Copied to
clipboard

G = C8.C8order 64 = 26

1st non-split extension by C8 of C8 acting via C8/C4=C2

p-group, metacyclic, nilpotent (class 3), monomial

Aliases: C8.1C8, C8.6Q8, C8.29D4, C42.8C4, M5(2).4C2, C22.5M4(2), (C2×C8).9C4, C4.8(C2×C8), C2.5(C4⋊C8), (C4×C8).10C2, C4.20(C4⋊C4), (C2×C8).96C22, (C2×C4).68(C2×C4), SmallGroup(64,45)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.C8
C1C2C4C8C2×C8C4×C8 — C8.C8
C1C2C4 — C8.C8
C1C8C2×C8 — C8.C8
C1C2C2C2C2C4C4C2×C8 — C8.C8

Generators and relations for C8.C8
 G = < a,b | a8=1, b8=a4, bab-1=a3 >

2C2
2C4
2C4
2C2×C4
2C16
2C16

Character table of C8.C8

 class 12A2B4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J16A16B16C16D16E16F16G16H
 size 1121122222111122222244444444
ρ11111111111111111111111111111    trivial
ρ211111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-11-1-1-11111-1-11-1-11-1-1111-11-1    linear of order 2
ρ411111-11-1-1-11111-1-11-1-1111-1-1-11-11    linear of order 2
ρ51111111111-1-1-1-1-1-1-1-1-1-1-i-i-ii-iiii    linear of order 4
ρ61111111111-1-1-1-1-1-1-1-1-1-1iii-ii-i-i-i    linear of order 4
ρ711111-11-1-1-1-1-1-1-111-111-1ii-ii-i-ii-i    linear of order 4
ρ811111-11-1-1-1-1-1-1-111-111-1-i-ii-iii-ii    linear of order 4
ρ911-1-1-1-i1ii-iii-i-i1-1-i-11iζ83ζ87ζ8ζ87ζ85ζ85ζ83ζ8    linear of order 8
ρ1011-1-1-1-i1ii-iii-i-i1-1-i-11iζ87ζ83ζ85ζ83ζ8ζ8ζ87ζ85    linear of order 8
ρ1111-1-1-1-i1ii-i-i-iii-11i1-1-iζ8ζ85ζ87ζ8ζ83ζ87ζ85ζ83    linear of order 8
ρ1211-1-1-1-i1ii-i-i-iii-11i1-1-iζ85ζ8ζ83ζ85ζ87ζ83ζ8ζ87    linear of order 8
ρ1311-1-1-1i1-i-ii-i-iii1-1i-11-iζ85ζ8ζ87ζ8ζ83ζ83ζ85ζ87    linear of order 8
ρ1411-1-1-1i1-i-iiii-i-i-11-i1-1iζ87ζ83ζ8ζ87ζ85ζ8ζ83ζ85    linear of order 8
ρ1511-1-1-1i1-i-ii-i-iii1-1i-11-iζ8ζ85ζ83ζ85ζ87ζ87ζ8ζ83    linear of order 8
ρ1611-1-1-1i1-i-iiii-i-i-11-i1-1iζ83ζ87ζ85ζ83ζ8ζ85ζ87ζ8    linear of order 8
ρ1722-2220-2000222200-200-200000000    orthogonal lifted from D4
ρ1822-2220-2000-2-2-2-200200200000000    symplectic lifted from Q8, Schur index 2
ρ19222-2-20-20002i2i-2i-2i002i00-2i00000000    complex lifted from M4(2)
ρ20222-2-20-2000-2i-2i2i2i00-2i002i00000000    complex lifted from M4(2)
ρ212-202i-2i1+i01-i-1+i-1-i8387858-2--20-22000000000    complex faithful
ρ222-202i-2i1+i01-i-1+i-1-i87838852-20--2-2000000000    complex faithful
ρ232-20-2i2i-1+i0-1-i1+i1-i8858783-2-20--22000000000    complex faithful
ρ242-20-2i2i1-i01+i-1-i-1+i8588387-2-20--22000000000    complex faithful
ρ252-202i-2i-1-i0-1+i1-i1+i8783885-2--20-22000000000    complex faithful
ρ262-202i-2i-1-i0-1+i1-i1+i83878582-20--2-2000000000    complex faithful
ρ272-20-2i2i-1+i0-1-i1+i1-i85883872--20-2-2000000000    complex faithful
ρ282-20-2i2i1-i01+i-1-i-1+i88587832--20-2-2000000000    complex faithful

Permutation representations of C8.C8
On 16 points - transitive group 16T124
Generators in S16
(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,124);

C8.C8 is a maximal subgroup of
C8≀C2  C8.32D8  C8.24D8  C8.25D8  C8.29D8  C8.1Q16  M4(2).1C8  C8.5M4(2)  C8.19M4(2)  C8.3D8  D83D4  C8.5D8  D83Q8  D8.2Q8  C42.9F5
 C8p.C8: C16.C8  C16.3C8  C24.1C8  C40.7C8  C40.1C8  C56.16Q8 ...
 C8p.D4: C16○D8  D8.C8  C24.97D4  C40.9Q8  C56.9Q8 ...
C8.C8 is a maximal quotient of
C82C16
 C8.D4p: C8.36D8  C24.1C8  C40.7C8  C56.16Q8 ...
 C2p.(C4⋊C8): C42.7C8  C24.97D4  C40.9Q8  C42.9F5  C40.1C8  C56.9Q8 ...

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

90
015
,
01
20
G:=sub<GL(2,GF(17))| [9,0,0,15],[0,2,1,0] >;

C8.C8 in GAP, Magma, Sage, TeX

C_8.C_8
% in TeX

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

G:=SmallGroup(64,45);
// by ID

G=gap.SmallGroup(64,45);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,31,650,158,69,88]);
// Polycyclic

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

Export

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

׿
×
𝔽