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## G = C2×D5≀C2order 400 = 24·52

### Direct product of C2 and D5≀C2

Aliases: C2×D5≀C2, C5⋊D5⋊D4, (C5×C10)⋊D4, D52⋊C22, C52⋊(C2×D4), C5⋊D5.3C23, C52⋊C42C22, (C2×D52)⋊5C2, (C2×C52⋊C4)⋊5C2, (C2×C5⋊D5).10C22, SmallGroup(400,211)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5⋊D5 — C2×D5≀C2
 Chief series C1 — C52 — C5⋊D5 — D52 — D5≀C2 — C2×D5≀C2
 Lower central C52 — C5⋊D5 — C2×D5≀C2
 Upper central C1 — C2

Generators and relations for C2×D5≀C2
G = < a,b,c,d,e | a2=b5=c5=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ece=b2, ebe=dcd-1=c3, ede=d-1 >

Subgroups: 926 in 94 conjugacy classes, 21 normal (9 characteristic)
C1, C2, C2 [×6], C4 [×2], C22 [×9], C5 [×3], C2×C4, D4 [×4], C23 [×2], D5 [×10], C10 [×7], C2×D4, F5 [×2], D10 [×13], C2×C10 [×2], C52, C2×F5, C22×D5 [×2], C5×D5 [×4], C5⋊D5 [×2], C5×C10, C52⋊C4 [×2], D52 [×4], D52 [×2], D5×C10 [×2], C2×C5⋊D5, D5≀C2 [×4], C2×C52⋊C4, C2×D52 [×2], C2×D5≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D5≀C2, C2×D5≀C2

Character table of C2×D5≀C2

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 5E 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M size 1 1 10 10 10 10 25 25 50 50 4 4 4 4 8 4 4 4 4 8 20 20 20 20 20 20 20 20 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 -2 0 0 0 0 -2 2 0 0 2 2 2 2 2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 0 0 -2 -2 0 0 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 2 2 0 0 0 0 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 -1+√5 3-√5/2 -1-√5 3+√5/2 -1 -1+√5/2 0 0 -1-√5/2 0 0 -1+√5/2 -1-√5/2 orthogonal lifted from D5≀C2 ρ12 4 4 -2 -2 0 0 0 0 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 -1-√5 3+√5/2 -1+√5 3-√5/2 -1 1+√5/2 0 0 1-√5/2 0 0 1+√5/2 1-√5/2 orthogonal lifted from D5≀C2 ρ13 4 4 2 2 0 0 0 0 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 -1-√5 3+√5/2 -1+√5 3-√5/2 -1 -1-√5/2 0 0 -1+√5/2 0 0 -1-√5/2 -1+√5/2 orthogonal lifted from D5≀C2 ρ14 4 -4 0 0 -2 2 0 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 -3-√5/2 1-√5 -3+√5/2 1+√5 1 0 1+√5/2 1-√5/2 0 -1-√5/2 -1+√5/2 0 0 orthogonal faithful ρ15 4 -4 -2 2 0 0 0 0 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 1-√5 -3+√5/2 1+√5 -3-√5/2 1 -1+√5/2 0 0 1+√5/2 0 0 1-√5/2 -1-√5/2 orthogonal faithful ρ16 4 4 0 0 2 2 0 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 3+√5/2 -1+√5 3-√5/2 -1-√5 -1 0 -1-√5/2 -1+√5/2 0 -1-√5/2 -1+√5/2 0 0 orthogonal lifted from D5≀C2 ρ17 4 4 0 0 2 2 0 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 3-√5/2 -1-√5 3+√5/2 -1+√5 -1 0 -1+√5/2 -1-√5/2 0 -1+√5/2 -1-√5/2 0 0 orthogonal lifted from D5≀C2 ρ18 4 -4 0 0 2 -2 0 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 -3+√5/2 1+√5 -3-√5/2 1-√5 1 0 -1+√5/2 -1-√5/2 0 1-√5/2 1+√5/2 0 0 orthogonal faithful ρ19 4 -4 0 0 2 -2 0 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 -3-√5/2 1-√5 -3+√5/2 1+√5 1 0 -1-√5/2 -1+√5/2 0 1+√5/2 1-√5/2 0 0 orthogonal faithful ρ20 4 4 0 0 -2 -2 0 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 3-√5/2 -1-√5 3+√5/2 -1+√5 -1 0 1-√5/2 1+√5/2 0 1-√5/2 1+√5/2 0 0 orthogonal lifted from D5≀C2 ρ21 4 -4 -2 2 0 0 0 0 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 1+√5 -3-√5/2 1-√5 -3+√5/2 1 -1-√5/2 0 0 1-√5/2 0 0 1+√5/2 -1+√5/2 orthogonal faithful ρ22 4 4 0 0 -2 -2 0 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 3+√5/2 -1+√5 3-√5/2 -1-√5 -1 0 1+√5/2 1-√5/2 0 1+√5/2 1-√5/2 0 0 orthogonal lifted from D5≀C2 ρ23 4 -4 0 0 -2 2 0 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 -3+√5/2 1+√5 -3-√5/2 1-√5 1 0 1-√5/2 1+√5/2 0 -1+√5/2 -1-√5/2 0 0 orthogonal faithful ρ24 4 -4 2 -2 0 0 0 0 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 1-√5 -3+√5/2 1+√5 -3-√5/2 1 1-√5/2 0 0 -1-√5/2 0 0 -1+√5/2 1+√5/2 orthogonal faithful ρ25 4 -4 2 -2 0 0 0 0 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 1+√5 -3-√5/2 1-√5 -3+√5/2 1 1+√5/2 0 0 -1+√5/2 0 0 -1-√5/2 1-√5/2 orthogonal faithful ρ26 4 4 -2 -2 0 0 0 0 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 -1+√5 3-√5/2 -1-√5 3+√5/2 -1 1-√5/2 0 0 1+√5/2 0 0 1-√5/2 1+√5/2 orthogonal lifted from D5≀C2 ρ27 8 -8 0 0 0 0 0 0 0 0 -2 -2 -2 -2 3 2 2 2 2 -3 0 0 0 0 0 0 0 0 orthogonal faithful ρ28 8 8 0 0 0 0 0 0 0 0 -2 -2 -2 -2 3 -2 -2 -2 -2 3 0 0 0 0 0 0 0 0 orthogonal lifted from D5≀C2

