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## G = D5.2+ 1+4order 320 = 26·5

### The non-split extension by D5 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5.2+ 1+4
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — D4×F5 — D5.2+ 1+4
 Lower central C5 — C10 — D5.2+ 1+4
 Upper central C1 — C2 — C4○D4

Generators and relations for D5.2+ 1+4
G = < a,b,c,d,e,f | a5=b2=c4=d2=1, e2=c2, f2=a-1b, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=a3, bc=cb, bd=db, be=eb, fbf-1=a2b, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >

Subgroups: 1018 in 294 conjugacy classes, 138 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, F5, D10, D10, D10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C2×F5, C2×F5, C22×D5, C23.33C23, C4×F5, C4⋊F5, C4⋊F5, C22⋊F5, C2×C4×D5, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, C22×F5, C2×C4⋊F5, D10.C23, D4×F5, Q8×F5, D5×C4○D4, D5.2+ 1+4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C23×C4, 2+ 1+4, 2- 1+4, C2×F5, C23.33C23, C22×F5, C23×F5, D5.2+ 1+4

Smallest permutation representation of D5.2+ 1+4
On 40 points
Generators in S40
```(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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)
(1 34 9 39)(2 35 10 40)(3 31 6 36)(4 32 7 37)(5 33 8 38)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(2 3 5 4)(6 8 7 10)(11 18 12 20)(13 17 15 16)(14 19)(21 28 22 30)(23 27 25 26)(24 29)(31 33 32 35)(36 38 37 40)```

`G:=sub<Sym(40)| (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39), (1,34,9,39)(2,35,10,40)(3,31,6,36)(4,32,7,37)(5,33,8,38)(11,26,16,21)(12,27,17,22)(13,28,18,23)(14,29,19,24)(15,30,20,25), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (2,3,5,4)(6,8,7,10)(11,18,12,20)(13,17,15,16)(14,19)(21,28,22,30)(23,27,25,26)(24,29)(31,33,32,35)(36,38,37,40)>;`

`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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39), (1,34,9,39)(2,35,10,40)(3,31,6,36)(4,32,7,37)(5,33,8,38)(11,26,16,21)(12,27,17,22)(13,28,18,23)(14,29,19,24)(15,30,20,25), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (2,3,5,4)(6,8,7,10)(11,18,12,20)(13,17,15,16)(14,19)(21,28,22,30)(23,27,25,26)(24,29)(31,33,32,35)(36,38,37,40) );`

`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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39)], [(1,34,9,39),(2,35,10,40),(3,31,6,36),(4,32,7,37),(5,33,8,38),(11,26,16,21),(12,27,17,22),(13,28,18,23),(14,29,19,24),(15,30,20,25)], [(1,34),(2,35),(3,31),(4,32),(5,33),(6,36),(7,37),(8,38),(9,39),(10,40),(11,26),(12,27),(13,28),(14,29),(15,30),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35)], [(2,3,5,4),(6,8,7,10),(11,18,12,20),(13,17,15,16),(14,19),(21,28,22,30),(23,27,25,26),(24,29),(31,33,32,35),(36,38,37,40)]])`

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 8 type + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 F5 2+ 1+4 2- 1+4 C2×F5 C2×F5 C2×F5 D5.2+ 1+4 kernel D5.2+ 1+4 C2×C4⋊F5 D10.C23 D4×F5 Q8×F5 D5×C4○D4 C4○D20 D4⋊2D5 Q8⋊2D5 C5×C4○D4 C4○D4 D5 D5 C2×C4 D4 Q8 C1 # reps 1 3 3 6 2 1 6 6 2 2 1 1 1 3 3 1 2

Matrix representation of D5.2+ 1+4 in GL8(ℤ)

 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 0 0 0
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1

`G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1] >;`

D5.2+ 1+4 in GAP, Magma, Sage, TeX

`D_5.2_+^{1+4}`
`% in TeX`

`G:=Group("D5.ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1604);`
`// by ID`

`G=gap.SmallGroup(320,1604);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,136,6278,818]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^5=b^2=c^4=d^2=1,e^2=c^2,f^2=a^-1*b,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^3,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^2*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c^2*e>;`
`// generators/relations`

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