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G = C8.C4order 32 = 25

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

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

Aliases: C8.1C4, C22.Q8, C4.19D4, M4(2).2C2, C4.8(C2×C4), (C2×C8).5C2, C2.5(C4⋊C4), (C2×C4).19C22, SmallGroup(32,15)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.C4
C1C2C4C2×C4C2×C8 — C8.C4
C1C2C4 — C8.C4
C1C4C2×C4 — C8.C4
C1C2C2C2×C4 — C8.C4

Generators and relations for C8.C4
 G = < a,b | a8=1, b4=a4, bab-1=a-1 >

2C2
2C8
2C8

Character table of C8.C4

 class 12A2B4A4B4C8A8B8C8D8E8F8G8H
 size 11211222224444
ρ111111111111111    trivial
ρ2111111-1-1-1-1-11-11    linear of order 2
ρ3111111-1-1-1-11-11-1    linear of order 2
ρ41111111111-1-1-1-1    linear of order 2
ρ511-1-1-11-11-11-i-iii    linear of order 4
ρ611-1-1-111-11-1i-i-ii    linear of order 4
ρ711-1-1-111-11-1-iii-i    linear of order 4
ρ811-1-1-11-11-11ii-i-i    linear of order 4
ρ922-222-200000000    orthogonal lifted from D4
ρ10222-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ112-202i-2i0-2-22--20000    complex faithful
ρ122-202i-2i02--2-2-20000    complex faithful
ρ132-20-2i2i0-2--22-20000    complex faithful
ρ142-20-2i2i02-2-2--20000    complex faithful

Permutation representations of C8.C4
On 16 points - transitive group 16T49
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 9 3 15 5 13 7 11)(2 16 4 14 6 12 8 10)

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

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

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

G:=TransitiveGroup(16,49);

Matrix representation of C8.C4 in GL2(𝔽17) generated by

90
72
,
215
415
G:=sub<GL(2,GF(17))| [9,7,0,2],[2,4,15,15] >;

C8.C4 in GAP, Magma, Sage, TeX

C_8.C_4
% in TeX

G:=Group("C8.C4");
// GroupNames label

G:=SmallGroup(32,15);
// by ID

G=gap.SmallGroup(32,15);
# by ID

G:=PCGroup([5,-2,2,-2,2,-2,40,61,26,302,72,58]);
// Polycyclic

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

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