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## G = C2×C52⋊C4order 200 = 23·52

### Direct product of C2 and C52⋊C4

Aliases: C2×C52⋊C4, C102F5, C5⋊D54C4, C53(C2×F5), (C5×C10)⋊4C4, C526(C2×C4), C5⋊D5.5C22, (C2×C5⋊D5).3C2, SmallGroup(200,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C52⋊C4
 Chief series C1 — C5 — C52 — C5⋊D5 — C52⋊C4 — C2×C52⋊C4
 Lower central C52 — C2×C52⋊C4
 Upper central C1 — C2

Generators and relations for C2×C52⋊C4
G = < a,b,c,d | a2=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2, dcd-1=c3 >

25C2
25C2
2C5
2C5
25C22
25C4
25C4
2C10
2C10
5D5
5D5
5D5
5D5
10D5
10D5
10D5
10D5
25C2×C4
5D10
5D10
5F5
5F5
5F5
5F5
10D10
10D10

Character table of C2×C52⋊C4

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E 5F 10A 10B 10C 10D 10E 10F size 1 1 25 25 25 25 25 25 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 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 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 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 linear of order 2 ρ5 1 -1 -1 1 i -i -i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 -1 i i -i -i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 -1 -1 1 -i i i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ8 1 1 -1 -1 -i -i i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 4 4 0 0 0 0 0 0 4 -1 -1 -1 -1 -1 -1 -1 4 -1 -1 -1 orthogonal lifted from F5 ρ10 4 -4 0 0 0 0 0 0 4 -1 -1 -1 -1 -1 1 1 -4 1 1 1 orthogonal lifted from C2×F5 ρ11 4 -4 0 0 0 0 0 0 -1 -1 -1 -1 -1 4 1 -4 1 1 1 1 orthogonal lifted from C2×F5 ρ12 4 4 0 0 0 0 0 0 -1 -1 -1 -1 -1 4 -1 4 -1 -1 -1 -1 orthogonal lifted from F5 ρ13 4 -4 0 0 0 0 0 0 -1 -1-√5 3+√5/2 3-√5/2 -1+√5 -1 1-√5 1 1 1+√5 -3-√5/2 -3+√5/2 orthogonal faithful ρ14 4 4 0 0 0 0 0 0 -1 3-√5/2 -1-√5 -1+√5 3+√5/2 -1 3+√5/2 -1 -1 3-√5/2 -1-√5 -1+√5 orthogonal lifted from C52⋊C4 ρ15 4 4 0 0 0 0 0 0 -1 -1-√5 3+√5/2 3-√5/2 -1+√5 -1 -1+√5 -1 -1 -1-√5 3+√5/2 3-√5/2 orthogonal lifted from C52⋊C4 ρ16 4 -4 0 0 0 0 0 0 -1 3+√5/2 -1+√5 -1-√5 3-√5/2 -1 -3+√5/2 1 1 -3-√5/2 1-√5 1+√5 orthogonal faithful ρ17 4 4 0 0 0 0 0 0 -1 3+√5/2 -1+√5 -1-√5 3-√5/2 -1 3-√5/2 -1 -1 3+√5/2 -1+√5 -1-√5 orthogonal lifted from C52⋊C4 ρ18 4 -4 0 0 0 0 0 0 -1 -1+√5 3-√5/2 3+√5/2 -1-√5 -1 1+√5 1 1 1-√5 -3+√5/2 -3-√5/2 orthogonal faithful ρ19 4 -4 0 0 0 0 0 0 -1 3-√5/2 -1-√5 -1+√5 3+√5/2 -1 -3-√5/2 1 1 -3+√5/2 1+√5 1-√5 orthogonal faithful ρ20 4 4 0 0 0 0 0 0 -1 -1+√5 3-√5/2 3+√5/2 -1-√5 -1 -1-√5 -1 -1 -1+√5 3-√5/2 3+√5/2 orthogonal lifted from C52⋊C4

Permutation representations of C2×C52⋊C4
On 20 points - transitive group 20T49
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(11 20)(12 16)(13 17)(14 18)(15 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 13 6 17)(2 11 10 19)(3 14 9 16)(4 12 8 18)(5 15 7 20)

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

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

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

G:=TransitiveGroup(20,49);

On 20 points - transitive group 20T52
Generators in S20
(1 8)(2 9)(3 10)(4 6)(5 7)(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 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)
(1 16)(2 19 5 18)(3 17 4 20)(6 15 10 12)(7 13 9 14)(8 11)

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

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

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

G:=TransitiveGroup(20,52);

C2×C52⋊C4 is a maximal subgroup of
C523C42  D10⋊F5  Dic5⋊F5  D52⋊C4  C2.D5≀C2  (C5×C10).Q8  C202F5  C1024C4
C2×C52⋊C4 is a maximal quotient of
C20.11F5  C528M4(2)  C202F5  C5214M4(2)  C1024C4

Matrix representation of C2×C52⋊C4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 0 1 0 0 40 34 0 0 0 0 7 7 0 0 34 40
,
 40 34 0 0 7 7 0 0 0 0 34 40 0 0 1 0
,
 0 0 40 0 0 0 0 40 40 0 0 0 7 1 0 0
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,34,0,0,0,0,7,34,0,0,7,40],[40,7,0,0,34,7,0,0,0,0,34,1,0,0,40,0],[0,0,40,7,0,0,0,1,40,0,0,0,0,40,0,0] >;

C2×C52⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes C_4
% in TeX

G:=Group("C2xC5^2:C4");
// GroupNames label

G:=SmallGroup(200,48);
// by ID

G=gap.SmallGroup(200,48);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,20,483,173,2004,1014]);
// Polycyclic

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

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