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G = D52⋊C4order 400 = 24·52

1st semidirect product of D52 and C4 acting via C4/C2=C2

non-abelian, soluble, monomial

Aliases: D521C4, C2.1D5≀C2, C5⋊D5.3D4, (C5×C10).1D4, C523(C22⋊C4), Dic52D54C2, (C2×D52).3C2, C5⋊D5.5(C2×C4), (C2×C52⋊C4)⋊2C2, (C2×C5⋊D5).6C22, SmallGroup(400,129)

Series: Derived Chief Lower central Upper central

C1C52C5⋊D5 — D52⋊C4
C1C52C5⋊D5C2×C5⋊D5C2×D52 — D52⋊C4
C52C5⋊D5 — D52⋊C4
C1C2

Generators and relations for D52⋊C4
 G = < a,b,c,d,e | a5=b2=c5=d2=e4=1, bab=a-1, ac=ca, ad=da, eae-1=c, bc=cb, bd=db, ebe-1=d, dcd=c-1, ece-1=a, ede-1=b >

Subgroups: 630 in 66 conjugacy classes, 13 normal (11 characteristic)
C1, C2, C2 [×4], C4 [×2], C22 [×5], C5 [×3], C2×C4 [×2], C23, D5 [×8], C10 [×5], C22⋊C4, Dic5, C20, F5 [×2], D10 [×8], C2×C10, C52, C4×D5, C2×F5, C22×D5, C5×D5 [×2], C5⋊D5 [×2], C5×C10, C5×Dic5, C52⋊C4, D52 [×2], D52, D5×C10, C2×C5⋊D5, Dic52D5, C2×C52⋊C4, C2×D52, D52⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D5≀C2, D52⋊C4

Character table of D52⋊C4

 class 12A2B2C2D2E4A4B4C4D5A5B5C5D5E10A10B10C10D10E10F10G10H10I20A20B20C20D
 size 11101025251010505044448444482020202020202020
ρ11111111111111111111111111111    trivial
ρ211-1-11111-1-11111111111-1-1-1-11111    linear of order 2
ρ311-1-111-1-1111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111-1-1-1-111111111111111-1-1-1-1    linear of order 2
ρ51-1-111-1i-i-ii11111-1-1-1-1-111-1-1-iii-i    linear of order 4
ρ61-11-11-1i-ii-i11111-1-1-1-1-1-1-111-iii-i    linear of order 4
ρ71-1-111-1-iii-i11111-1-1-1-1-111-1-1i-i-ii    linear of order 4
ρ81-11-11-1-ii-ii11111-1-1-1-1-1-1-111i-i-ii    linear of order 4
ρ92-200-22000022222-2-2-2-2-200000000    orthogonal lifted from D4
ρ102200-2-20000222222222200000000    orthogonal lifted from D4
ρ114-42-20000003+5/23-5/2-1+5-1-5-11+5-3+5/21-5-3-5/211+5/21-5/2-1+5/2-1-5/20000    orthogonal faithful
ρ1244220000003-5/23+5/2-1-5-1+5-1-1+53+5/2-1-53-5/2-1-1+5/2-1-5/2-1-5/2-1+5/20000    orthogonal lifted from D5≀C2
ρ134-42-20000003-5/23+5/2-1-5-1+5-11-5-3-5/21+5-3+5/211-5/21+5/2-1-5/2-1+5/20000    orthogonal faithful
ρ1444-2-20000003+5/23-5/2-1+5-1-5-1-1-53-5/2-1+53+5/2-11+5/21-5/21-5/21+5/20000    orthogonal lifted from D5≀C2
ρ1544220000003+5/23-5/2-1+5-1-5-1-1-53-5/2-1+53+5/2-1-1-5/2-1+5/2-1+5/2-1-5/20000    orthogonal lifted from D5≀C2
ρ1644-2-20000003-5/23+5/2-1-5-1+5-1-1+53+5/2-1-53-5/2-11-5/21+5/21+5/21-5/20000    orthogonal lifted from D5≀C2
ρ174-4-220000003-5/23+5/2-1-5-1+5-11-5-3-5/21+5-3+5/21-1+5/2-1-5/21+5/21-5/20000    orthogonal faithful
ρ184400002200-1+5-1-53-5/23+5/2-13+5/2-1-53-5/2-1+5-10000-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5≀C2
ρ19440000-2-200-1+5-1-53-5/23+5/2-13+5/2-1-53-5/2-1+5-100001-5/21-5/21+5/21+5/2    orthogonal lifted from D5≀C2
ρ204400002200-1-5-1+53+5/23-5/2-13-5/2-1+53+5/2-1-5-10000-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5≀C2
ρ21440000-2-200-1-5-1+53+5/23-5/2-13-5/2-1+53+5/2-1-5-100001+5/21+5/21-5/21-5/2    orthogonal lifted from D5≀C2
ρ224-4-220000003+5/23-5/2-1+5-1-5-11+5-3+5/21-5-3-5/21-1-5/2-1+5/21-5/21+5/20000    orthogonal faithful
ρ234-40000-2i2i00-1-5-1+53+5/23-5/2-1-3+5/21-5-3-5/21+510000ζ4ζ534ζ52ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ544ζ5    complex faithful
ρ244-40000-2i2i00-1+5-1-53-5/23+5/2-1-3-5/21+5-3+5/21-510000ζ4ζ544ζ5ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ534ζ52    complex faithful
ρ254-400002i-2i00-1-5-1+53+5/23-5/2-1-3+5/21-5-3-5/21+510000ζ43ζ5343ζ52ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5443ζ5    complex faithful
ρ264-400002i-2i00-1+5-1-53-5/23+5/2-1-3-5/21+5-3+5/21-510000ζ43ζ5443ζ5ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5343ζ52    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 D52⋊C4
On 20 points - transitive group 20T93
Generators in S20
(11 12 13 14 15)(16 17 18 19 20)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 20)(12 19)(13 18)(14 17)(15 16)
(1 2 3 4 5)(6 7 8 9 10)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)

