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G = C42⋊C4  order 64 = 26

2nd semidirect product of C42 and C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C42⋊2C4, C23.3D4, (C2×D4)⋊2C4, C23⋊C4⋊2C2, C4⋊1D4.2C2, C2.8(C23⋊C4), (C2×D4).3C22, C22.11(C22⋊C4), (C2×C4).1(C2×C4), SmallGroup(64,34)

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

Derived series
C1 — C2×C4 — C42⋊C4
Chief series
C1 — C2 — C22 — C23 — C2×D4 — C4⋊1D4 — C42⋊C4
Lower central
C1 — C2 — C22 — C2×C4 — C42⋊C4
Upper central
C1 — C2 — C22 — C2×D4 — C42⋊C4
Jennings
C1 — C2 — C22 — C2×D4 — C42⋊C4

Generators and relations for C42⋊C4
 G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=a-1b, cbc-1=a2b >

C1
C2
2C2
4C2
4C2
8C2
C22
2C4
2C22
2C22
2C4
2C4
4C22
4C22
4C22
4C22
4C22
8C4
8C4
C23
C2×C4
C23
2C23
2C2×C4
4D4
4D4
4D4
4D4
4D4
4C2×C4
4D4
4C2×C4
C2×D4
C42
C2×D4
2C22⋊C4
2C2×D4
2C2×D4
2C22⋊C4
C23⋊C4
C23⋊C4
C4⋊1D4
C42⋊C4

Character table of C42⋊C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G
 size 1124484448888
ρ11111111111111    trivial
ρ211111-11-1-1-1-111    linear of order 2
ρ3111111111-1-1-1-1    linear of order 2
ρ411111-11-1-111-1-1    linear of order 2
ρ5111-1-1-1111i-ii-i    linear of order 4
ρ6111-1-111-1-1-iii-i    linear of order 4
ρ7111-1-1-1111-ii-ii    linear of order 4
ρ8111-1-111-1-1i-i-ii    linear of order 4
ρ92222-20-2000000    orthogonal lifted from D4
ρ10222-220-2000000    orthogonal lifted from D4
ρ114-400000-220000    orthogonal faithful
ρ124-4000002-20000    orthogonal faithful
ρ1344-40000000000    orthogonal lifted from C23⋊C4

Permutation representations of C42⋊C4
►On 8 points - transitive group 8T30
Generators in S8
(5 6 7 8)

(1 4 2 3)(5 6 7 8)

(1 5)(2 7)(3 8 4 6)
 
G:=sub<Sym(8)| (5,6,7,8), (1,4,2,3)(5,6,7,8), (1,5)(2,7)(3,8,4,6)>;
 
G:=Group( (5,6,7,8), (1,4,2,3)(5,6,7,8), (1,5)(2,7)(3,8,4,6) );
 
G=PermutationGroup([[(5,6,7,8)], [(1,4,2,3),(5,6,7,8)], [(1,5),(2,7),(3,8,4,6)]])
 
G:=TransitiveGroup(8,30);
 

►On 16 points - transitive group 16T143
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

(1 10 11 2)(3 7 9 14)(4 13 12 8)(5 6 16 15)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11,15,6)(2,12,16,7)(3,9,13,8)(4,10,14,5), (1,10,11,2)(3,7,9,14)(4,13,12,8)(5,6,16,15)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11,15,6)(2,12,16,7)(3,9,13,8)(4,10,14,5), (1,10,11,2)(3,7,9,14)(4,13,12,8)(5,6,16,15) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,11,15,6),(2,12,16,7),(3,9,13,8),(4,10,14,5)], [(1,10,11,2),(3,7,9,14),(4,13,12,8),(5,6,16,15)]])
 
G:=TransitiveGroup(16,143);
 

►On 16 points - transitive group 16T167
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

