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G = C4×D4order 32 = 25

Direct product of C4 and D4

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

Aliases: C4×D4, C424C2, C22.7C23, C23.8C22, C4⋊C47C2, C41(C2×C4), C42(C4⋊C4), C2.3(C2×D4), C22⋊C46C2, (C22×C4)⋊2C2, C221(C2×C4), (C2×D4).7C2, C42(C22⋊C4), C2.2(C4○D4), C2.4(C22×C4), (C2×C4).11C22, (C2×C4)(C2×D4), (C2×C4)(C4⋊C4), (C2×C4)(C22⋊C4), 2-Sylow(CO(3,5)), SmallGroup(32,25)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×D4
C1C2C22C2×C4C22×C4 — C4×D4
C1C2 — C4×D4
C1C2×C4 — C4×D4
C1C22 — C4×D4

Generators and relations for C4×D4
 G = < a,b,c | a4=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C2
2C2
2C4
2C22
2C22
2C22
2C4
2C4
2C22
2C2×C4
2C2×C4
2C2×C4
2C2×C4

Character table of C4×D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L
 size 11112222111122222222
ρ111111111111111111111    trivial
ρ2111111-1-111111-1-1-11-1-1-1    linear of order 2
ρ31111-1-1111111-11-11-1-1-1-1    linear of order 2
ρ41111-1-1-1-11111-1-11-1-1111    linear of order 2
ρ51111-1-111-1-1-1-11-11-11-11-1    linear of order 2
ρ61111-1-1-1-1-1-1-1-111-1111-11    linear of order 2
ρ711111111-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ8111111-1-1-1-1-1-1-1111-1-11-1    linear of order 2
ρ91-11-1-111-1-ii-ii-ii-i-ii1i-1    linear of order 4
ρ101-11-1-11-11-ii-ii-i-iiii-1-i1    linear of order 4
ρ111-11-11-11-1i-ii-i-i-i-iii-1i1    linear of order 4
ρ121-11-11-1-11i-ii-i-iii-ii1-i-1    linear of order 4
ρ131-11-11-11-1-ii-iiiii-i-i-1-i1    linear of order 4
ρ141-11-11-1-11-ii-iii-i-ii-i1i-1    linear of order 4
ρ151-11-1-111-1i-ii-ii-iii-i1-i-1    linear of order 4
ρ161-11-1-11-11i-ii-iii-i-i-i-1i1    linear of order 4
ρ172-2-220000-222-200000000    orthogonal lifted from D4
ρ182-2-2200002-2-2200000000    orthogonal lifted from D4
ρ1922-2-20000-2i-2i2i2i00000000    complex lifted from C4○D4
ρ2022-2-200002i2i-2i-2i00000000    complex lifted from C4○D4

Permutation representations of C4×D4
On 16 points - transitive group 16T19
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16 11 7)(2 13 12 8)(3 14 9 5)(4 15 10 6)
(5 14)(6 15)(7 16)(8 13)

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

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

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

G:=TransitiveGroup(16,19);

Matrix representation of C4×D4 in GL3(𝔽5) generated by

200
040
004
,
100
004
010
,
100
010
004
G:=sub<GL(3,GF(5))| [2,0,0,0,4,0,0,0,4],[1,0,0,0,0,1,0,4,0],[1,0,0,0,1,0,0,0,4] >;

C4×D4 in GAP, Magma, Sage, TeX

C_4\times D_4
% in TeX

G:=Group("C4xD4");
// GroupNames label

G:=SmallGroup(32,25);
// by ID

G=gap.SmallGroup(32,25);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,80,101,72]);
// Polycyclic

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

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