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G = C2.D5≀C2order 400 = 24·52

2nd central extension by C2 of D5≀C2

non-abelian, soluble, monomial

Aliases: C2.2D5≀C2, C5⋊D5.2Q8, C523(C4⋊C4), C52⋊C43C4, (C5×C10).2D4, Dic52D5.2C2, C5⋊D5.6(C2×C4), (C2×C52⋊C4).3C2, (C2×C5⋊D5).7C22, SmallGroup(400,130)

Series: Derived Chief Lower central Upper central

C1C52C5⋊D5 — C2.D5≀C2
C1C52C5⋊D5C2×C5⋊D5Dic52D5 — C2.D5≀C2
C52C5⋊D5 — C2.D5≀C2
C1C2

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

25C2
25C2
2C5
2C5
2C5
10C4
10C4
25C4
25C22
25C4
2C10
2C10
2C10
10D5
10D5
10D5
10D5
10D5
10D5
25C2×C4
25C2×C4
25C2×C4
2Dic5
2Dic5
10F5
10C20
10F5
10C20
10D10
10D10
10D10
25C4⋊C4
10C2×F5
10C4×D5
10C4×D5
2C5×Dic5
2C5×Dic5

Character table of C2.D5≀C2

 class 12A2B2C4A4B4C4D4E4F5A5B5C5D5E10A10B10C10D10E20A20B20C20D20E20F20G20H
 size 11252510101010505044448444482020202020202020
ρ11111111111111111111111111111    trivial
ρ21111-1-111-1-11111111111-111-111-1-1    linear of order 2
ρ3111111-1-1-1-111111111111-1-11-1-111    linear of order 2
ρ41111-1-1-1-1111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-11-1i-i-ii-1111111-1-1-1-1-1-i-i-iiiii-i    linear of order 4
ρ61-11-1-ii-ii1-111111-1-1-1-1-1i-i-i-iii-ii    linear of order 4
ρ71-11-1i-ii-i1-111111-1-1-1-1-1-iiii-i-ii-i    linear of order 4
ρ81-11-1-iii-i-1111111-1-1-1-1-1iii-i-i-i-ii    linear of order 4
ρ922-2-2000000222222222200000000    orthogonal lifted from D4
ρ102-2-2200000022222-2-2-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ1144000022003+5/23-5/2-1+5-1-5-1-1-53-5/2-1+53+5/2-10-1-5/2-1+5/20-1-5/2-1+5/200    orthogonal lifted from D5≀C2
ρ12440000-2-2003+5/23-5/2-1+5-1-5-1-1-53-5/2-1+53+5/2-101+5/21-5/201+5/21-5/200    orthogonal lifted from D5≀C2
ρ134400-2-20000-1-5-1+53+5/23-5/2-13-5/2-1+53+5/2-1-5-11+5/2001+5/2001-5/21-5/2    orthogonal lifted from D5≀C2
ρ14440000-2-2003-5/23+5/2-1-5-1+5-1-1+53+5/2-1-53-5/2-101-5/21+5/201-5/21+5/200    orthogonal lifted from D5≀C2
ρ154400-2-20000-1+5-1-53-5/23+5/2-13+5/2-1-53-5/2-1+5-11-5/2001-5/2001+5/21+5/2    orthogonal lifted from D5≀C2
ρ1644000022003-5/23+5/2-1-5-1+5-1-1+53+5/2-1-53-5/2-10-1+5/2-1-5/20-1+5/2-1-5/200    orthogonal lifted from D5≀C2
ρ174400220000-1+5-1-53-5/23+5/2-13+5/2-1-53-5/2-1+5-1-1+5/200-1+5/200-1-5/2-1-5/2    orthogonal lifted from D5≀C2
ρ184400220000-1-5-1+53+5/23-5/2-13-5/2-1+53+5/2-1-5-1-1-5/200-1-5/200-1+5/2-1+5/2    orthogonal lifted from D5≀C2
ρ194-400002i-2i003-5/23+5/2-1-5-1+5-11-5-3-5/21+5-3+5/210ζ4ζ544ζ5ζ4ζ534ζ520ζ43ζ5443ζ5ζ43ζ5343ζ5200    complex faithful
ρ204-400-2i2i0000-1+5-1-53-5/23+5/2-1-3-5/21+5-3+5/21-51ζ4ζ544ζ500ζ43ζ5443ζ500ζ43ζ5343ζ52ζ4ζ534ζ52    complex faithful
ρ214-40000-2i2i003-5/23+5/2-1-5-1+5-11-5-3-5/21+5-3+5/210ζ43ζ5443ζ5ζ43ζ5343ζ520ζ4ζ544ζ5ζ4ζ534ζ5200    complex faithful
ρ224-4002i-2i0000-1-5-1+53+5/23-5/2-1-3+5/21-5-3-5/21+51ζ43ζ5343ζ5200ζ4ζ534ζ5200ζ4ζ544ζ5ζ43ζ5443ζ5    complex faithful
ρ234-400002i-2i003+5/23-5/2-1+5-1-5-11+5-3+5/21-5-3-5/210ζ4ζ534ζ52ζ4ζ544ζ50ζ43ζ5343ζ52ζ43ζ5443ζ500    complex faithful
ρ244-400-2i2i0000-1-5-1+53+5/23-5/2-1-3+5/21-5-3-5/21+51ζ4ζ534ζ5200ζ43ζ5343ζ5200ζ43ζ5443ζ5ζ4ζ544ζ5    complex faithful
ρ254-40000-2i2i003+5/23-5/2-1+5-1-5-11+5-3+5/21-5-3-5/210ζ43ζ5343ζ52ζ43ζ5443ζ50ζ4ζ534ζ52ζ4ζ544ζ500    complex faithful
ρ264-4002i-2i0000-1+5-1-53-5/23+5/2-1-3-5/21+5-3+5/21-51ζ43ζ5443ζ500ζ4ζ544ζ500ζ4ζ534ζ52ζ43ζ5343ζ52    complex faithful
ρ278-800000000-2-2-2-232222-300000000    orthogonal faithful
ρ288800000000-2-2-2-23-2-2-2-2300000000    orthogonal lifted from D5≀C2

Permutation representations of C2.D5≀C2
On 20 points - transitive group 20T105
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(11 18)(12 19)(13 20)(14 16)(15 17)
(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 8 10 7 9)
(1 6)(2 8 5 9)(3 10 4 7)(11 18)(12 16 15 20)(13 19 14 17)
(1 18 6 11)(2 19 7 12)(3 20 8 13)(4 16 9 14)(5 17 10 15)

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

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

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

G:=TransitiveGroup(20,105);

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

4000000
0400000
001000
000100
000010
000001
,
100000
010000
0073400
0074000
00004036
00004035
,
100000
010000
0040700
0034700
00004036
00004035
,
0400000
100000
000016
000001
0035100
001000
,
090000
900000
000001
000016
0035100
001000

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,34,40,0,0,0,0,0,0,40,40,0,0,0,0,36,35],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,40,0,0,0,0,36,35],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,6,1,0,0],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,0,35,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,6,0,0] >;

C2.D5≀C2 in GAP, Magma, Sage, TeX

C_2.D_5\wr C_2
% in TeX

G:=Group("C2.D5wrC2");
// GroupNames label

G:=SmallGroup(400,130);
// by ID

G=gap.SmallGroup(400,130);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,5,48,73,55,7204,1210,496,1157,299,2897]);
// Polycyclic

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

Export

Subgroup lattice of C2.D5≀C2 in TeX
Character table of C2.D5≀C2 in TeX

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