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## G = D5.Q32order 320 = 26·5

### The non-split extension by D5 of Q32 acting via Q32/Q16=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D5.Q32
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8×D5 — D5.D8 — D5.Q32
 Lower central C5 — C10 — C20 — C40 — D5.Q32
 Upper central C1 — C2 — C4 — C8 — Q16

Generators and relations for D5.Q32
G = < a,b,c,d | a5=b2=c16=1, d2=c8, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd-1=a-1bc-1 >

Subgroups: 298 in 58 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, Q8, D5, C10, C16, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C2.D8, C2×C16, C2×Q16, C52C8, C40, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C2.Q32, C5⋊C16, C8×D5, Dic20, C5⋊Q16, C5×Q16, C4⋊F5, Q8×D5, D5⋊C16, D5.D8, D5×Q16, D5.Q32
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, F5, D4⋊C4, SD32, Q32, C2×F5, C2.Q32, C22⋊F5, D20⋊C4, D5.Q32

Character table of D5.Q32

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5 8A 8B 8C 8D 10 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 40A 40B size 1 1 5 5 2 8 10 40 40 40 4 2 2 10 10 4 10 10 10 10 10 10 10 10 8 16 16 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 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 -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 -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 1 1 -1 -1 1 1 linear of order 2 ρ5 1 1 -1 -1 1 -1 -1 -i 1 i 1 1 1 -1 -1 1 i -i -i -i -i i i i 1 -1 -1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 1 -1 -i -1 i 1 1 1 -1 -1 1 -i i i i i -i -i -i 1 1 1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 -1 -1 i 1 -i 1 1 1 -1 -1 1 -i i i i i -i -i -i 1 -1 -1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 1 -1 i -1 -i 1 1 1 -1 -1 1 i -i -i -i -i i i i 1 1 1 1 1 linear of order 4 ρ9 2 2 2 2 2 0 2 0 0 0 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 2 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 0 -2 0 0 0 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 2 0 0 -2 -2 orthogonal lifted from D4 ρ11 2 2 2 2 -2 0 -2 0 0 0 2 0 0 0 0 2 -√2 √2 √2 -√2 -√2 √2 √2 -√2 -2 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 2 2 -2 0 -2 0 0 0 2 0 0 0 0 2 √2 -√2 -√2 √2 √2 -√2 -√2 √2 -2 0 0 0 0 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 0 0 0 0 0 2 √2 -√2 -√2 √2 -2 ζ167-ζ16 -ζ165+ζ163 ζ165-ζ163 ζ167-ζ16 -ζ167+ζ16 ζ165-ζ163 -ζ165+ζ163 -ζ167+ζ16 0 0 0 √2 -√2 symplectic lifted from Q32, Schur index 2 ρ14 2 -2 -2 2 0 0 0 0 0 0 2 √2 -√2 -√2 √2 -2 -ζ167+ζ16 ζ165-ζ163 -ζ165+ζ163 -ζ167+ζ16 ζ167-ζ16 -ζ165+ζ163 ζ165-ζ163 ζ167-ζ16 0 0 0 √2 -√2 symplectic lifted from Q32, Schur index 2 ρ15 2 -2 -2 2 0 0 0 0 0 0 2 -√2 √2 √2 -√2 -2 ζ165-ζ163 ζ167-ζ16 -ζ167+ζ16 ζ165-ζ163 -ζ165+ζ163 -ζ167+ζ16 ζ167-ζ16 -ζ165+ζ163 0 0 0 -√2 √2 symplectic lifted from Q32, Schur index 2 ρ16 2 -2 -2 2 0 0 0 0 0 0 2 -√2 √2 √2 -√2 -2 -ζ165+ζ163 -ζ167+ζ16 ζ167-ζ16 -ζ165+ζ163 ζ165-ζ163 ζ167-ζ16 -ζ167+ζ16 ζ165-ζ163 0 0 0 -√2 √2 symplectic lifted from Q32, Schur index 2 ρ17 2 2 -2 -2 -2 0 2 0 0 0 2 0 0 0 0 2 √-2 √-2 √-2 -√-2 -√-2 -√-2 -√-2 √-2 -2 0 0 0 0 complex lifted from SD16 ρ18 2 -2 2 -2 0 0 0 0 0 0 2 -√2 √2 -√2 √2 -2 ζ167+ζ16 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 0 0 0 -√2 √2 complex lifted from SD32 ρ19 2 2 -2 -2 -2 0 2 0 0 0 2 0 0 0 0 2 -√-2 -√-2 -√-2 √-2 √-2 √-2 √-2 -√-2 -2 0 0 0 0 complex lifted from SD16 ρ20 2 -2 2 -2 0 0 0 0 0 0 2 √2 -√2 √2 -√2 -2 ζ165+ζ163 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 0 0 0 √2 -√2 complex lifted from SD32 ρ21 2 -2 2 -2 0 0 0 0 0 0 2 -√2 √2 -√2 √2 -2 ζ1615+ζ169 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 0 0 0 -√2 √2 complex lifted from SD32 ρ22 2 -2 2 -2 0 0 0 0 0 0 2 √2 -√2 √2 -√2 -2 ζ1613+ζ1611 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 0 0 0 √2 -√2 complex lifted from SD32 ρ23 4 4 0 0 4 -4 0 0 0 0 -1 4 4 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 -1 -1 orthogonal lifted from C2×F5 ρ24 4 4 0 0 4 4 0 0 0 0 -1 4 4 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ25 4 4 0 0 4 0 0 0 0 0 -1 -4 -4 0 0 -1 0 0 0 0 0 0 0 0 -1 √5 -√5 1 1 orthogonal lifted from C22⋊F5 ρ26 4 4 0 0 4 0 0 0 0 0 -1 -4 -4 0 0 -1 0 0 0 0 0 0 0 0 -1 -√5 √5 1 1 orthogonal lifted from C22⋊F5 ρ27 8 8 0 0 -8 0 0 0 0 0 -2 0 0 0 0 -2 0 0 0 0 0 0 0 0 2 0 0 0 0 orthogonal lifted from D20⋊C4, Schur index 2 ρ28 8 -8 0 0 0 0 0 0 0 0 -2 -4√2 4√2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 symplectic faithful, Schur index 2 ρ29 8 -8 0 0 0 0 0 0 0 0 -2 4√2 -4√2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 symplectic faithful, Schur index 2

