Copied to
clipboard

G = C41D4⋊C4order 128 = 27

2nd semidirect product of C41D4 and C4 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: (C4×C8)⋊4C4, C41D42C4, (C2×D4).5D4, C84D4.2C2, C42⋊C42C2, C42.15(C2×C4), C41D4.2C22, C2.8(C42⋊C4), C22.18(C23⋊C4), (C2×C4).34(C22⋊C4), SmallGroup(128,140)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C41D4⋊C4
C1C2C22C2×C4C2×D4C41D4C84D4 — C41D4⋊C4
C1C2C22C2×C4C42 — C41D4⋊C4
C1C2C22C2×C4C41D4 — C41D4⋊C4
C1C2C2C22C2×C4C41D4 — C41D4⋊C4

Generators and relations for C41D4⋊C4
 G = < a,b,c,d | a4=b4=c2=d4=1, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a2b-1, dcd-1=ac >

Subgroups: 256 in 60 conjugacy classes, 14 normal (10 characteristic)
C1, C2, C2 [×4], C4 [×5], C22, C22 [×7], C8 [×2], C2×C4, C2×C4 [×3], D4 [×8], C23 [×3], C42, C22⋊C4 [×2], C2×C8, D8 [×4], C2×D4 [×2], C2×D4 [×3], C4×C8, C23⋊C4 [×2], C41D4 [×2], C2×D8 [×2], C42⋊C4 [×2], C84D4, C41D4⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C23⋊C4, C42⋊C4, C41D4⋊C4

Character table of C41D4⋊C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G8A8B8C8D
 size 1128816444161616164444
ρ111111111111111111    trivial
ρ211111-1111-111-1-1-1-1-1    linear of order 2
ρ311111-11111-1-11-1-1-1-1    linear of order 2
ρ4111111111-1-1-1-11111    linear of order 2
ρ5111-1-11111i-ii-i-1-1-1-1    linear of order 4
ρ6111-1-1-1111-i-iii1111    linear of order 4
ρ7111-1-1-1111ii-i-i1111    linear of order 4
ρ8111-1-11111-ii-ii-1-1-1-1    linear of order 4
ρ92222-202-2-200000000    orthogonal lifted from D4
ρ10222-2202-2-200000000    orthogonal lifted from D4
ρ1144-40000000000-22-22    orthogonal lifted from C42⋊C4
ρ1244-400000000002-22-2    orthogonal lifted from C42⋊C4
ρ13444000-40000000000    orthogonal lifted from C23⋊C4
ρ144-4000002-20000-220220    orthogonal faithful
ρ154-4000002-20000220-220    orthogonal faithful
ρ164-400000-2200000-22022    orthogonal faithful
ρ174-400000-2200000220-22    orthogonal faithful

Permutation representations of C41D4⋊C4
On 16 points - transitive group 16T377
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(13 16 15 14)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)
(1 11 2 9)(3 10)(4 12)(5 13 8 14)(6 15 7 16)

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

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

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

G:=TransitiveGroup(16,377);

On 16 points - transitive group 16T413
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 7 6 8)(9 12 11 10)(13 16 15 14)
(1 2)(5 8)(6 7)(9 11)(13 14)(15 16)
(1 12 8 16)(2 10 7 14)(3 11 5 13)(4 9 6 15)

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

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

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

G:=TransitiveGroup(16,413);

Matrix representation of C41D4⋊C4 in GL4(𝔽7) generated by

6361
0254
0566
0235
,
0126
4021
4416
6146
,
2245
6535
6140
2213
,
4121
4060
2223
5211
G:=sub<GL(4,GF(7))| [6,0,0,0,3,2,5,2,6,5,6,3,1,4,6,5],[0,4,4,6,1,0,4,1,2,2,1,4,6,1,6,6],[2,6,6,2,2,5,1,2,4,3,4,1,5,5,0,3],[4,4,2,5,1,0,2,2,2,6,2,1,1,0,3,1] >;

C41D4⋊C4 in GAP, Magma, Sage, TeX

C_4\rtimes_1D_4\rtimes C_4
% in TeX

G:=Group("C4:1D4:C4");
// GroupNames label

G:=SmallGroup(128,140);
// by ID

G=gap.SmallGroup(128,140);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,1059,520,794,745,1684,1411,375,172,4037]);
// Polycyclic

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

Export

Character table of C41D4⋊C4 in TeX

׿
×
𝔽