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G = C2×C52⋊C4  order 200 = 23·52

Direct product of C2 and C52⋊C4

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×C52⋊C4, C10⋊2F5, C5⋊D5⋊4C4, C5⋊3(C2×F5), (C5×C10)⋊4C4, C52⋊6(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 >

C1
C2
25C2
25C2
C5
C5
2C5
2C5
25C22
25C4
25C4
C10
C10
2C10
2C10
5D5
5D5
5D5
5D5
10D5
10D5
10D5
10D5
C52
25C2×C4
5D10
5D10
5F5
5F5
5F5
5F5
10D10
10D10
C5×C10
C5⋊D5
C5⋊D5
5C2×F5
5C2×F5
C2×C5⋊D5
C52⋊C4
C52⋊C4
C2×C52⋊C4

Character table of C2×C52⋊C4

 class 12A2B2C4A4B4C4D5A5B5C5D5E5F10A10B10C10D10E10F
 size 11252525252525444444444444
ρ111111111111111111111    trivial
ρ21-11-11-11-1111111-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-1-1111111111111    linear of order 2
ρ41-11-1-11-11111111-1-1-1-1-1-1    linear of order 2
ρ51-1-11i-i-ii111111-1-1-1-1-1-1    linear of order 4
ρ611-1-1ii-i-i111111111111    linear of order 4
ρ71-1-11-iii-i111111-1-1-1-1-1-1    linear of order 4
ρ811-1-1-i-iii111111111111    linear of order 4
ρ9440000004-1-1-1-1-1-1-14-1-1-1    orthogonal lifted from F5
ρ104-40000004-1-1-1-1-111-4111    orthogonal lifted from C2×F5
ρ114-4000000-1-1-1-1-141-41111    orthogonal lifted from C2×F5
ρ1244000000-1-1-1-1-14-14-1-1-1-1    orthogonal lifted from F5
ρ134-4000000-1-1-√53+√5/23-√5/2-1+√5-11-√5111+√5-3-√5/2-3+√5/2    orthogonal faithful
ρ1444000000-13-√5/2-1-√5-1+√53+√5/2-13+√5/2-1-13-√5/2-1-√5-1+√5    orthogonal lifted from C52⋊C4
ρ1544000000-1-1-√53+√5/23-√5/2-1+√5-1-1+√5-1-1-1-√53+√5/23-√5/2    orthogonal lifted from C52⋊C4
ρ164-4000000-13+√5/2-1+√5-1-√53-√5/2-1-3+√5/211-3-√5/21-√51+√5    orthogonal faithful
ρ1744000000-13+√5/2-1+√5-1-√53-√5/2-13-√5/2-1-13+√5/2-1+√5-1-√5    orthogonal lifted from C52⋊C4
ρ184-4000000-1-1+√53-√5/23+√5/2-1-√5-11+√5111-√5-3+√5/2-3-√5/2    orthogonal faithful
ρ194-4000000-13-√5/2-1-√5-1+√53+√5/2-1-3-√5/211-3+√5/21+√51-√5    orthogonal faithful
ρ2044000000-1-1+√53-√5/23+√5/2-1-√5-1-1-√5-1-1-1+√53-√5/23+√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
 C52⋊3C42  D10⋊F5  Dic5⋊F5  D52⋊C4  C2.D5≀C2  (C5×C10).Q8  C20⋊2F5  C102⋊4C4
C2×C52⋊C4 is a maximal quotient of
 C20.11F5  C52⋊8M4(2)  C20⋊2F5  C52⋊14M4(2)  C102⋊4C4

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

40000
04000
00400
00040
,
0100
403400
0077
003440
,
403400
7700
003440
0010
,
00400
00040
40000
7100
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
 

Export

Subgroup lattice of C2×C52⋊C4 in TeX
Character table of C2×C52⋊C4 in TeX

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