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## G = D40.C4order 320 = 26·5

### 1st non-split extension by D40 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D40.C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8×D5 — C40.C4 — D40.C4
 Lower central C5 — C10 — C20 — C40 — D40.C4
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for D40.C4
G = < a,b,c | a40=b2=1, c4=a20, bab=a-1, cac-1=a3, cbc-1=a37b >

Subgroups: 410 in 62 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, C23, D5, C10, C10, C16, C2×C8, M4(2), D8, D8, C2×D4, Dic5, C20, D10, D10, C2×C10, C8.C4, M5(2), C2×D8, C52C8, C40, C5⋊C8, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, M5(2)⋊C2, C5⋊C16, C8×D5, D40, D4⋊D5, C5×D8, C4.F5, D4×D5, C8.F5, C40.C4, D5×D8, D40.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, F5, D4⋊C4, C2×F5, M5(2)⋊C2, C22⋊F5, D20⋊C4, D40.C4

Character table of D40.C4

 class 1 2A 2B 2C 2D 4A 4B 5 8A 8B 8C 8D 8E 10A 10B 10C 16A 16B 16C 16D 20 40A 40B size 1 1 8 10 40 2 10 4 4 10 10 40 40 4 16 16 20 20 20 20 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ4 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 -1 -1 1 -1 1 1 -1 -1 -i i 1 1 1 -i i i -i 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 1 -1 1 1 -1 -1 -i i 1 -1 -1 i -i -i i 1 1 1 linear of order 4 ρ7 1 1 1 -1 -1 1 -1 1 1 -1 -1 i -i 1 1 1 i -i -i i 1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 1 -1 1 1 -1 -1 i -i 1 -1 -1 -i i i -i 1 1 1 linear of order 4 ρ9 2 2 0 2 0 2 2 2 -2 -2 -2 0 0 2 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ10 2 2 0 -2 0 2 -2 2 -2 2 2 0 0 2 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ11 2 2 0 -2 0 -2 2 2 0 0 0 0 0 2 0 0 √2 -√2 √2 -√2 -2 0 0 orthogonal lifted from D8 ρ12 2 2 0 -2 0 -2 2 2 0 0 0 0 0 2 0 0 -√2 √2 -√2 √2 -2 0 0 orthogonal lifted from D8 ρ13 2 2 0 2 0 -2 -2 2 0 0 0 0 0 2 0 0 √-2 √-2 -√-2 -√-2 -2 0 0 complex lifted from SD16 ρ14 2 2 0 2 0 -2 -2 2 0 0 0 0 0 2 0 0 -√-2 -√-2 √-2 √-2 -2 0 0 complex lifted from SD16 ρ15 4 4 4 0 0 4 0 -1 4 0 0 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 orthogonal lifted from F5 ρ16 4 4 -4 0 0 4 0 -1 4 0 0 0 0 -1 1 1 0 0 0 0 -1 -1 -1 orthogonal lifted from C2×F5 ρ17 4 4 0 0 0 4 0 -1 -4 0 0 0 0 -1 -√5 √5 0 0 0 0 -1 1 1 orthogonal lifted from C22⋊F5 ρ18 4 4 0 0 0 4 0 -1 -4 0 0 0 0 -1 √5 -√5 0 0 0 0 -1 1 1 orthogonal lifted from C22⋊F5 ρ19 4 -4 0 0 0 0 0 4 0 2√2 -2√2 0 0 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from M5(2)⋊C2 ρ20 4 -4 0 0 0 0 0 4 0 -2√2 2√2 0 0 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from M5(2)⋊C2 ρ21 8 8 0 0 0 -8 0 -2 0 0 0 0 0 -2 0 0 0 0 0 0 2 0 0 orthogonal lifted from D20⋊C4, Schur index 2 ρ22 8 -8 0 0 0 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 0 0 -√10 √10 orthogonal faithful, Schur index 2 ρ23 8 -8 0 0 0 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 0 0 √10 -√10 orthogonal faithful, Schur index 2

