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G = C8.Q8order 64 = 26

The non-split extension by C8 of Q8 acting via Q8/C2=C22

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

Aliases: C8.Q8, C161C4, C4.9SD16, M5(2).1C2, C22.5SD16, C4.6(C4⋊C4), C8.19(C2×C4), (C2×C4).13D4, C4.Q8.1C2, C2.3(C4.Q8), C8.C4.2C2, (C2×C8).12C22, 2-Sylow(AGammaL(1,81)), SmallGroup(64,46)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.Q8
C1C2C4C2×C4C2×C8M5(2) — C8.Q8
C1C2C4C8 — C8.Q8
C1C2C2×C4C2×C8 — C8.Q8
C1C2C2C2C2C4C4C2×C8 — C8.Q8

Generators and relations for C8.Q8
 G = < a,b,c | a8=1, b4=a2, c2=a-1b2, bab-1=a5, cac-1=a3, cbc-1=a4b3 >

2C2
8C4
4C2×C4
4C8
2M4(2)
2C4⋊C4

Character table of C8.Q8

 class 12A2B4A4B4C4D8A8B8C8D8E16A16B16C16D
 size 1122288224884444
ρ11111111111111111    trivial
ρ211111-1-111111-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-1    linear of order 2
ρ411111-1-1111-1-11111    linear of order 2
ρ511-11-1i-i11-1-ii1-1-11    linear of order 4
ρ611-11-1-ii11-1-ii-111-1    linear of order 4
ρ711-11-1i-i11-1i-i-111-1    linear of order 4
ρ811-11-1-ii11-1i-i1-1-11    linear of order 4
ρ92222200-2-2-2000000    orthogonal lifted from D4
ρ1022-22-200-2-22000000    symplectic lifted from Q8, Schur index 2
ρ1122-2-220000000-2--2-2--2    complex lifted from SD16
ρ12222-2-20000000--2--2-2-2    complex lifted from SD16
ρ1322-2-220000000--2-2--2-2    complex lifted from SD16
ρ14222-2-20000000-2-2--2--2    complex lifted from SD16
ρ154-4000002-2-2-20000000    complex faithful
ρ164-400000-2-22-20000000    complex faithful

Permutation representations of C8.Q8
On 16 points - transitive group 16T136
Generators in S16
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)

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

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

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

G:=TransitiveGroup(16,136);

C8.Q8 is a maximal subgroup of
M5(2)⋊3C4  Q32⋊C4  D16⋊C4  M5(2).C22  C23.10SD16  C804C4  C805C4
 C4p.SD16: D83Q8  D8.2Q8  C8.Dic6  C24.6Q8  C24.Q8  C8.Dic10  C40.6Q8  C40.Q8 ...
C8.Q8 is a maximal quotient of
C161C8
 C4p.(C4⋊C4): C8.11C42  C8.Dic6  C24.6Q8  C24.Q8  C8.Dic10  C40.6Q8  C40.Q8  C804C4 ...

Matrix representation of C8.Q8 in GL4(𝔽3) generated by

2010
0201
2020
0100
,
0001
0020
0201
2010
,
1000
0002
0020
0100
G:=sub<GL(4,GF(3))| [2,0,2,0,0,2,0,1,1,0,2,0,0,1,0,0],[0,0,0,2,0,0,2,0,0,2,0,1,1,0,1,0],[1,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0] >;

C8.Q8 in GAP, Magma, Sage, TeX

C_8.Q_8
% in TeX

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

G:=SmallGroup(64,46);
// by ID

G=gap.SmallGroup(64,46);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,31,362,86,489,1444,88]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^4=a^2,c^2=a^-1*b^2,b*a*b^-1=a^5,c*a*c^-1=a^3,c*b*c^-1=a^4*b^3>;
// generators/relations

Export

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

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