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## G = D5×A4order 120 = 23·3·5

### Direct product of D5 and A4

Aliases: D5×A4, C5⋊(C2×A4), (C2×C10)⋊C6, (C22×D5)⋊C3, C22⋊(C3×D5), (C5×A4)⋊2C2, SmallGroup(120,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D5×A4
 Chief series C1 — C5 — C2×C10 — C5×A4 — D5×A4
 Lower central C2×C10 — D5×A4
 Upper central C1

Generators and relations for D5×A4
G = < a,b,c,d,e | a5=b2=c2=d2=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

3C2
5C2
15C2
4C3
15C22
15C22
20C6
3C10
3D5
4C15
5C23
3D10
3D10

Character table of D5×A4

 class 1 2A 2B 2C 3A 3B 5A 5B 6A 6B 10A 10B 15A 15B 15C 15D size 1 3 5 15 4 4 2 2 20 20 6 6 8 8 8 8 ρ1 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 linear of order 2 ρ3 1 1 -1 -1 ζ3 ζ32 1 1 ζ6 ζ65 1 1 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ4 1 1 -1 -1 ζ32 ζ3 1 1 ζ65 ζ6 1 1 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ5 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ7 2 2 0 0 2 2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ8 2 2 0 0 2 2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 2 0 0 -1-√-3 -1+√-3 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 complex lifted from C3×D5 ρ10 2 2 0 0 -1+√-3 -1-√-3 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 complex lifted from C3×D5 ρ11 2 2 0 0 -1+√-3 -1-√-3 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 complex lifted from C3×D5 ρ12 2 2 0 0 -1-√-3 -1+√-3 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 complex lifted from C3×D5 ρ13 3 -1 -3 1 0 0 3 3 0 0 -1 -1 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 -1 3 -1 0 0 3 3 0 0 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ15 6 -2 0 0 0 0 -3-3√5/2 -3+3√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 orthogonal faithful ρ16 6 -2 0 0 0 0 -3+3√5/2 -3-3√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 orthogonal faithful

Permutation representations of D5×A4
On 20 points - transitive group 20T37
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)
(6 11 16)(7 12 17)(8 13 18)(9 14 19)(10 15 20)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20), (6,11,16)(7,12,17)(8,13,18)(9,14,19)(10,15,20)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20), (6,11,16)(7,12,17)(8,13,18)(9,14,19)(10,15,20) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20)], [(6,11,16),(7,12,17),(8,13,18),(9,14,19),(10,15,20)]])

G:=TransitiveGroup(20,37);

On 30 points - transitive group 30T20
Generators in S30
(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)
(1 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)
(1 9)(2 10)(3 6)(4 7)(5 8)(21 26)(22 27)(23 28)(24 29)(25 30)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)

G:=sub<Sym(30)| (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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27), (1,9)(2,10)(3,6)(4,7)(5,8)(21,26)(22,27)(23,28)(24,29)(25,30), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)>;

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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27), (1,9)(2,10)(3,6)(4,7)(5,8)(21,26)(22,27)(23,28)(24,29)(25,30), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20) );

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)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27)], [(1,9),(2,10),(3,6),(4,7),(5,8),(21,26),(22,27),(23,28),(24,29),(25,30)], [(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,24,14),(2,25,15),(3,21,11),(4,22,12),(5,23,13),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20)]])

G:=TransitiveGroup(30,20);

On 30 points - transitive group 30T28
Generators in S30
(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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)
(1 9)(2 10)(3 6)(4 7)(5 8)(21 26)(22 27)(23 28)(24 29)(25 30)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)

G:=sub<Sym(30)| (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29), (1,9)(2,10)(3,6)(4,7)(5,8)(21,26)(22,27)(23,28)(24,29)(25,30), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)>;

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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29), (1,9)(2,10)(3,6)(4,7)(5,8)(21,26)(22,27)(23,28)(24,29)(25,30), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20) );

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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29)], [(1,9),(2,10),(3,6),(4,7),(5,8),(21,26),(22,27),(23,28),(24,29),(25,30)], [(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,24,14),(2,25,15),(3,21,11),(4,22,12),(5,23,13),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20)]])

G:=TransitiveGroup(30,28);

D5×A4 is a maximal subgroup of   A4⋊F5  C204D4⋊C3  (C4×C20)⋊C6  (C22×D5)⋊A4
D5×A4 is a maximal quotient of   Dic5.A4  C204D4⋊C3  (C4×C20)⋊C6  (C22×D5)⋊A4

Matrix representation of D5×A4 in GL5(𝔽31)

 0 1 0 0 0 30 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 18 30 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 30 30 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 30 30 30 0 0 1 0 0
,
 5 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 30 30 30 0 0 0 1 0

G:=sub<GL(5,GF(31))| [0,30,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,18,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,30,0,0,1,0,30,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,0,30,1,0,0,0,30,0,0,0,1,30,0],[5,0,0,0,0,0,5,0,0,0,0,0,1,30,0,0,0,0,30,1,0,0,0,30,0] >;

D5×A4 in GAP, Magma, Sage, TeX

D_5\times A_4
% in TeX

G:=Group("D5xA4");
// GroupNames label

G:=SmallGroup(120,39);
// by ID

G=gap.SmallGroup(120,39);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-5,142,68,2404]);
// Polycyclic

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

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