Smallest permutation representation of D40.C4
On 80 points
Generators in S80
```(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)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(41 77)(42 76)(43 75)(44 74)(45 73)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 60)(78 80)
(1 57 11 47 21 77 31 67)(2 44 20 50 22 64 40 70)(3 71 29 53 23 51 9 73)(4 58 38 56 24 78 18 76)(5 45 7 59 25 65 27 79)(6 72 16 62 26 52 36 42)(8 46 34 68 28 66 14 48)(10 60 12 74 30 80 32 54)(13 61 39 43 33 41 19 63)(15 75 17 49 35 55 37 69)```

`G:=sub<Sym(80)| (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(41,77)(42,76)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(78,80), (1,57,11,47,21,77,31,67)(2,44,20,50,22,64,40,70)(3,71,29,53,23,51,9,73)(4,58,38,56,24,78,18,76)(5,45,7,59,25,65,27,79)(6,72,16,62,26,52,36,42)(8,46,34,68,28,66,14,48)(10,60,12,74,30,80,32,54)(13,61,39,43,33,41,19,63)(15,75,17,49,35,55,37,69)>;`

`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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(41,77)(42,76)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(78,80), (1,57,11,47,21,77,31,67)(2,44,20,50,22,64,40,70)(3,71,29,53,23,51,9,73)(4,58,38,56,24,78,18,76)(5,45,7,59,25,65,27,79)(6,72,16,62,26,52,36,42)(8,46,34,68,28,66,14,48)(10,60,12,74,30,80,32,54)(13,61,39,43,33,41,19,63)(15,75,17,49,35,55,37,69) );`

`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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(41,77),(42,76),(43,75),(44,74),(45,73),(46,72),(47,71),(48,70),(49,69),(50,68),(51,67),(52,66),(53,65),(54,64),(55,63),(56,62),(57,61),(58,60),(78,80)], [(1,57,11,47,21,77,31,67),(2,44,20,50,22,64,40,70),(3,71,29,53,23,51,9,73),(4,58,38,56,24,78,18,76),(5,45,7,59,25,65,27,79),(6,72,16,62,26,52,36,42),(8,46,34,68,28,66,14,48),(10,60,12,74,30,80,32,54),(13,61,39,43,33,41,19,63),(15,75,17,49,35,55,37,69)]])`

Matrix representation of D40.C4 in GL8(𝔽241)

 240 52 0 0 0 0 0 0 189 52 0 0 0 0 0 0 0 141 240 190 0 0 0 0 100 141 51 190 0 0 0 0 0 0 0 0 0 71 0 0 0 0 0 0 112 22 0 0 0 0 0 0 0 0 219 71 0 0 0 0 0 0 112 0
,
 1 0 0 0 0 0 0 0 52 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 141 0 190 240 0 0 0 0 0 0 0 0 0 170 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 0 1 125 0 0 0 0 0 0 0 240
,
 222 236 166 31 0 0 0 0 222 0 166 0 0 0 0 0 143 7 19 5 0 0 0 0 143 0 19 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 240 116 0 0 0 0 0 0 54 1 0 0

`G:=sub<GL(8,GF(241))| [240,189,0,100,0,0,0,0,52,52,141,141,0,0,0,0,0,0,240,51,0,0,0,0,0,0,190,190,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,71,22,0,0,0,0,0,0,0,0,219,112,0,0,0,0,0,0,71,0],[1,52,0,141,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,190,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,170,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,125,240],[222,222,143,143,0,0,0,0,236,0,7,0,0,0,0,0,166,166,19,19,0,0,0,0,31,0,5,0,0,0,0,0,0,0,0,0,0,0,240,54,0,0,0,0,0,0,116,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

D40.C4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(320,244);`
`// by ID`

`G=gap.SmallGroup(320,244);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,184,675,794,80,1684,851,102,6278,3156]);`
`// 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^3,c*b*c^-1=a^37*b>;`
`// generators/relations`

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