Copied to
clipboard

G = C4.10C42  order 64 = 26

2nd central stem extension by C4 of C42

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C23.1Q8, C4.10C42, (C2×C8).1C4, (C2×C4).9D4, C22.2(C4⋊C4), C4.18(C22⋊C4), (C2×M4(2)).5C2, (C22×C4).19C22, C2.3(C2.C42), (C2×C4).62(C2×C4), SmallGroup(64,19)

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

Derived series
C1 — C4 — C4.10C42
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C4.10C42
Lower central
C1 — C4 — C4.10C42
Upper central
C1 — C4 — C4.10C42
Jennings
C1 — C2 — C2 — C22×C4 — C4.10C42

Generators and relations for C4.10C42
 G = < a,b,c | a4=1, b4=c4=a2, cbc-1=ab=ba, ac=ca >

C1
C2
2C2
2C2
2C2
C4
C22
C22
C4
C22
C4
C4
4C22
C2×C4
C2×C4
C23
C2×C4
C2×C4
C2×C4
C2×C4
2C8
2C8
2C8
2C8
2C8
2C8
C2×C8
C2×C8
C2×C8
C2×C8
C2×C8
C2×C8
C22×C4
2M4(2)
2M4(2)
2M4(2)
2M4(2)
2M4(2)
2M4(2)
C2×M4(2)
C2×M4(2)
C2×M4(2)
C4.10C42

Character table of C4.10C42

 class 12A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G8H8I8J8K8L
 size 1122211222444444444444
ρ11111111111111111111111    trivial
ρ21111111111-1-1-1-1-1-1-11111-1    linear of order 2
ρ31111111111-1111-1-1-1-1-1-1-11    linear of order 2
ρ411111111111-1-1-1111-1-1-1-1-1    linear of order 2
ρ5111-1-111-11-1i1-1-1i-i-ii-i-ii1    linear of order 4
ρ611-11-1111-1-1-1i-ii1-11-ii-ii-i    linear of order 4
ρ711-1-1111-1-11-i-i-iiii-i-1-111i    linear of order 4
ρ8111-1-111-11-1-i-111-iiii-i-ii-1    linear of order 4
ρ911-1-1111-1-11-iii-iii-i11-1-1-i    linear of order 4
ρ1011-11-1111-1-1-1-ii-i1-11i-ii-ii    linear of order 4
ρ1111-1-1111-1-11i-i-ii-i-ii11-1-1i    linear of order 4
ρ1211-11-1111-1-11i-ii-11-1i-ii-i-i    linear of order 4
ρ13111-1-111-11-1-i1-1-1-iii-iii-i1    linear of order 4
ρ1411-11-1111-1-11-ii-i-11-1-ii-iii    linear of order 4
ρ15111-1-111-11-1i-111i-i-i-iii-i-1    linear of order 4
ρ1611-1-1111-1-11iii-i-i-ii-1-111-i    linear of order 4
ρ1722-2-22-2-222-2000000000000    orthogonal lifted from D4
ρ18222-2-2-2-22-22000000000000    orthogonal lifted from D4
ρ1922-22-2-2-2-222000000000000    orthogonal lifted from D4
ρ2022222-2-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ214-40004i-4i000000000000000    complex faithful
ρ224-4000-4i4i000000000000000    complex faithful

Permutation representations of C4.10C42
►On 16 points - transitive group 16T108
Generators in S16
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)

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

(1 12 7 14 5 16 3 10)(2 15 4 13 6 11 8 9)
 
G:=sub<Sym(16)| (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,12,7,14,5,16,3,10)(2,15,4,13,6,11,8,9)>;
 
G:=Group( (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,12,7,14,5,16,3,10)(2,15,4,13,6,11,8,9) );
 
G=PermutationGroup([[(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,12,7,14,5,16,3,10),(2,15,4,13,6,11,8,9)]])
 
G:=TransitiveGroup(16,108);
 

C4.10C42 is a maximal subgroup of
 C23.SD16  C23.2SD16  (C2×C8).103D4  C8⋊C4⋊17C4  M4(2).46D4  M4(2).47D4  (C2×C8).2D4  C42.131D4  C24.Q8  M4(2).15D4  C24.11Q8  C22⋊C4.Q8  C23.SL2(𝔽3)
 C4p.C42: C8.16C42  C12.21C42  C20.51C42  C20.25C42  C28.21C42 ...
C4.10C42 is a maximal quotient of
 C42.20D4  C42.26D4  C24.2Q8  C20.25C42
 (C2×C4).D4p: C42.27D4  C12.21C42  C20.51C42  C28.21C42 ...

Matrix representation of C4.10C42 ►in GL4(𝔽5) generated by

2000
0200
0020
0002
,
0200
1000
0004
0020
,
0030
0002
1000
0100
G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,1,0,0,2,0,0,0,0,0,0,2,0,0,4,0],[0,0,1,0,0,0,0,1,3,0,0,0,0,2,0,0] >;
 

C4.10C42 in GAP, Magma, Sage, TeX 

C_4._{10}C_4^2
 
% in TeX
 
G:=Group("C4.10C4^2");
 
// GroupNames label
 
G:=SmallGroup(64,19);
 
// by ID
 
G=gap.SmallGroup(64,19);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,158,489,117,1444,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=1,b^4=c^4=a^2,c*b*c^-1=a*b=b*a,a*c=c*a>;
 
// generators/relations
 

Export

Subgroup lattice of C4.10C42 in TeX
Character table of C4.10C42 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