Permutation representations of C2×D5≀C2
On 20 points - transitive group 20T92
Generators in S20
(1 7)(2 8)(3 9)(4 10)(5 6)(11 19)(12 20)(13 16)(14 17)(15 18)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 4 2 5 3)(6 9 7 10 8)(11 13 15 12 14)(16 18 20 17 19)
(1 11 7 19)(2 14 6 16)(3 12 10 18)(4 15 9 20)(5 13 8 17)
(2 5)(3 4)(6 8)(9 10)(11 19)(12 20)(13 16)(14 17)(15 18)

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

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

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

G:=TransitiveGroup(20,92);

On 20 points - transitive group 20T98
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(11 12 13 14 15)(16 17 18 19 20)
(1 2 3 4 5)(6 7 8 9 10)
(1 6)(2 8 5 9)(3 10 4 7)(11 16)(12 19 15 18)(13 17 14 20)
(1 16)(2 18)(3 20)(4 17)(5 19)(6 11)(7 13)(8 15)(9 12)(10 14)

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

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

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

G:=TransitiveGroup(20,98);

On 20 points - transitive group 20T100
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 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 12)(2 15 5 14)(3 13 4 11)(6 17)(7 20 10 19)(8 18 9 16)
(1 12)(2 14)(3 11)(4 13)(5 15)(6 17)(7 19)(8 16)(9 18)(10 20)

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

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

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

G:=TransitiveGroup(20,100);

Matrix representation of C2×D5≀C2 in GL6(𝔽41)

 40 0 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 1 6 0 0 0 0 0 0 35 6 0 0 0 0 35 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 1 6 0 0 0 0 0 0 40 35 0 0 0 0 6 35
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 6 35 0 0 0 0 40 35 0 0
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

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

C2×D5≀C2 in GAP, Magma, Sage, TeX

C_2\times D_5\wr C_2
% in TeX

G:=Group("C2xD5wrC2");
// GroupNames label

G:=SmallGroup(400,211);
// by ID

G=gap.SmallGroup(400,211);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,5,121,7204,1210,262,1157,299,1463]);
// Polycyclic

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

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