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G = C426D4order 128 = 27

6th semidirect product of C42 and D4 acting faithfully

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

Aliases: C426D4, C24.41D4, 2+ 1+4.4C22, C2≀C44C2, (C2×D4)⋊4D4, C22⋊C44D4, (C22×C4)⋊4D4, C2≀C223C2, D44D42C2, C42⋊C47C2, C2.24C2≀C22, (C2×D4).5C23, C23.17(C2×D4), C23.7D43C2, C23.D43C2, C23⋊C4.4C22, C22≀C2.6C22, C22.48C22≀C2, C41D4.57C22, C4.D4.4C22, C22.54C241C2, C22.D4.5C22, (C2×C4).17(C2×D4), 2-Sylow(M12`2), SmallGroup(128,932)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×D4 — C426D4
C1C2C22C23C2×D4C22≀C2C22.54C24 — C426D4
C1C2C22C2×D4 — C426D4
C1C2C22C2×D4 — C426D4
C1C2C22C2×D4 — C426D4

Generators and relations for C426D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b-1, dad=a-1b, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 408 in 130 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2 [×6], C4 [×8], C22, C22 [×16], C8, C2×C4, C2×C4 [×11], D4 [×13], Q8, C23 [×2], C23 [×6], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], M4(2), D8, SD16, C22×C4, C22×C4, C2×D4 [×2], C2×D4 [×9], C4○D4 [×2], C24, C23⋊C4 [×2], C23⋊C4 [×2], C4.D4, C4≀C2, C22≀C2, C22≀C2 [×2], C4⋊D4 [×3], C22.D4, C22.D4 [×2], C422C2, C41D4, C8⋊C22, 2+ 1+4, C2≀C4, C23.D4, C42⋊C4, D44D4, C2≀C22, C23.7D4, C22.54C24, C426D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, C426D4

Character table of C426D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8
 size 11244888488888161616
ρ111111111111111111    trivial
ρ2111111-1111111-1-1-1-1    linear of order 2
ρ3111111-1-11-11-1-1-11-11    linear of order 2
ρ41111111-11-11-1-11-11-1    linear of order 2
ρ511111-1111-1-1-111-1-11    linear of order 2
ρ611111-1-111-1-1-11-111-1    linear of order 2
ρ711111-11-111-11-111-1-1    linear of order 2
ρ811111-1-1-111-11-1-1-111    linear of order 2
ρ9222-2-20022000-20000    orthogonal lifted from D4
ρ102222-2-200-202000000    orthogonal lifted from D4
ρ11222-22000-2-20200000    orthogonal lifted from D4
ρ12222-2-200-2200020000    orthogonal lifted from D4
ρ13222-22000-220-200000    orthogonal lifted from D4
ρ142222-2200-20-2000000    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 C426D4
On 16 points - transitive group 16T336
Generators in S16
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 2 3 4)(5 7 6 8)(9 12 11 10)(13 14 15 16)
(1 10 4 11)(2 9 3 12)(5 13 8 16)(6 15 7 14)
(1 13)(2 14)(3 15)(4 16)(5 10)(6 12)(7 9)(8 11)

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

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

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

G:=TransitiveGroup(16,336);

On 16 points - transitive group 16T340
Generators in S16
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 7 6 8)(9 10 11 12)(13 16 15 14)
(1 15)(2 13)(3 16 4 14)(5 12)(6 10)(7 9 8 11)
(1 12)(2 10)(3 11)(4 9)(5 15)(6 13)(7 14)(8 16)

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

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

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

G:=TransitiveGroup(16,340);

On 16 points - transitive group 16T341
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 13 6 9)(2 14 7 10)(3 15 8 11)(4 16 5 12)
(2 14 4 12)(3 8)(5 16 7 10)(9 15 13 11)
(2 16)(3 8)(4 10)(5 14)(7 12)(11 15)

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

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

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

G:=TransitiveGroup(16,341);

On 16 points - transitive group 16T352
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 7 4 6)(2 8 3 5)(9 16 11 14)(10 13 12 15)
(1 13 5 14)(2 9 6 10)(3 11 7 12)(4 15 8 16)
(1 16)(2 10)(3 12)(4 14)(5 15)(6 9)(7 11)(8 13)

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

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

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

G:=TransitiveGroup(16,352);

On 16 points - transitive group 16T359
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 7 4 6)(2 8 3 5)(9 15 11 13)(10 16 12 14)
(1 11 7 13)(2 14 8 10)(3 16 5 12)(4 9 6 15)
(1 9)(2 14)(3 16)(4 11)(5 12)(6 13)(7 15)(8 10)

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

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

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

G:=TransitiveGroup(16,359);

On 16 points - transitive group 16T366
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 7 5 4)(2 8 6 3)(9 16 11 14)(10 13 12 15)
(1 12 2 16)(3 9 7 15)(4 13 8 11)(5 10 6 14)
(1 11)(2 13)(3 10)(4 16)(5 9)(6 15)(7 14)(8 12)

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

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

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

G:=TransitiveGroup(16,366);

On 16 points - transitive group 16T402
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4 5 8)(2 3 6 7)(9 16 11 14)(10 13 12 15)
(1 14)(2 10)(3 13 7 15)(4 9 8 11)(5 16)(6 12)
(1 16)(2 10)(3 13)(4 11)(5 14)(6 12)(7 15)(8 9)

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

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

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

G:=TransitiveGroup(16,402);

Matrix representation of C426D4 in GL8(ℤ)

00100000
000-10000
10000000
0-1000000
00000-100
00001000
00000001
000000-10
,
00010000
00100000
0-1000000
-10000000
00000001
00000010
00000-100
0000-1000
,
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,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[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] >;

C426D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_6D_4
% in TeX

G:=Group("C4^2:6D4");
// GroupNames label

G:=SmallGroup(128,932);
// by ID

G=gap.SmallGroup(128,932);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,723,297,1971,375,4037]);
// Polycyclic

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

Export

Character table of C426D4 in TeX

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