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G = C423C4order 64 = 26

3rd semidirect product of C42 and C4 acting faithfully

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

Aliases: C423C4, 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

C1C2×C4 — C423C4
C1C2C22C23C2×D4C4.4D4 — C423C4
C1C2C22C2×C4 — C423C4
C1C2C22C2×D4 — C423C4
C1C2C22C2×D4 — C423C4

Generators and relations for C423C4
 G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=a-1b-1, cbc-1=a2b-1 >

2C2
4C2
4C2
2C4
2C22
2C22
4C4
4C22
4C4
4C22
8C4
8C4
2C2×C4
2C2×C4
4D4
4C2×C4
4Q8
4C2×C4
2C22⋊C4
2C22⋊C4
2C22⋊C4
2C22⋊C4

Character table of C423C4

 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 C423C4
On 16 points - transitive group 16T154
Generators in S16
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 3 4 2)(5 8 6 7)(9 10 11 12)(13 16 15 14)
(1 11 6 15)(2 12 7 16)(3 10 8 14)(4 9 5 13)

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

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

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

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

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

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

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

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);

C423C4 is a maximal subgroup of
C424D4  C425D4  C42.14D4  (C2×Q8)⋊F5
 (C4×C4p)⋊C4: (C4×C8)⋊C4  C424Dic3  C42⋊Dic5  C422F5  C422Dic7 ...
 (C2×D4).D2p: C42.3D4  C8⋊C4⋊C4  (C2×D4).D4  C4⋊Q829C4  C24.39D4  C4⋊Q8⋊C4  C23.D12  C23.D20 ...
C423C4 is a maximal quotient of
(C2×C42).C4  C42⋊C8  C23.Q16  C24.6D4  C8⋊C45C4  C8⋊C4.C4  (C2×Q8)⋊F5
 C23.D4p: C23.4D8  C23.D12  C23.D20  C23.D28 ...
 (C4×C4p)⋊C4: (C4×C8)⋊C4  C424Dic3  C42⋊Dic5  C422F5  C422Dic7 ...

Matrix representation of C423C4 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] >;

C423C4 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 C423C4 in TeX
Character table of C423C4 in TeX

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