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## G = D5×2+ 1+4order 320 = 26·5

### Direct product of D5 and 2+ 1+4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5×2+ 1+4
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — C2×D4×D5 — D5×2+ 1+4
 Lower central C5 — C10 — D5×2+ 1+4
 Upper central C1 — C2 — 2+ 1+4

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

Subgroups: 3118 in 898 conjugacy classes, 445 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, D5, C10, C10, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, Dic5, C20, D10, D10, D10, C2×C10, C2×C10, C22×D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C22×C10, C2×2+ 1+4, C2×C4×D5, C2×D20, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C2×C5⋊D4, D4×C10, C5×C4○D4, C23×D5, C2×D4×D5, D46D10, D5×C4○D4, D48D10, C5×2+ 1+4, D5×2+ 1+4
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C25, C22×D5, C2×2+ 1+4, C23×D5, D5×C24, 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 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 24 9 29)(2 25 10 30)(3 21 6 26)(4 22 7 27)(5 23 8 28)(11 31 16 36)(12 32 17 37)(13 33 18 38)(14 34 19 39)(15 35 20 40)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40), (1,24,9,29)(2,25,10,30)(3,21,6,26)(4,22,7,27)(5,23,8,28)(11,31,16,36)(12,32,17,37)(13,33,18,38)(14,34,19,39)(15,35,20,40), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)>;

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

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

68 conjugacy classes

 class 1 2A 2B ··· 2J 2K 2L 2M ··· 2U 4A ··· 4F 4G ··· 4L 5A 5B 10A 10B 10C ··· 10T 20A ··· 20L order 1 2 2 ··· 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 10 10 ··· 10 20 ··· 20 size 1 1 2 ··· 2 5 5 10 ··· 10 2 ··· 2 10 ··· 10 2 2 2 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D5 D10 D10 2+ 1+4 D5×2+ 1+4 kernel D5×2+ 1+4 C2×D4×D5 D4⋊6D10 D5×C4○D4 D4⋊8D10 C5×2+ 1+4 2+ 1+4 C2×D4 C4○D4 D5 C1 # reps 1 9 9 6 6 1 2 18 12 2 2

Matrix representation of D5×2+ 1+4 in GL6(𝔽41)

 40 1 0 0 0 0 5 35 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
,
 1 0 0 0 0 0 36 40 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
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 1 1 1 2 0 0 40 0 40 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 40 0 40 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 1 1 1 2 0 0 1 0 0 0 0 0 40 40 0 40
,
 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 0 0 0 40 0 0 0 40 40 0 40

G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,1,40,0,0,40,0,1,0,0,0,0,0,1,40,0,0,0,0,2,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,40,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,1,40,0,0,0,1,0,40,0,0,40,1,0,0,0,0,0,2,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,40,0,0,0,0,0,0,40] >;

D5×2+ 1+4 in GAP, Magma, Sage, TeX

D_5\times 2_+^{1+4}
% in TeX

G:=Group("D5xES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1622);
// by ID

G=gap.SmallGroup(320,1622);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,297,851,12550]);
// Polycyclic

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

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