Smallest permutation representation of D5.Q32
On 80 points
Generators in S80
```(1 61 69 42 17)(2 43 62 18 70)(3 19 44 71 63)(4 72 20 64 45)(5 49 73 46 21)(6 47 50 22 74)(7 23 48 75 51)(8 76 24 52 33)(9 53 77 34 25)(10 35 54 26 78)(11 27 36 79 55)(12 80 28 56 37)(13 57 65 38 29)(14 39 58 30 66)(15 31 40 67 59)(16 68 32 60 41)
(1 25)(2 78)(3 55)(4 37)(5 29)(6 66)(7 59)(8 41)(9 17)(10 70)(11 63)(12 45)(13 21)(14 74)(15 51)(16 33)(18 35)(19 79)(20 28)(22 39)(23 67)(24 32)(26 43)(27 71)(30 47)(31 75)(34 61)(36 44)(38 49)(40 48)(42 53)(46 57)(50 58)(52 68)(54 62)(56 72)(60 76)(64 80)(65 73)(69 77)
(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 2 9 10)(3 16 11 8)(4 7 12 15)(5 14 13 6)(17 70 25 78)(18 34 26 42)(19 68 27 76)(20 48 28 40)(21 66 29 74)(22 46 30 38)(23 80 31 72)(24 44 32 36)(33 63 41 55)(35 61 43 53)(37 59 45 51)(39 57 47 49)(50 73 58 65)(52 71 60 79)(54 69 62 77)(56 67 64 75)```

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

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

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

Matrix representation of D5.Q32 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 240 0 1 0 0 0 240 0 0 1 0 0 240 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 240 0 0 1 0 0 240 0 1 0 0 0 240 1 0 0
,
 85 214 0 0 0 0 27 85 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
,
 138 41 0 0 0 0 41 103 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0],[85,27,0,0,0,0,214,85,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[138,41,0,0,0,0,41,103,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

D5.Q32 in GAP, Magma, Sage, TeX

`D_5.Q_{32}`
`% in TeX`

`G:=Group("D5.Q32");`
`// GroupNames label`

`G:=SmallGroup(320,246);`
`// by ID`

`G=gap.SmallGroup(320,246);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,675,346,192,1684,851,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^2=c^16=1,d^2=c^8,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^-1*b*c^-1>;`
`// generators/relations`

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