Copied to
clipboard

G = C8⋊C4⋊C4  order 128 = 27

1st semidirect product of C8⋊C4 and C4 acting faithfully

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

Aliases: C8⋊C4⋊1C4, C4⋊1D4⋊1C4, (C2×D4).3D4, (C2×Q8).3D4, C8⋊3D4.1C2, C42.3(C2×C4), C42⋊3C4⋊2C2, C42.C4⋊5C2, C2.6(C42⋊C4), C4.4D4.3C22, C22.16(C23⋊C4), (C2×C4).32(C22⋊C4), SmallGroup(128,138)

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

Derived series
C1 — C42 — C8⋊C4⋊C4
Chief series
C1 — C2 — C22 — C2×C4 — C2×D4 — C4.4D4 — C8⋊3D4 — C8⋊C4⋊C4
Lower central
C1 — C2 — C22 — C2×C4 — C42 — C8⋊C4⋊C4
Upper central
C1 — C2 — C22 — C2×C4 — C4.4D4 — C8⋊C4⋊C4
Jennings
C1 — C2 — C2 — C22 — C2×C4 — C4.4D4 — C8⋊C4⋊C4

Generators and relations for C8⋊C4⋊C4
 G = < a,b,c | a8=b4=c4=1, bab-1=a5, cac-1=a5b, cbc-1=a6b-1 >

C1
C2
2C2
8C2
16C2
C22
2C4
4C4
4C4
4C22
8C22
8C22
8C22
8C22
16C4
C2×C4
2C2×C4
2C23
2C8
2C2×C4
2C8
4Q8
4D4
4D4
4C23
4D4
8D4
8D4
8C2×C4
8C8
C2×Q8
C2×D4
C42
2C2×D4
2C2×C8
4M4(2)
4C2×D4
4SD16
4SD16
4D8
4D8
4C22⋊C4
4C22⋊C4
C4.4D4
C4⋊1D4
C8⋊C4
2C4.10D4
2C2×D8
2C2×SD16
2C23⋊C4
C42.C4
C8⋊3D4
C42⋊3C4
C8⋊C4⋊C4

Character table of C8⋊C4⋊C4

 class 12A2B2C2D4A4B4C4D4E8A8B8C8D
 size 1128164881616881616
ρ111111111111111    trivial
ρ21111-1111-1-1-1-111    linear of order 2
ρ311111111-1-111-1-1    linear of order 2
ρ41111-111111-1-1-1-1    linear of order 2
ρ5111-111-11i-i-1-1-ii    linear of order 4
ρ6111-1-11-11-ii11-ii    linear of order 4
ρ7111-111-11-ii-1-1i-i    linear of order 4
ρ8111-1-11-11i-i11i-i    linear of order 4
ρ9222202-2-2000000    orthogonal lifted from D4
ρ10222-2022-2000000    orthogonal lifted from D4
ρ1144400-400000000    orthogonal lifted from C23⋊C4
ρ1244-400000002-200    orthogonal lifted from C42⋊C4
ρ1344-40000000-2200    orthogonal lifted from C42⋊C4
ρ148-8000000000000    orthogonal faithful

Permutation representations of C8⋊C4⋊C4
►On 16 points - transitive group 16T362
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(2 6)(4 8)(9 15 13 11)(10 12 14 16)

(1 14 6 11)(2 15 5 10)(3 12 4 13)(7 16 8 9)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,15,13,11)(10,12,14,16), (1,14,6,11)(2,15,5,10)(3,12,4,13)(7,16,8,9)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,15,13,11)(10,12,14,16), (1,14,6,11)(2,15,5,10)(3,12,4,13)(7,16,8,9) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,15,13,11),(10,12,14,16)], [(1,14,6,11),(2,15,5,10),(3,12,4,13),(7,16,8,9)]])
 
G:=TransitiveGroup(16,362);
 

Matrix representation of C8⋊C4⋊C4 ►in GL8(ℤ)

00010000
00100000
10000000
0-1000000
000000-10
00000001
00000100
00001000
,
-10000000
0-1000000
00100000
00010000
00000-100
00001000
0000000-1
00000010
,
00001000
00000100
00000010
00000001
00100000
00010000
10000000
01000000

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

C8⋊C4⋊C4 in GAP, Magma, Sage, TeX 

C_8\rtimes C_4\rtimes C_4
 
% in TeX
 
G:=Group("C8:C4:C4");
 
// GroupNames label
 
G:=SmallGroup(128,138);
 
// by ID
 
G=gap.SmallGroup(128,138);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,387,184,1690,745,1684,1411,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^5,c*a*c^-1=a^5*b,c*b*c^-1=a^6*b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C8⋊C4⋊C4 in TeX
Character table of C8⋊C4⋊C4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