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G = (C4×C8)⋊6C4  order 128 = 27

6th semidirect product of C4×C8 and C4 acting faithfully

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

Aliases: (C4×C8)⋊6C4, C4⋊Q8⋊2C4, (C2×D4).6D4, C8⋊5D4.6C2, C42.16(C2×C4), C42⋊C4.1C2, C4⋊1D4.3C22, C2.9(C42⋊C4), C22.19(C23⋊C4), (C2×C4).35(C22⋊C4), SmallGroup(128,141)

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

Derived series
C1 — C42 — (C4×C8)⋊6C4
Chief series
C1 — C2 — C22 — C2×C4 — C2×D4 — C4⋊1D4 — C8⋊5D4 — (C4×C8)⋊6C4
Lower central
C1 — C2 — C22 — C2×C4 — C42 — (C4×C8)⋊6C4
Upper central
C1 — C2 — C22 — C2×C4 — C4⋊1D4 — (C4×C8)⋊6C4
Jennings
C1 — C2 — C2 — C22 — C2×C4 — C4⋊1D4 — (C4×C8)⋊6C4

Generators and relations for (C4×C8)⋊6C4
 G = < a,b,c | a4=b8=c4=1, ab=ba, cac-1=ab2, cbc-1=ab3 >

C1
C2
2C2
8C2
8C2
C22
2C4
2C4
2C4
4C22
4C22
8C22
8C4
8C22
16C4
16C4
C2×C4
2C8
2C2×C4
2C23
2C23
2C8
4D4
4C2×C4
4Q8
4Q8
4D4
8D4
8C2×C4
8D4
8C2×C4
C2×D4
C42
C2×D4
2C2×Q8
2C2×C8
4C4⋊C4
4C22⋊C4
4SD16
4SD16
4SD16
4SD16
4C22⋊C4
4C2×D4
C4⋊1D4
C4⋊Q8
C4×C8
2C23⋊C4
2C2×SD16
2C2×SD16
2C23⋊C4
C8⋊5D4
C42⋊C4
C42⋊C4
(C4×C8)⋊6C4

Character table of (C4×C8)⋊6C4

 class 12A2B2C2D4A4B4C4D4E4F4G4H8A8B8C8D
 size 1128844416161616164444
ρ111111111111111111    trivial
ρ21111111111-1-1-1-1-1-1-1    linear of order 2
ρ311111111-1-11-11-1-1-1-1    linear of order 2
ρ411111111-1-1-11-11111    linear of order 2
ρ5111-1-1111-ii-i-1i1111    linear of order 4
ρ6111-1-1111-iii1-i-1-1-1-1    linear of order 4
ρ7111-1-1111i-i-i1i-1-1-1-1    linear of order 4
ρ8111-1-1111i-ii-1-i1111    linear of order 4
ρ92222-2-22-2000000000    orthogonal lifted from D4
ρ10222-22-22-2000000000    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
ρ144-400020-2000002√-20-2√-20    complex faithful
ρ154-400020-200000-2√-202√-20    complex faithful
ρ164-4000-202000000-2√-202√-2    complex faithful
ρ174-4000-2020000002√-20-2√-2    complex faithful

Permutation representations of (C4×C8)⋊6C4
►On 16 points - transitive group 16T363
Generators in S16
(1 5)(2 6)(3 7)(4 8)(9 11 13 15)(10 12 14 16)

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

(1 11)(2 10 4 16)(3 9 7 13)(5 15)(6 14 8 12)
 
G:=sub<Sym(16)| (1,5)(2,6)(3,7)(4,8)(9,11,13,15)(10,12,14,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11)(2,10,4,16)(3,9,7,13)(5,15)(6,14,8,12)>;
 
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,11,13,15)(10,12,14,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11)(2,10,4,16)(3,9,7,13)(5,15)(6,14,8,12) );
 
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,11,13,15),(10,12,14,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,11),(2,10,4,16),(3,9,7,13),(5,15),(6,14,8,12)]])
 
G:=TransitiveGroup(16,363);
 

Matrix representation of (C4×C8)⋊6C4 ►in GL4(𝔽3) generated by

1020
1012
0221
2121
,
0202
0220
0002
1101
,
1021
0021
0001
0112
G:=sub<GL(4,GF(3))| [1,1,0,2,0,0,2,1,2,1,2,2,0,2,1,1],[0,0,0,1,2,2,0,1,0,2,0,0,2,0,2,1],[1,0,0,0,0,0,0,1,2,2,0,1,1,1,1,2] >;
 

(C4×C8)⋊6C4 in GAP, Magma, Sage, TeX 

(C_4\times C_8)\rtimes_6C_4
 
% in TeX
 
G:=Group("(C4xC8):6C4");
 
// GroupNames label
 
G:=SmallGroup(128,141);
 
// by ID
 
G=gap.SmallGroup(128,141);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,456,422,1059,520,794,745,1684,1411,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^8=c^4=1,a*b=b*a,c*a*c^-1=a*b^2,c*b*c^-1=a*b^3>;
 
// generators/relations
 

Export

Subgroup lattice of (C4×C8)⋊6C4 in TeX
Character table of (C4×C8)⋊6C4 in TeX

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