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G = D5≀C2⋊C2  order 400 = 24·52

The semidirect product of D5≀C2 and C2 acting faithfully

non-abelian, soluble, monomial

Aliases: D5≀C2⋊C2, D5⋊F5⋊C2, C52⋊Q8⋊C2, C52⋊(C4○D4), C5⋊D5.C23, D52.2C22, C52⋊C4.C22, C5⋊F5.2C22, SmallGroup(400,207)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C5⋊D5 — D5≀C2⋊C2
Chief series
C1 — C52 — C5⋊D5 — C5⋊F5 — D5⋊F5 — D5≀C2⋊C2
Lower central
C52 — C5⋊D5 — D5≀C2⋊C2
Upper central
C1

Generators and relations for D5≀C2⋊C2
 G = < a,b,c,d,e | a5=b5=c4=d2=e2=1, ab=ba, cac-1=dbd=a2, dad=cbc-1=b3, ae=ea, ebe=b-1, dcd=c-1, ce=ec, ede=c2d >

Subgroups: 646 in 64 conjugacy classes, 18 normal (6 characteristic)
C1, C2, C4, C22, C5, C2×C4, D4, Q8, D5, C10, C4○D4, F5, D10, C52, C2×F5, C5×D5, C5⋊D5, C5⋊F5, C52⋊C4, D52, D5⋊F5, D5≀C2, C52⋊Q8, D5≀C2⋊C2
Quotients: C1, C2, C22, C23, C4○D4, D5≀C2⋊C2

Character table of D5≀C2⋊C2

 class 12A2B2C2D4A4B4C4D4E5A5B5C10A10B10C
 size 1101010252525505050888404040
ρ11111111111111111    trivial
ρ21-1111-1-11-1-1111-111    linear of order 2
ρ31-1-11111-11-1111-1-11    linear of order 2
ρ411-111-1-1-1-111111-11    linear of order 2
ρ51-1-1-11-1-1111111-1-1-1    linear of order 2
ρ611-1-11111-1-11111-1-1    linear of order 2
ρ7111-11-1-1-11-111111-1    linear of order 2
ρ81-11-1111-1-11111-11-1    linear of order 2
ρ92000-22i-2i000222000    complex lifted from C4○D4
ρ102000-2-2i2i000222000    complex lifted from C4○D4
ρ118040000000-2-230-10    orthogonal faithful
ρ12800-4000000-23-2001    orthogonal faithful
ρ138004000000-23-200-1    orthogonal faithful
ρ1480-40000000-2-23010    orthogonal faithful
ρ158-4000000003-2-2100    orthogonal faithful
ρ1684000000003-2-2-100    orthogonal faithful

Permutation representations of D5≀C2⋊C2
►On 10 points - transitive group 10T27
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)

(1 2 3 4 5)(6 10 9 8 7)

(1 7)(2 10 5 9)(3 8 4 6)

(1 7)(2 9)(3 6)(4 8)(5 10)

(1 7)(2 8)(3 9)(4 10)(5 6)
 
G:=sub<Sym(10)| (1,2,3,4,5)(6,7,8,9,10), (1,2,3,4,5)(6,10,9,8,7), (1,7)(2,10,5,9)(3,8,4,6), (1,7)(2,9)(3,6)(4,8)(5,10), (1,7)(2,8)(3,9)(4,10)(5,6)>;
 
G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,2,3,4,5)(6,10,9,8,7), (1,7)(2,10,5,9)(3,8,4,6), (1,7)(2,9)(3,6)(4,8)(5,10), (1,7)(2,8)(3,9)(4,10)(5,6) );
 
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,2,3,4,5),(6,10,9,8,7)], [(1,7),(2,10,5,9),(3,8,4,6)], [(1,7),(2,9),(3,6),(4,8),(5,10)], [(1,7),(2,8),(3,9),(4,10),(5,6)]])
 
G:=TransitiveGroup(10,27);
 

►On 20 points - transitive group 20T90
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 2 3 4 5)(6 7 8 9 10)(11 15 14 13 12)(16 20 19 18 17)