(1 12 7 10)(2 14 8 16)(3 9)(4 15)(5 11)(6 13)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,7,6)(2,3,8,5)(9,16,11,14)(10,13,12,15), (1,12,7,10)(2,14,8,16)(3,9)(4,15)(5,11)(6,13)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,7,6)(2,3,8,5)(9,16,11,14)(10,13,12,15), (1,12,7,10)(2,14,8,16)(3,9)(4,15)(5,11)(6,13) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,7,6),(2,3,8,5),(9,16,11,14),(10,13,12,15)], [(1,12,7,10),(2,14,8,16),(3,9),(4,15),(5,11),(6,13)]])
 
G:=TransitiveGroup(16,167);
 

►On 16 points - transitive group 16T168
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

(1 13 3 9)(2 15 4 11)(5 14 8 12)(6 16 7 10)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,6,2,5)(3,8,4,7)(9,12,11,10)(13,16,15,14), (1,13,3,9)(2,15,4,11)(5,14,8,12)(6,16,7,10)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,6,2,5)(3,8,4,7)(9,12,11,10)(13,16,15,14), (1,13,3,9)(2,15,4,11)(5,14,8,12)(6,16,7,10) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,6,2,5),(3,8,4,7),(9,12,11,10),(13,16,15,14)], [(1,13,3,9),(2,15,4,11),(5,14,8,12),(6,16,7,10)]])
 
G:=TransitiveGroup(16,168);
 

►On 16 points - transitive group 16T169
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

(1 16 8 13)(2 12 7 9)(3 14 5 15)(4 10 6 11)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,7,3,6)(2,8,4,5)(9,14,11,16)(10,15,12,13), (1,16,8,13)(2,12,7,9)(3,14,5,15)(4,10,6,11)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,7,3,6)(2,8,4,5)(9,14,11,16)(10,15,12,13), (1,16,8,13)(2,12,7,9)(3,14,5,15)(4,10,6,11) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,7,3,6),(2,8,4,5),(9,14,11,16),(10,15,12,13)], [(1,16,8,13),(2,12,7,9),(3,14,5,15),(4,10,6,11)]])
 
G:=TransitiveGroup(16,169);
 

C42⋊C4 is a maximal subgroup of
 D4≀C2  C42⋊6D4  C42⋊Dic3  (C2×D4)⋊F5
 (C4×C4p)⋊C4: C4⋊1D4⋊C4  (C4×C8)⋊6C4  C42⋊5Dic3  C42⋊3Dic5  C42⋊F5  C42⋊3Dic7 ...
 (C2×D4).D2p: C42.D4  C8⋊C4⋊5C4  C4⋊Q8⋊29C4  C24.39D4  C4.4D4⋊C4  C42.13D4  C23.2D12  C23.2D20 ...
C42⋊C4 is a maximal quotient of
 (C2×D4)⋊C8  C42⋊3C8  C24.4D4  C24.6D4  C8⋊C4⋊C4  (C2×D4).D4  (C4×C8).C4  (C2×Q8).D4  (C2×D4)⋊F5
 C23.D4p: C24.D4  C23.2D12  C23.2D20  C23.2D28 ...
 (C4×C4p)⋊C4: C4⋊1D4⋊C4  (C4×C8)⋊6C4  C42⋊5Dic3  C42⋊3Dic5  C42⋊F5  C42⋊3Dic7 ...

Polynomial with Galois group C42⋊C4 over ℚ
actionf(x)Disc(f)
8T30x8-4x7-4x6+26x5-x4-46x3+13x2+15x+155·113·313

Matrix representation of C42⋊C4 ►in GL4(ℤ) generated by

0100
-1000
00-10
000-1
,
0-100
1000
000-1
0010
,
0010
0001
0100
1000
G:=sub<GL(4,Integers())| [0,-1,0,0,1,0,0,0,0,0,-1,0,0,0,0,-1],[0,1,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
 

C42⋊C4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes C_4
 
% in TeX
 
G:=Group("C4^2:C4");
 
// GroupNames label
 
G:=SmallGroup(64,34);
 
// by ID
 
G=gap.SmallGroup(64,34);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,332,158,681,255,1444]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^2*b>;
 
// generators/relations
 

Export

Subgroup lattice of C42⋊C4 in TeX
Character table of C42⋊C4 in TeX

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