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G = C425D4order 128 = 27

5th semidirect product of C42 and D4 acting faithfully

p-group, non-abelian, nilpotent (class 4), monomial, rational

Aliases: C425D4, C243D4, 2+ 1+4.3C22, C2≀C43C2, (C2×Q8)⋊4D4, C22⋊C43D4, C2≀C222C2, C423C46C2, C2.23C2≀C22, D4.9D42C2, (C2×D4).4C23, C23.16(C2×D4), C23⋊C4.3C22, C24⋊C222C2, C22≀C2.5C22, C22.47C22≀C2, C4.D4.3C22, C4.4D4.19C22, (C2×C4).16(C2×D4), 2-Sylow(PSL(4,5)), SmallGroup(128,931)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×D4 — C425D4
C1C2C22C23C2×D4C22≀C2C24⋊C22 — C425D4
C1C2C22C2×D4 — C425D4
C1C2C22C2×D4 — C425D4
C1C2C22C2×D4 — C425D4

Generators and relations for C425D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, faf=ab=ba, ac=ca, ad=da, eae-1=abd, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=e-1 >

Subgroups: 440 in 135 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×8], C22, C22 [×19], C8, C2×C4, C2×C4 [×9], D4 [×13], Q8 [×3], C23 [×2], C23 [×8], C42, C42, C22⋊C4 [×2], C22⋊C4 [×12], M4(2), SD16, Q16, C2×D4, C2×D4 [×9], C2×Q8, C2×Q8, C4○D4 [×2], C24 [×2], C23⋊C4 [×2], C23⋊C4 [×2], C4.D4, C4≀C2, C22≀C2 [×2], C22≀C2 [×4], C4.4D4, C4.4D4 [×4], C8.C22, 2+ 1+4, C2≀C4 [×2], C423C4, D4.9D4, C2≀C22 [×2], C24⋊C22, C425D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, C425D4

Character table of C425D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8
 size 11244888488888161616
ρ111111111111111111    trivial
ρ2111111-1111111-1-1-1-1    linear of order 2
ρ311111-1111-11-1-11-11-1    linear of order 2
ρ411111-1-111-11-1-1-11-11    linear of order 2
ρ511111-1-1-111-1-11-111-1    linear of order 2
ρ611111-11-111-1-111-1-11    linear of order 2
ρ71111111-11-1-11-111-1-1    linear of order 2
ρ8111111-1-11-1-11-1-1-111    linear of order 2
ρ92222-200-2-202000000    orthogonal lifted from D4
ρ10222-22200-200-200000    orthogonal lifted from D4
ρ11222-2-20002200-20000    orthogonal lifted from D4
ρ12222-22-200-200200000    orthogonal lifted from D4
ρ13222-2-20002-20020000    orthogonal lifted from D4
ρ142222-2002-20-2000000    orthogonal lifted from D4
ρ1544-40002000000-2000    orthogonal lifted from C2≀C22
ρ1644-4000-20000002000    orthogonal lifted from C2≀C22
ρ178-8000000000000000    orthogonal faithful

Permutation representations of C425D4
On 16 points - transitive group 16T342
Generators in S16
(1 10)(2 6)(3 15)(5 12)(8 13)(9 11)
(1 10)(2 9)(3 13)(4 16)(5 12)(6 11)(7 14)(8 15)
(2 6)(4 7)(9 11)(14 16)
(1 5)(2 6)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 7)(2 8)(3 6)(4 5)(9 15)(10 14)(11 13)(12 16)

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

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

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

G:=TransitiveGroup(16,342);

On 16 points - transitive group 16T365
Generators in S16
(2 5)(3 13)(4 14)(6 12)(7 9)(11 16)
(2 16)(3 6)(4 9)(5 11)(7 14)(12 13)
(1 10)(2 16)(3 12)(4 14)(5 11)(6 13)(7 9)(8 15)
(1 8)(2 5)(3 6)(4 7)(9 14)(10 15)(11 16)(12 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 4)(5 7)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(16,365);

On 16 points - transitive group 16T381
Generators in S16
(1 9)(2 14)(3 5)(4 16)(6 8)(7 15)(10 12)(11 13)
(1 5)(2 12)(3 9)(4 8)(6 16)(7 13)(10 14)(11 15)
(2 16)(4 14)(6 12)(8 10)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(16,381);

On 16 points - transitive group 16T386
Generators in S16
(1 5)(2 16)(3 7)(6 11)(10 15)(12 13)
(1 5)(2 6)(3 12)(4 9)(7 13)(8 14)(10 15)(11 16)
(1 15)(3 13)(5 10)(7 12)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 16)(14 15)

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

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

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

G:=TransitiveGroup(16,386);

On 16 points - transitive group 16T394
Generators in S16
(2 15)(3 16)(4 13)(6 11)
(2 15)(3 16)(7 12)(8 9)
(2 15)(4 13)(5 10)(7 12)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)

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

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

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

G:=TransitiveGroup(16,394);

Matrix representation of C425D4 in GL8(ℤ)

00100000
000-10000
10000000
0-1000000
00001000
00000-100
00000010
0000000-1
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00001000
00000100
000000-10
0000000-1
01000000
10000000
00010000
00100000
,
00001000
00000100
00000010
00000001
10000000
01000000
00100000
00010000

G:=sub<GL(8,Integers())| [0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C425D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_5D_4
% in TeX

G:=Group("C4^2:5D4");
// GroupNames label

G:=SmallGroup(128,931);
// by ID

G=gap.SmallGroup(128,931);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,723,297,1971,375,4037]);
// Polycyclic

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

Export

Character table of C425D4 in TeX

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