(1 17)(2 20 5 19)(3 18 4 16)(6 12)(7 15 10 14)(8 13 9 11)

(1 12)(2 14)(3 11)(4 13)(5 15)(6 17)(7 19)(8 16)(9 18)(10 20)

(1 12)(2 13)(3 14)(4 15)(5 11)(6 17)(7 18)(8 19)(9 20)(10 16)
 
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,2,3,4,5)(6,7,8,9,10)(11,15,14,13,12)(16,20,19,18,17), (1,17)(2,20,5,19)(3,18,4,16)(6,12)(7,15,10,14)(8,13,9,11), (1,12)(2,14)(3,11)(4,13)(5,15)(6,17)(7,19)(8,16)(9,18)(10,20), (1,12)(2,13)(3,14)(4,15)(5,11)(6,17)(7,18)(8,19)(9,20)(10,16)>;
 
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,2,3,4,5)(6,7,8,9,10)(11,15,14,13,12)(16,20,19,18,17), (1,17)(2,20,5,19)(3,18,4,16)(6,12)(7,15,10,14)(8,13,9,11), (1,12)(2,14)(3,11)(4,13)(5,15)(6,17)(7,19)(8,16)(9,18)(10,20), (1,12)(2,13)(3,14)(4,15)(5,11)(6,17)(7,18)(8,19)(9,20)(10,16) );
 
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,15,14,13,12),(16,20,19,18,17)], [(1,17),(2,20,5,19),(3,18,4,16),(6,12),(7,15,10,14),(8,13,9,11)], [(1,12),(2,14),(3,11),(4,13),(5,15),(6,17),(7,19),(8,16),(9,18),(10,20)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,17),(7,18),(8,19),(9,20),(10,16)]])
 
G:=TransitiveGroup(20,90);
 

►On 20 points - transitive group 20T96
Generators in S20
(11 12 13 14 15)(16 17 18 19 20)

(1 10 3 4 5)(2 7 6 9 8)

(1 2)(3 8 4 7)(5 9 10 6)(11 16)(12 19 15 18)(13 17 14 20)

(1 11)(2 16)(3 15)(4 12)(5 14)(6 20)(7 18)(8 19)(9 17)(10 13)

(3 4)(5 10)(6 9)(7 8)
 
G:=sub<Sym(20)| (11,12,13,14,15)(16,17,18,19,20), (1,10,3,4,5)(2,7,6,9,8), (1,2)(3,8,4,7)(5,9,10,6)(11,16)(12,19,15,18)(13,17,14,20), (1,11)(2,16)(3,15)(4,12)(5,14)(6,20)(7,18)(8,19)(9,17)(10,13), (3,4)(5,10)(6,9)(7,8)>;
 
G:=Group( (11,12,13,14,15)(16,17,18,19,20), (1,10,3,4,5)(2,7,6,9,8), (1,2)(3,8,4,7)(5,9,10,6)(11,16)(12,19,15,18)(13,17,14,20), (1,11)(2,16)(3,15)(4,12)(5,14)(6,20)(7,18)(8,19)(9,17)(10,13), (3,4)(5,10)(6,9)(7,8) );
 
G=PermutationGroup([[(11,12,13,14,15),(16,17,18,19,20)], [(1,10,3,4,5),(2,7,6,9,8)], [(1,2),(3,8,4,7),(5,9,10,6),(11,16),(12,19,15,18),(13,17,14,20)], [(1,11),(2,16),(3,15),(4,12),(5,14),(6,20),(7,18),(8,19),(9,17),(10,13)], [(3,4),(5,10),(6,9),(7,8)]])
 
G:=TransitiveGroup(20,96);
 

►On 20 points - transitive group 20T97
Generators in S20
(11 12 13 14 15)(16 17 18 19 20)

(1 2 3 4 10)(5 7 6 8 9)

(1 5)(2 6 10 8)(3 9 4 7)(11 16)(12 19 15 18)(13 17 14 20)

