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G = C8⋊C4⋊5C4  order 128 = 27

5th semidirect product of C8⋊C4 and C4 acting faithfully

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

Aliases: C8⋊C4⋊5C4, (C2×D4).7D4, C42.5(C2×C4), C42.C2⋊1C4, C42⋊C4.2C2, C4⋊1D4.4C22, C2.6(C42⋊3C4), C22.22(C23⋊C4), C42.29C22.3C2, (C2×C4).38(C22⋊C4), SmallGroup(128,144)

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

Derived series
C1 — C42 — C8⋊C4⋊5C4
Chief series
C1 — C2 — C22 — C2×C4 — C2×D4 — C4⋊1D4 — C42.29C22 — C8⋊C4⋊5C4
Lower central
C1 — C2 — C22 — C2×C4 — C42 — C8⋊C4⋊5C4
Upper central
C1 — C2 — C22 — C2×C4 — C4⋊1D4 — C8⋊C4⋊5C4
Jennings
C1 — C2 — C2 — C22 — C2×C4 — C4⋊1D4 — C8⋊C4⋊5C4

Generators and relations for C8⋊C4⋊5C4
 G = < a,b,c | a8=b4=c4=1, bab-1=a5, cac-1=ab, cbc-1=a6b >

C1
C2
2C2
8C2
8C2
C22
2C4
4C22
4C22
4C4
8C4
8C22
8C22
16C4
16C4
C2×C4
2C2×C4
2C23
2C23
4C8
4C2×C4
4D4
4D4
8C2×C4
8D4
8D4
8C2×C4
C2×D4
C42
C2×D4
2C4⋊C4
2C2×C8
4C2×D4
4C22⋊C4
4C4⋊C4
4C22⋊C4
C4⋊1D4
C8⋊C4
C42.C2
2D4⋊C4
2C23⋊C4
2D4⋊C4
2C23⋊C4
C42⋊C4
C42⋊C4
C42.29C22
C8⋊C4⋊5C4

Character table of C8⋊C4⋊5C4

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B
 size 1128848161616161688
ρ111111111111111    trivial
ρ21111111-111-1-1-1-1    linear of order 2
ρ311111111-1-11-1-1-1    linear of order 2
ρ41111111-1-1-1-1111    linear of order 2
ρ5111-1-111-ii-ii1-1-1    linear of order 4
ρ6111-1-111ii-i-i-111    linear of order 4
ρ7111-1-111-i-iii-111    linear of order 4
ρ8111-1-111i-ii-i1-1-1    linear of order 4
ρ92222-22-20000000    orthogonal lifted from D4
ρ10222-222-20000000    orthogonal lifted from D4
ρ1144400-400000000    orthogonal lifted from C23⋊C4
ρ1244-4000000000-2i2i    complex lifted from C42⋊3C4
ρ1344-40000000002i-2i    complex lifted from C42⋊3C4
ρ148-8000000000000    orthogonal faithful

Permutation representations of C8⋊C4⋊5C4
►On 16 points - transitive group 16T375
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

(1 13)(2 10 8 12)(3 11 7 15)(4 16 6 14)(5 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)(9,11,13,15)(10,16,14,12), (1,13)(2,10,8,12)(3,11,7,15)(4,16,6,14)(5,9)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,11,13,15)(10,16,14,12), (1,13)(2,10,8,12)(3,11,7,15)(4,16,6,14)(5,9) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,11,13,15),(10,16,14,12)], [(1,13),(2,10,8,12),(3,11,7,15),(4,16,6,14),(5,9)]])
 
G:=TransitiveGroup(16,375);
 

Matrix representation of C8⋊C4⋊5C4 ►in GL8(ℤ)

00-100000
00010000
0-1000000
-10000000
00000001
00000010
0000-1000
00000100
,
-10000000
0-1000000
00100000
00010000
00000-100
00001000
0000000-1
00000010
,
00001000
00000100
00000010
00000001
10000000
0-1000000
00010000
00100000

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

C8⋊C4⋊5C4 in GAP, Magma, Sage, TeX 

C_8\rtimes C_4\rtimes_5C_4
 
% in TeX
 
G:=Group("C8:C4:5C4");
 
// GroupNames label
 
G:=SmallGroup(128,144);
 
// by ID
 
G=gap.SmallGroup(128,144);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,232,422,387,520,794,745,1684,1411,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^5,c*a*c^-1=a*b,c*b*c^-1=a^6*b>;
 
// generators/relations
 

Export

Subgroup lattice of C8⋊C4⋊5C4 in TeX
Character table of C8⋊C4⋊5C4 in TeX

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