Copied to
clipboard

G = C8.4Q8order 64 = 26

3rd non-split extension by C8 of Q8 acting via Q8/C4=C2

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

Aliases: C8.4Q8, C16.1C4, C4.19D8, C22.1Q16, C4.9(C4⋊C4), (C2×C16).5C2, C8.16(C2×C4), (C2×C4).65D4, C2.5(C2.D8), C8.C4.3C2, (C2×C8).87C22, SmallGroup(64,49)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.4Q8
C1C2C4C2×C4C2×C8C2×C16 — C8.4Q8
C1C2C4C8 — C8.4Q8
C1C4C2×C4C2×C8 — C8.4Q8
C1C2C2C2C2C4C4C2×C8 — C8.4Q8

Generators and relations for C8.4Q8
 G = < a,b,c | a8=1, b4=a2, c2=ab2, ab=ba, cac-1=a3, cbc-1=a6b3 >

2C2
4C8
4C8
2M4(2)
2M4(2)

Character table of C8.4Q8

 class 12A2B4A4B4C8A8B8C8D8E8F8G8H16A16B16C16D16E16F16G16H
 size 1121122222888822222222
ρ11111111111111111111111    trivial
ρ21111111111-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-111111111    linear of order 2
ρ511-1-1-11-1-111-i-iii-111-1-1-111    linear of order 4
ρ611-1-1-11-1-111ii-i-i-111-1-1-111    linear of order 4
ρ711-1-1-11-1-111-ii-ii1-1-1111-1-1    linear of order 4
ρ811-1-1-11-1-111i-ii-i1-1-1111-1-1    linear of order 4
ρ9222222-2-2-2-2000000000000    orthogonal lifted from D4
ρ1022-222-2000000002-22-22-22-2    orthogonal lifted from D8
ρ1122-222-200000000-22-22-22-22    orthogonal lifted from D8
ρ1222-2-2-2222-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ13222-2-2-200000000-2-222-222-2    symplectic lifted from Q16, Schur index 2
ρ14222-2-2-20000000022-2-22-2-22    symplectic lifted from Q16, Schur index 2
ρ152-20-2i2i0-2--2-220000ζ1615169ζ16716ζ165163ζ16131611ζ16716ζ16516316516316716    complex faithful
ρ162-202i-2i0--2-2-220000ζ16716ζ16716ζ165163ζ165163ζ1615169ζ1613161116516316716    complex faithful
ρ172-20-2i2i0--2-22-20000ζ165163ζ16516316716ζ1615169ζ16131611ζ16716ζ16716165163    complex faithful
ρ182-202i-2i0--2-2-220000ζ161516916716165163ζ16131611ζ16716ζ165163ζ165163ζ16716    complex faithful
ρ192-20-2i2i0-2--2-220000ζ1671616716165163ζ165163ζ1615169ζ16131611ζ165163ζ16716    complex faithful
ρ202-202i-2i0-2--22-20000ζ16131611ζ16516316716ζ16716ζ165163ζ1615169ζ16716165163    complex faithful
ρ212-202i-2i0-2--22-20000ζ165163165163ζ16716ζ1615169ζ16131611ζ1671616716ζ165163    complex faithful
ρ222-20-2i2i0--2-22-20000ζ16131611165163ζ16716ζ16716ζ165163ζ161516916716ζ165163    complex faithful

Smallest permutation representation of C8.4Q8
On 32 points
Generators in S32
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)
(1 30 13 18 9 22 5 26)(2 29 14 17 10 21 6 25)(3 28 15 32 11 20 7 24)(4 27 16 31 12 19 8 23)

G:=sub<Sym(32)| (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,30,13,18,9,22,5,26)(2,29,14,17,10,21,6,25)(3,28,15,32,11,20,7,24)(4,27,16,31,12,19,8,23)>;

G:=Group( (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,30,13,18,9,22,5,26)(2,29,14,17,10,21,6,25)(3,28,15,32,11,20,7,24)(4,27,16,31,12,19,8,23) );

G=PermutationGroup([(1,11,5,15,9,3,13,7),(2,12,6,16,10,4,14,8),(17,19,21,23,25,27,29,31),(18,20,22,24,26,28,30,32)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(1,30,13,18,9,22,5,26),(2,29,14,17,10,21,6,25),(3,28,15,32,11,20,7,24),(4,27,16,31,12,19,8,23)])

C8.4Q8 is a maximal subgroup of
D16.C4  M6(2)⋊C2  C16.18D4  M5(2).1C4  C8○D16  D165C4  D4.3D8  D4.4D8  D4.5D8
 C16p.C4: C32.C4  C48.C4  C80.6C4  C16.F5  C80.2C4  C112.C4 ...
 C8p.Q8: C8.Q16  C24.7Q8  C40.7Q8  C8.7Dic14 ...
C8.4Q8 is a maximal quotient of
C164C8
 C4.D8p: C163C8  C48.C4  C80.6C4  C112.C4 ...
 C4p.(C4⋊C4): C8.8C42  C24.7Q8  C40.7Q8  C16.F5  C80.2C4  C8.7Dic14 ...

Matrix representation of C8.4Q8 in GL2(𝔽17) generated by

90
015
,
50
07
,
01
40
G:=sub<GL(2,GF(17))| [9,0,0,15],[5,0,0,7],[0,4,1,0] >;

C8.4Q8 in GAP, Magma, Sage, TeX

C_8._4Q_8
% in TeX

G:=Group("C8.4Q8");
// GroupNames label

G:=SmallGroup(64,49);
// by ID

G=gap.SmallGroup(64,49);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,127,362,230,117,1444,88]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^4=a^2,c^2=a*b^2,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=a^6*b^3>;
// generators/relations

Export

Subgroup lattice of C8.4Q8 in TeX
Character table of C8.4Q8 in TeX

׿
×
𝔽