(1 11)(2 13)(3 15)(4 12)(5 16)(6 20)(7 18)(8 17)(9 19)(10 14)

(1 5)(2 9)(3 8)(4 6)(7 10)(11 16)(12 17)(13 18)(14 19)(15 20)
 
G:=sub<Sym(20)| (11,12,13,14,15)(16,17,18,19,20), (1,2,3,4,10)(5,7,6,8,9), (1,5)(2,6,10,8)(3,9,4,7)(11,16)(12,19,15,18)(13,17,14,20), (1,11)(2,13)(3,15)(4,12)(5,16)(6,20)(7,18)(8,17)(9,19)(10,14), (1,5)(2,9)(3,8)(4,6)(7,10)(11,16)(12,17)(13,18)(14,19)(15,20)>;
 
G:=Group( (11,12,13,14,15)(16,17,18,19,20), (1,2,3,4,10)(5,7,6,8,9), (1,5)(2,6,10,8)(3,9,4,7)(11,16)(12,19,15,18)(13,17,14,20), (1,11)(2,13)(3,15)(4,12)(5,16)(6,20)(7,18)(8,17)(9,19)(10,14), (1,5)(2,9)(3,8)(4,6)(7,10)(11,16)(12,17)(13,18)(14,19)(15,20) );
 
G=PermutationGroup([[(11,12,13,14,15),(16,17,18,19,20)], [(1,2,3,4,10),(5,7,6,8,9)], [(1,5),(2,6,10,8),(3,9,4,7),(11,16),(12,19,15,18),(13,17,14,20)], [(1,11),(2,13),(3,15),(4,12),(5,16),(6,20),(7,18),(8,17),(9,19),(10,14)], [(1,5),(2,9),(3,8),(4,6),(7,10),(11,16),(12,17),(13,18),(14,19),(15,20)]])
 
G:=TransitiveGroup(20,97);
 

►On 25 points: primitive - transitive group 25T30
Generators in S25
(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)

(1 12 17 22 7)(2 13 18 23 8)(3 14 19 24 9)(4 15 20 25 10)(5 11 16 21 6)

(2 4 5 3)(6 24 13 20)(7 22 12 17)(8 25 11 19)(9 23 15 16)(10 21 14 18)

(2 22)(3 12)(4 7)(5 17)(6 20)(8 25)(9 15)(11 19)(13 24)(18 21)

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

Polynomial with Galois group D5≀C2⋊C2 over ℚ
actionf(x)Disc(f)
10T27x10-20x8+140x6-16x5-400x4+160x3+400x2-320x+62219·510·74·172

Matrix representation of D5≀C2⋊C2 ►in GL8(ℤ)

00100000
00010000
-1-1-1-10000
10000000
0000-1-1-1-1
00001000
00000100
00000010
,
-1-1-1-10000
10000000
01000000
00100000
00000010
00000001
0000-1-1-1-1
00001000
,
00001000
0000-1-1-1-1
00000001
00000010
10000000
01000000
00100000
00010000
,
10000000
01000000
00100000
00010000
00001000
0000-1-1-1-1
00000001
00000010
,
00001000
00000010
0000-1-1-1-1
00000100
10000000
00010000
01000000
-1-1-1-10000

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

D5≀C2⋊C2 in GAP, Magma, Sage, TeX 

D_5\wr C_2\rtimes C_2
 
% in TeX
 
G:=Group("D5wrC2:C2");
 
// GroupNames label
 
G:=SmallGroup(400,207);
 
// by ID
 
G=gap.SmallGroup(400,207);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,5,48,116,964,3610,616,262,5765,587,161,1463]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^5=c^4=d^2=e^2=1,a*b=b*a,c*a*c^-1=d*b*d=a^2,d*a*d=c*b*c^-1=b^3,a*e=e*a,e*b*e=b^-1,d*c*d=c^-1,c*e=e*c,e*d*e=c^2*d>;
 
// generators/relations
 

Export

Character table of D5≀C2⋊C2 in TeX

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