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G = C8○D4order 32 = 25

Central product of C8 and D4

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

Aliases: C8D4, C8Q8, D4.C4, Q8.C4, C8M4(2), C8.7C22, M4(2)⋊5C2, C4.12C23, (C2×C8)⋊7C2, C8(C4○D4), C4.5(C2×C4), C4○D4.3C2, C22.1(C2×C4), C2.7(C22×C4), (C2×C4).25C22, SmallGroup(32,38)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C8○D4
C1C2C4C2×C4C4○D4 — C8○D4
C1C2 — C8○D4
C1C8 — C8○D4
C1C2C2C4 — C8○D4

Generators and relations for C8○D4
 G = < a,b,c | a8=c2=1, b2=a4, ab=ba, ac=ca, cbc=a4b >

2C2
2C2
2C2

Character table of C8○D4

 class 12A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G8H8I8J
 size 11222112221111222222
ρ111111111111111111111    trivial
ρ2111-1-1111-1-11111-11-1-1-11    linear of order 2
ρ3111-1-1111-1-1-1-1-1-11-1111-1    linear of order 2
ρ41111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ511-1-1111-11-1-1-1-1-1-11-1111    linear of order 2
ρ611-11-111-1-11-1-1-1-1111-1-11    linear of order 2
ρ711-11-111-1-111111-1-1-111-1    linear of order 2
ρ811-1-1111-11-111111-11-1-1-1    linear of order 2
ρ9111-11-1-1-1-11-i-iiiii-ii-i-i    linear of order 4
ρ1011-111-1-11-1-1ii-i-i-iiii-i-i    linear of order 4
ρ111111-1-1-1-11-1-i-iii-iii-ii-i    linear of order 4
ρ1211-1-1-1-1-1111ii-i-iii-i-ii-i    linear of order 4
ρ131111-1-1-1-11-1ii-i-ii-i-ii-ii    linear of order 4
ρ1411-1-1-1-1-1111-i-iii-i-iii-ii    linear of order 4
ρ15111-11-1-1-1-11ii-i-i-i-ii-iii    linear of order 4
ρ1611-111-1-11-1-1-i-iiii-i-i-iii    linear of order 4
ρ172-2000-2i2i0008588387000000    complex faithful
ρ182-20002i-2i0008387858000000    complex faithful
ρ192-2000-2i2i0008858783000000    complex faithful
ρ202-20002i-2i0008783885000000    complex faithful

Permutation representations of C8○D4
On 16 points - transitive group 16T16
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)

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

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

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

G:=TransitiveGroup(16,16);

Matrix representation of C8○D4 in GL2(𝔽17) generated by

80
08
,
01
160
,
01
10
G:=sub<GL(2,GF(17))| [8,0,0,8],[0,16,1,0],[0,1,1,0] >;

C8○D4 in GAP, Magma, Sage, TeX

C_8\circ D_4
% in TeX

G:=Group("C8oD4");
// GroupNames label

G:=SmallGroup(32,38);
// by ID

G=gap.SmallGroup(32,38);
# by ID

G:=PCGroup([5,-2,2,2,-2,-2,40,157,58]);
// Polycyclic

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

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