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G = C42⋊6D4  order 128 = 27

6th semidirect product of C42 and D4 acting faithfully

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

Aliases: C42⋊6D4, C24.41D4, 2+ 1+4.4C22, C2≀C4⋊4C2, (C2×D4)⋊4D4, C22⋊C4⋊4D4, (C22×C4)⋊4D4, C2≀C22⋊3C2, D4⋊4D4⋊2C2, C42⋊C4⋊7C2, C2.24C2≀C22, (C2×D4).5C23, C23.17(C2×D4), C23.7D4⋊3C2, C23.D4⋊3C2, C23⋊C4.4C22, C22≀C2.6C22, C22.48C22≀C2, C4⋊1D4.57C22, C4.D4.4C22, C22.54C24⋊1C2, C22.D4.5C22, (C2×C4).17(C2×D4), 2-Sylow(M12`2), SmallGroup(128,932)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2 — C2×D4 — C42⋊6D4
Chief series
C1 — C2 — C22 — C23 — C2×D4 — C22≀C2 — C22.54C24 — C42⋊6D4
Lower central
C1 — C2 — C22 — C2×D4 — C42⋊6D4
Upper central
C1 — C2 — C22 — C2×D4 — C42⋊6D4
Jennings
C1 — C2 — C22 — C2×D4 — C42⋊6D4

Generators and relations for C42⋊6D4
 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, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C4○D4, C24, C23⋊C4, C23⋊C4, C4.D4, C4≀C2, C22≀C2, C22≀C2, C4⋊D4, C22.D4, C22.D4, C42⋊2C2, C4⋊1D4, C8⋊C22, 2+ 1+4, C2≀C4, C23.D4, C42⋊C4, D4⋊4D4, C2≀C22, C23.7D4, C22.54C24, C42⋊6D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C2≀C22, C42⋊6D4

Character table of C42⋊6D4

 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 C42⋊6D4
►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 8 6 7)(9 12 11 10)(13 14 15 16)

(1 12 2 11)(3 10 4 9)(5 15 7 14)(6 13 8 16)

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

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

(1 14 3 13)(2 10 4 9)(5 12 8 11)(6 16 7 15)

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

(1 15 6 11)(2 10 5 16)(3 14 7 12)(4 9 8 13)

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

(1 9 2 15)(3 12 7 16)(4 14 8 10)(5 13 6 11)

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

(1 12 6 10)(2 16 5 14)(3 11)(4 15)(7 9)(8 13)

(1 10)(2 16)(3 11)(4 13)(5 14)(6 12)(7 9)(8 15)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,6,8)(2,3,5,7)(9,16,11,14)(10,13,12,15), (1,12,6,10)(2,16,5,14)(3,11)(4,15)(7,9)(8,13), (1,10)(2,16)(3,11)(4,13)(5,14)(6,12)(7,9)(8,15)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,6,8)(2,3,5,7)(9,16,11,14)(10,13,12,15), (1,12,6,10)(2,16,5,14)(3,11)(4,15)(7,9)(8,13), (1,10)(2,16)(3,11)(4,13)(5,14)(6,12)(7,9)(8,15) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,6,8),(2,3,5,7),(9,16,11,14),(10,13,12,15)], [(1,12,6,10),(2,16,5,14),(3,11),(4,15),(7,9),(8,13)], [(1,10),(2,16),(3,11),(4,13),(5,14),(6,12),(7,9),(8,15)]])
 
G:=TransitiveGroup(16,402);
 

Matrix representation of C42⋊6D4 ►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] >;
 

C42⋊6D4 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 C42⋊6D4 in TeX

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