 150 46)(15 100 151 77)(16 91 152 68)(17 82 153 59)(18 113 154 50)(19 104 155 41)(20 95 156 72)(21 86 157 63)(22 117 158 54)(23 108 159 45)(24 99 160 76)(25 90 121 67)(26 81 122 58)(27 112 123 49)(28 103 124 80)(29 94 125 71)(30 85 126 62)(31 116 127 53)(32 107 128 44)(33 98 129 75)(34 89 130 66)(35 120 131 57)(36 111 132 48)(37 102 133 79)(38 93 134 70)(39 84 135 61)(40 115 136 52)
(1 68 26 53 11 78 36 63 21 48 6 73 31 58 16 43)(2 77 27 62 12 47 37 72 22 57 7 42 32 67 17 52)(3 46 28 71 13 56 38 41 23 66 8 51 33 76 18 61)(4 55 29 80 14 65 39 50 24 75 9 60 34 45 19 70)(5 64 30 49 15 74 40 59 25 44 10 69 35 54 20 79)(81 152 106 137 91 122 116 147 101 132 86 157 111 142 96 127)(82 121 107 146 92 131 117 156 102 141 87 126 112 151 97 136)(83 130 108 155 93 140 118 125 103 150 88 135 113 160 98 145)(84 139 109 124 94 149 119 134 104 159 89 144 114 129 99 154)(85 148 110 133 95 158 120 143 105 128 90 153 115 138 100 123)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D4 D5 SD16 D8 D10 D16 SD32 C4×D5 D20 C5⋊D4 Q8⋊D5 D4⋊D5 C5⋊D16 C5⋊SD32 kernel C40.5D4 C2×C5⋊2C16 C5×C2.D8 C2×D40 D40 C40 C2×C20 C2.D8 C20 C2×C10 C2×C8 C10 C10 C8 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 4 4 2 2 4 4

Matrix representation of C40.5D4 in GL6(𝔽241)

 11 230 0 0 0 0 11 11 0 0 0 0 0 0 11 230 0 0 0 0 11 11 0 0 0 0 0 0 190 52 0 0 0 0 190 0
,
 103 200 0 0 0 0 200 138 0 0 0 0 0 0 85 27 0 0 0 0 27 156 0 0 0 0 0 0 165 103 0 0 0 0 89 76
,
 103 200 0 0 0 0 41 103 0 0 0 0 0 0 27 156 0 0 0 0 85 27 0 0 0 0 0 0 69 27 0 0 0 0 20 172

`G:=sub<GL(6,GF(241))| [11,11,0,0,0,0,230,11,0,0,0,0,0,0,11,11,0,0,0,0,230,11,0,0,0,0,0,0,190,190,0,0,0,0,52,0],[103,200,0,0,0,0,200,138,0,0,0,0,0,0,85,27,0,0,0,0,27,156,0,0,0,0,0,0,165,89,0,0,0,0,103,76],[103,41,0,0,0,0,200,103,0,0,0,0,0,0,27,85,0,0,0,0,156,27,0,0,0,0,0,0,69,20,0,0,0,0,27,172] >;`

C40.5D4 in GAP, Magma, Sage, TeX

`C_{40}._5D_4`
`% in TeX`

`G:=Group("C40.5D4");`
`// GroupNames label`

`G:=SmallGroup(320,49);`
`// by ID`

`G=gap.SmallGroup(320,49);`
`# by ID`

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

`G:=Group<a,b,c|a^40=b^4=1,c^2=a^25,b*a*b^-1=a^31,c*a*c^-1=a^9,c*b*c^-1=a^25*b^-1>;`
`// generators/relations`

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