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

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

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

G:=TransitiveGroup(20,93);

On 20 points - transitive group 20T94
Generators in S20
(11 12 13 14 15)(16 17 18 19 20)
(11 20)(12 19)(13 18)(14 17)(15 16)
(1 2 3 4 5)(6 7 8 9 10)
(1 10)(2 9)(3 8)(4 7)(5 6)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)

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

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

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

G:=TransitiveGroup(20,94);

On 20 points - transitive group 20T95
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 16)(7 20)(8 19)(9 18)(10 17)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 18)(2 19)(3 20)(4 16)(5 17)(6 12)(7 13)(8 14)(9 15)(10 11)
(1 16 7 11)(2 19 8 14)(3 17 9 12)(4 20 10 15)(5 18 6 13)

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

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

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

G:=TransitiveGroup(20,95);

Matrix representation of D52⋊C4 in GL6(𝔽41)

100000
010000
000001
000100
000010
00403556
,
40390000
010000
000001
000100
000010
001000
,
100000
010000
001010
0006340
000600
000011
,
120000
0400000
001010
0001400
0000400
000011
,
40390000
110000
001000
0034403535
0040001
001010

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,1,0,35,0,0,0,0,1,5,0,0,1,0,0,6],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,1,34,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,40,40,1,0,0,0,0,0,1],[40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,34,40,1,0,0,0,40,0,0,0,0,0,35,0,1,0,0,0,35,1,0] >;

D52⋊C4 in GAP, Magma, Sage, TeX

D_5^2\rtimes C_4
% in TeX

G:=Group("D5^2:C4");
// GroupNames label

G:=SmallGroup(400,129);
// by ID

G=gap.SmallGroup(400,129);
# by ID

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

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

Export

Character table of D52⋊C4 in TeX

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