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G = C42.C4  order 64 = 26

2nd non-split extension by C42 of C4 acting faithfully

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

Aliases: C42.2C4, (C2×C4).3D4, (C2×D4).3C4, C4.10D4⋊5C2, C4.4D4.3C2, (C2×Q8).1C22, C2.10(C23⋊C4), C22.13(C22⋊C4), (C2×C4).3(C2×C4), SmallGroup(64,36)

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

Derived series
C1 — C2×C4 — C42.C4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C4.4D4 — C42.C4
Lower central
C1 — C2 — C22 — C2×C4 — C42.C4
Upper central
C1 — C2 — C22 — C2×Q8 — C42.C4
Jennings
C1 — C2 — C22 — C2×Q8 — C42.C4

Generators and relations for C42.C4
 G = < a,b,c | a4=b4=1, c4=b2, ab=ba, cac-1=a-1b-1, cbc-1=a2b-1 >

C1
C2
2C2
8C2
C22
2C4
2C4
2C4
4C22
4C22
4C22
4C4
C2×C4
C2×C4
C2×C4
2C23
2C2×C4
4Q8
4C8
4D4
4C8
C42
C2×Q8
C2×D4
2M4(2)
2M4(2)
2C22⋊C4
2C22⋊C4
C4.10D4
C4.10D4
C4.4D4
C42.C4

Character table of C42.C4

 class 12A2B2C4A4B4C4D4E8A8B8C8D
 size 1128444448888
ρ11111111111111    trivial
ρ2111-1-1-11111-11-1    linear of order 2
ρ3111-1-1-1111-11-11    linear of order 2
ρ4111111111-1-1-1-1    linear of order 2
ρ5111-111-11-1-i-iii    linear of order 4
ρ61111-1-1-11-1-iii-i    linear of order 4
ρ71111-1-1-11-1i-i-ii    linear of order 4
ρ8111-111-11-1ii-i-i    linear of order 4
ρ9222000-2-220000    orthogonal lifted from D4
ρ102220002-2-20000    orthogonal lifted from D4
ρ1144-40000000000    orthogonal lifted from C23⋊C4
ρ124-400-2i2i0000000    complex faithful
ρ134-4002i-2i0000000    complex faithful

Permutation representations of C42.C4
►On 16 points - transitive group 16T131
Generators in S16
(2 16 6 12)(3 7)(4 10 8 14)(9 13)

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

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
 
G:=sub<Sym(16)| (2,16,6,12)(3,7)(4,10,8,14)(9,13), (1,11,5,15)(2,16,6,12)(3,9,7,13)(4,14,8,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;
 
G:=Group( (2,16,6,12)(3,7)(4,10,8,14)(9,13), (1,11,5,15)(2,16,6,12)(3,9,7,13)(4,14,8,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );
 
G=PermutationGroup([[(2,16,6,12),(3,7),(4,10,8,14),(9,13)], [(1,11,5,15),(2,16,6,12),(3,9,7,13),(4,14,8,10)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])
 
G:=TransitiveGroup(16,131);
 

►On 16 points - transitive group 16T151
Generators in S16
(1 3 5 7)(2 10)(4 16)(6 14)(8 12)(9 15 13 11)

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

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
 
G:=sub<Sym(16)| (1,3,5,7)(2,10)(4,16)(6,14)(8,12)(9,15,13,11), (1,11,5,15)(2,12,6,16)(3,9,7,13)(4,10,8,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;
 
G:=Group( (1,3,5,7)(2,10)(4,16)(6,14)(8,12)(9,15,13,11), (1,11,5,15)(2,12,6,16)(3,9,7,13)(4,10,8,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );
 
G=PermutationGroup([[(1,3,5,7),(2,10),(4,16),(6,14),(8,12),(9,15,13,11)], [(1,11,5,15),(2,12,6,16),(3,9,7,13),(4,10,8,14)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])
 
G:=TransitiveGroup(16,151);
 

C42.C4 is a maximal subgroup of
 (C2×D4).135D4  C4⋊1D4.C4  C42.15D4  C42.2F5  (D4×C10).C4
 (C2×Q8).D2p: C42.2D4  C8⋊C4⋊C4  (C2×D4).D4  (C4×C8)⋊C4  C4⋊Q8.C4  C42.16D4  C42.17D4  (C2×C4).D12 ...
C42.C4 is a maximal quotient of
 (C2×D4)⋊C8  C42⋊C8  (C2×C4).Q16  C42.2F5  (D4×C10).C4
 (C2×C4).D4p: C2.C2≀C4  (C2×C4).D12  (C2×C4).D20  (C2×C4).D28 ...
 (C2×Q8).D2p: (C2×Q8).Q8  C42.Dic3  C42.Dic5  C42.Dic7 ...

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

0010
0300
1000
0003
,
0020
0001
2000
0400
,
0003
1000
0100
0030
G:=sub<GL(4,GF(5))| [0,0,1,0,0,3,0,0,1,0,0,0,0,0,0,3],[0,0,2,0,0,0,0,4,2,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,3,3,0,0,0] >;
 

C42.C4 in GAP, Magma, Sage, TeX 

C_4^2.C_4
 
% in TeX
 
G:=Group("C4^2.C4");
 
// GroupNames label
 
G:=SmallGroup(64,36);
 
// by ID
 
G=gap.SmallGroup(64,36);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,332,158,681,255,117,1444]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=1,c^4=b^2,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.C4 in TeX
Character table of C42.C4 in TeX

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