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G = C42⋊3C4  order 64 = 26

3rd semidirect product of C42 and C4 acting faithfully

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

Aliases: C42⋊3C4, C23.4D4, (C2×Q8)⋊2C4, C23⋊C4.2C2, C2.9(C23⋊C4), C4.4D4.2C2, (C2×D4).4C22, C22.12(C22⋊C4), (C2×C4).2(C2×C4), SmallGroup(64,35)

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

Derived series
C1 — C2×C4 — C42⋊3C4
Chief series
C1 — C2 — C22 — C23 — C2×D4 — C4.4D4 — C42⋊3C4
Lower central
C1 — C2 — C22 — C2×C4 — C42⋊3C4
Upper central
C1 — C2 — C22 — C2×D4 — C42⋊3C4
Jennings
C1 — C2 — C22 — C2×D4 — C42⋊3C4

Generators and relations for C42⋊3C4
 G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=a-1b-1, cbc-1=a2b-1 >

C1
C2
2C2
4C2
4C2
C22
2C4
2C22
2C22
4C4
4C22
4C4
4C22
8C4
8C4
C23
C23
C2×C4
2C2×C4
2C2×C4
4D4
4C2×C4
4Q8
4C2×C4
C42
C2×D4
C2×Q8
2C22⋊C4
2C22⋊C4
2C22⋊C4
2C22⋊C4
C4.4D4
C23⋊C4
C23⋊C4
C42⋊3C4

Character table of C42⋊3C4

 class 12A2B2C2D4A4B4C4D4E4F4G4H
 size 1124444488888
ρ11111111111111    trivial
ρ2111111-1-1-111-1-1    linear of order 2
ρ3111111-1-11-1-1-11    linear of order 2
ρ411111111-1-1-11-1    linear of order 2
ρ5111-1-1111-ii-i-1i    linear of order 4
ρ6111-1-11-1-1ii-i1-i    linear of order 4
ρ7111-1-11-1-1-i-ii1i    linear of order 4
ρ8111-1-1111i-ii-1-i    linear of order 4
ρ9222-22-20000000    orthogonal lifted from D4
ρ102222-2-20000000    orthogonal lifted from D4
ρ1144-40000000000    orthogonal lifted from C23⋊C4
ρ124-40000-2i2i00000    complex faithful
ρ134-400002i-2i00000    complex faithful

Permutation representations of C42⋊3C4
►On 16 points - transitive group 16T154
Generators in S16
(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

(1 15 6 12)(2 14 7 11)(3 16 8 9)(4 13 5 10)
 
G:=sub<Sym(16)| (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,2,4,3)(5,7,6,8)(9,12,11,10)(13,14,15,16), (1,15,6,12)(2,14,7,11)(3,16,8,9)(4,13,5,10)>;
 
G:=Group( (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,2,4,3)(5,7,6,8)(9,12,11,10)(13,14,15,16), (1,15,6,12)(2,14,7,11)(3,16,8,9)(4,13,5,10) );
 
G=PermutationGroup([[(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,2,4,3),(5,7,6,8),(9,12,11,10),(13,14,15,16)], [(1,15,6,12),(2,14,7,11),(3,16,8,9),(4,13,5,10)]])
 
G:=TransitiveGroup(16,154);
 

►On 16 points - transitive group 16T174
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

(1 14 7 16)(2 11)(3 13 6 15)(4 10)(5 12)(8 9)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,7,5)(2,3,8,6)(9,13,11,15)(10,14,12,16), (1,14,7,16)(2,11)(3,13,6,15)(4,10)(5,12)(8,9)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,7,5)(2,3,8,6)(9,13,11,15)(10,14,12,16), (1,14,7,16)(2,11)(3,13,6,15)(4,10)(5,12)(8,9) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,7,5),(2,3,8,6),(9,13,11,15),(10,14,12,16)], [(1,14,7,16),(2,11),(3,13,6,15),(4,10),(5,12),(8,9)]])
 
G:=TransitiveGroup(16,174);
 

►On 16 points - transitive group 16T176
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

(1 5 6 15)(2 14 10 11)(3 12 8 4)(7 9 13 16)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11,14,6)(2,12,15,7)(3,9,16,8)(4,10,13,5), (1,5,6,15)(2,14,10,11)(3,12,8,4)(7,9,13,16)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11,14,6)(2,12,15,7)(3,9,16,8)(4,10,13,5), (1,5,6,15)(2,14,10,11)(3,12,8,4)(7,9,13,16) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,11,14,6),(2,12,15,7),(3,9,16,8),(4,10,13,5)], [(1,5,6,15),(2,14,10,11),(3,12,8,4),(7,9,13,16)]])
 
G:=TransitiveGroup(16,176);
 

C42⋊3C4 is a maximal subgroup of
 C42⋊4D4  C42⋊5D4  C42.14D4  (C2×Q8)⋊F5
 (C4×C4p)⋊C4: (C4×C8)⋊C4  C42⋊4Dic3  C42⋊Dic5  C42⋊2F5  C42⋊2Dic7 ...
 (C2×D4).D2p: C42.3D4  C8⋊C4⋊C4  (C2×D4).D4  C4⋊Q8⋊29C4  C24.39D4  C4⋊Q8⋊C4  C23.D12  C23.D20 ...
C42⋊3C4 is a maximal quotient of
 (C2×C42).C4  C42⋊C8  C23.Q16  C24.6D4  C8⋊C4⋊5C4  C8⋊C4.C4  (C2×Q8)⋊F5
 C23.D4p: C23.4D8  C23.D12  C23.D20  C23.D28 ...
 (C4×C4p)⋊C4: (C4×C8)⋊C4  C42⋊4Dic3  C42⋊Dic5  C42⋊2F5  C42⋊2Dic7 ...

Matrix representation of C42⋊3C4 ►in GL4(𝔽5) generated by

1411
4111
4414
4441
,
0010
0001
4000
0400
,
1000
0400
0004
0010
G:=sub<GL(4,GF(5))| [1,4,4,4,4,1,4,4,1,1,1,4,1,1,4,1],[0,0,4,0,0,0,0,4,1,0,0,0,0,1,0,0],[1,0,0,0,0,4,0,0,0,0,0,1,0,0,4,0] >;
 

C42⋊3C4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_3C_4
 
% in TeX
 
G:=Group("C4^2:3C4");
 
// GroupNames label
 
G:=SmallGroup(64,35);
 
// by ID
 
G=gap.SmallGroup(64,35);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,332,158,681,255,1444]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^2*b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C42⋊3C4 in TeX
Character table of C42⋊3C4 in TeX

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