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G = C424D4order 128 = 27

4th semidirect product of C42 and D4 acting faithfully

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

Aliases: C424D4, C24.40D4, 2+ 1+4.1C22, C2≀C42C2, (C2×Q8)⋊3D4, C2≀C22.C2, C22⋊C42D4, (C22×C4)⋊2D4, C423C45C2, C2.21C2≀C22, D4.9D41C2, (C2×D4).2C23, C23.14(C2×D4), C23.7D41C2, C23.D41C2, C23⋊C4.1C22, C22≀C2.4C22, C22.45C22≀C2, C4.D4.1C22, C22.45C241C2, C4.4D4.18C22, C22.D4.3C22, (C2×C4).14(C2×D4), SmallGroup(128,929)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×D4 — C424D4
C1C2C22C23C2×D4C22≀C2C22.45C24 — C424D4
C1C2C22C2×D4 — C424D4
C1C2C22C2×D4 — C424D4
C1C2C22C2×D4 — C424D4

Generators and relations for C424D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1b-1, cbc-1=a2b-1, bd=db, dcd=c-1 >

Subgroups: 368 in 129 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2 [×6], C4 [×9], C22, C22 [×14], C8, C2×C4, C2×C4 [×14], D4 [×9], Q8 [×2], C23 [×2], C23 [×6], C42, C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×5], M4(2), SD16, Q16, C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×5], C2×Q8, C4○D4 [×2], C24, C23⋊C4 [×2], C23⋊C4 [×2], C4.D4, C4≀C2, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22≀C2, C22⋊Q8, C22.D4, C22.D4 [×2], C4.4D4, C422C2, C8.C22, 2+ 1+4, C2≀C4, C23.D4, C423C4, D4.9D4, C2≀C22, C23.7D4, C22.45C24, C424D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, C424D4

Character table of C424D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8
 size 11244448444448888161616
ρ111111111111111111111    trivial
ρ21111111-1-11-1-1-1-1-1-11-111    linear of order 2
ρ3111-1-111-1111-1-1-11-1-111-1    linear of order 2
ρ4111-1-1111-11-1111-11-1-11-1    linear of order 2
ρ5111-1-111-1-11-1111-1-1-11-11    linear of order 2
ρ6111-1-1111111-1-1-111-1-1-11    linear of order 2
ρ711111111-11-1-1-1-1-1111-1-1    linear of order 2
ρ81111111-11111111-11-1-1-1    linear of order 2
ρ922200-2-20222000-200000    orthogonal lifted from D4
ρ10222222-200-2000000-2000    orthogonal lifted from D4
ρ1122200-2200-20-2-22000000    orthogonal lifted from D4
ρ1222200-2200-2022-2000000    orthogonal lifted from D4
ρ13222-2-22-200-20000002000    orthogonal lifted from D4
ρ1422200-2-20-22-2000200000    orthogonal lifted from D4
ρ1544-4000020000000-20000    orthogonal lifted from C2≀C22
ρ1644-40000-2000000020000    orthogonal lifted from C2≀C22
ρ174-402-20002i0-2i-2i2i0000000    complex faithful
ρ184-402-2000-2i02i2i-2i0000000    complex faithful
ρ194-40-220002i0-2i2i-2i0000000    complex faithful
ρ204-40-22000-2i02i-2i2i0000000    complex faithful

Permutation representations of C424D4
On 16 points - transitive group 16T345
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 5 7 3)(2 6 8 4)(9 13 11 15)(10 14 12 16)
(1 14 5 10)(2 11 4 15)(3 12 7 16)(6 13 8 9)
(1 12)(2 13)(3 14)(4 9)(5 16)(6 11)(7 10)(8 15)

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

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

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

G:=TransitiveGroup(16,345);

On 16 points - transitive group 16T399
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 7 15 12)(2 8 16 9)(3 5 13 10)(4 6 14 11)
(2 9 14 11)(3 13)(4 6 16 8)(5 12 10 7)
(2 11)(3 13)(4 8)(5 10)(6 16)(9 14)

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

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

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

G:=TransitiveGroup(16,399);

On 16 points - transitive group 16T400
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 7 5 3)(2 8 6 4)(9 15 11 13)(10 16 12 14)
(1 9)(2 14 6 16)(3 15 7 13)(4 10)(5 11)(8 12)
(1 9)(2 16)(3 13)(4 10)(5 11)(6 14)(7 15)(8 12)

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

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

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

G:=TransitiveGroup(16,400);

On 16 points - transitive group 16T410
Generators in S16
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 2 4 3)(5 8 6 7)(9 10 11 12)(13 16 15 14)
(1 11 8 13)(2 10 5 16)(3 12 6 14)(4 9 7 15)
(1 15)(2 14)(3 16)(4 13)(5 12)(6 10)(7 11)(8 9)

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

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

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

G:=TransitiveGroup(16,410);

Matrix representation of C424D4 in GL4(𝔽5) generated by

2004
0200
1200
4320
,
0030
3011
3000
2410
,
2404
4320
4000
0300
,
0040
0002
4000
0300
G:=sub<GL(4,GF(5))| [2,0,1,4,0,2,2,3,0,0,0,2,4,0,0,0],[0,3,3,2,0,0,0,4,3,1,0,1,0,1,0,0],[2,4,4,0,4,3,0,3,0,2,0,0,4,0,0,0],[0,0,4,0,0,0,0,3,4,0,0,0,0,2,0,0] >;

C424D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4D_4
% in TeX

G:=Group("C4^2:4D4");
// GroupNames label

G:=SmallGroup(128,929);
// by ID

G=gap.SmallGroup(128,929);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,297,1971,375,4037]);
// Polycyclic

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

Export

Character table of C424D4 in TeX

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